状态依赖重尾噪声下的鲁棒朗之万采样及其在大语言模型微调中的应用

# 1Introduction
Source: [https://arxiv.org/html/2610.12865](https://arxiv.org/html/2610.12865)
Robust Langevin Sampling under State\-Dependent Heavy\-Tailed Noise with Applications to LLM Fine\-Tuning

M\. Berk Sahin1Ege C\. Kaya1Arda Fazla1Behzad Sharif1,2Abolfazl Hashemi11Elmore Family School of Electrical and Computer Engineering, Purdue University2Weldon School of Biomedical Engineering, Purdue University

###### Abstract

Existing convergence guarantees for non\-log\-concave sampling with stochastic gradient Langevin Monte Carlo \(SG\-LMC\) typically assume globally Lipschitz gradients and uniformly bounded noise variance\. Both assumptions can fail even in elementary sampling problems, and relaxing them better reflects practical applications such as Bayesian fine\-tuning of large language models \(LLMs\)\. We relax both assumptions through generalized smoothness and our proposed state\-dependent heavy\-tailed oracle with finitepp\-th moments,<p≤21\\\!\\\!<\\\!\\\!p\\\!\\\!\\leq\\\!\\\!2\. The oracle permits infinite variance whenp<2p\\\!\\\!<\\\!\\\!2and includes the Blum–Gladyshev \(BG\-0\) condition\. We study these relaxations separately and jointly, proposing three methods\. Under globally Lipschitz gradients, robust LMC \(R\-LMC\) handles the proposed oracle through geometric\-median centering and residual clipping\. With exact gradients, adaptive LMC \(A\-LMC\) controls discretization error under generalized smoothness\. Robust adaptive LMC \(RA\-LMC\) addresses both relaxations through robust estimation and a noise\-aware step size\. For all three methods, we establish non\-asymptotic convergence guarantees in Fisher information \(FI\) without assuming dissipativity or a functional inequality\. Synthetic experiments demonstrate FI convergence and the benefit of each mechanism\. Bayesian fine\-tuning experiments with LLMs and vision\-language models support our oracle model and show that RA\-LMC matches or outperforms state\-of\-the\-art baselines in accuracy and uncertainty quantification\.

## 1Introduction

We consider the problem of sampling from a possibly non\-log\-concave target probability measureπ∝e−V\\pi\\\!\\propto\\\!e^\{\-V\}, whereV:ℝd→ℝV\\\!\\\!:\\\!\\\!\\mathbb\{R\}^\{d\}\\\!\\\!\\to\\\!\\\!\\mathbb\{R\}is the potential function, usingLangevin Monte Carlo \(LMC\)\. Earlier non\-asymptotic analyses beyond log\-concavity typically establish convergence under structural assumptions, including functional inequalities\([Vempala and Wibisono, 2019](https://arxiv.org/html/2610.12865#bib.bib9);[Wibisono, 2019](https://arxiv.org/html/2610.12865#bib.bib10);[Ma et al\., 2021](https://arxiv.org/html/2610.12865#bib.bib11);[Chewi et al\., 2022](https://arxiv.org/html/2610.12865#bib.bib12)\), tail growth or dissipativity conditions\([Durmus and Moulines, 2017](https://arxiv.org/html/2610.12865#bib.bib13);[Cheng et al\., 2018](https://arxiv.org/html/2610.12865#bib.bib14);[Majka et al\., 2020](https://arxiv.org/html/2610.12865#bib.bib15);[Erdogdu and Hosseinzadeh, 2021](https://arxiv.org/html/2610.12865#bib.bib16)\), or explicit mixture structure\([Lee et al\., 2018](https://arxiv.org/html/2610.12865#bib.bib17)\)\. Although these conditions enable convergence guarantees in strong distributional metrics, they do not provide a general measure of computational progress for targets with arbitrary multimodal geometry, where transitions between well\-separated modes can be prohibitively slow\.

To quantify this form of progress,[Balasubramanian et al\. \(2022\)](https://arxiv.org/html/2610.12865#bib.bib1)proposed a sampling analogue of first\-order stationarity in non\-convex optimization\. The analogy comes from viewing sampling as minimizing KL divergence over probability measures\([Jordan et al\., 1998](https://arxiv.org/html/2610.12865#bib.bib18);[Wibisono, 2018](https://arxiv.org/html/2610.12865#bib.bib19)\)\. In this formulation, the Langevin diffusion is the Wasserstein gradient flow of KL divergence, and relative Fisher information \(FI\), defined in Definition[2\.1](https://arxiv.org/html/2610.12865#S2.Thmtheorem1), plays the role of the squared gradient norm\. FI therefore serves as a measure of stationarity and can be small even when the law assigns incorrect weights to well\-separated modes\. Building on this perspective, subsequent work has clarified the statistical meaning and computational limits of FI guarantees\([Chewi et al\., 2023](https://arxiv.org/html/2610.12865#bib.bib4);[Cheng et al\., 2023](https://arxiv.org/html/2610.12865#bib.bib6);[Niles\-Weed, 2026](https://arxiv.org/html/2610.12865#bib.bib8)\)and developed parallel, proximal, variance\-reduced, and zeroth\-order sampling methods\([Zhou and Sugiyama, 2026](https://arxiv.org/html/2610.12865#bib.bib7);[Chewi and Wibisono, 2026](https://arxiv.org/html/2610.12865#bib.bib5);[Sahin et al\., 2026b](https://arxiv.org/html/2610.12865#bib.bib2);[Sahin et al\., 2026a](https://arxiv.org/html/2610.12865#bib.bib3)\)\. Standard direct SG\-LMC FI bounds use global smoothness and uniform second moments; existing extensions do not jointly address BG\-ppnoise and gradient\-dependent curvature \(Appendix[C\.1](https://arxiv.org/html/2610.12865#A3.SS1)\)\.

Although analytically convenient, these assumptions fail even for elementary models\. GlobalLL\-smoothness excludes superquadratic potentials, including quartic double\-well potentials and Ginzburg–Landau models\([Sun et al\., 2023](https://arxiv.org/html/2610.12865#bib.bib20);[Lytras and Mertikopoulos, 2024](https://arxiv.org/html/2610.12865#bib.bib21)\)\. The uniform variance bound can fail even for quadratic potentials, as minibatching in Bayesian linear regression can produce gradient noise whose variance grows quadratically with distance from the posterior mode\([Brosse et al\., 2018](https://arxiv.org/html/2610.12865#bib.bib24)\)\. Such state\-dependent variance is ensured by the Blum–Gladyshev \(BG\-0\) condition\([Blum, 1954](https://arxiv.org/html/2610.12865#bib.bib54);[Gladyshev, 1965](https://arxiv.org/html/2610.12865#bib.bib55)\)\.

Beyond these elementary examples, Bayesian fine\-tuning of large language models \(LLMs\) and vision\-language models \(VLMs\) provides important practical applications of non\-log\-concave sampling, where a sampler generates multiple parameter vectors that serve as approximate posterior samples\. Predictions are then averaged across these vectors to account for uncertainty in the fitted parameters\([Kim and Hospedales, 2023](https://arxiv.org/html/2610.12865#bib.bib52);[Yang et al\., 2024](https://arxiv.org/html/2610.12865#bib.bib48);[Samplawski et al\., 2026](https://arxiv.org/html/2610.12865#bib.bib53)\)\. Low\-rank adaptation \(LoRA\) makes such fine\-tuning more efficient by reducing the number of trainable parameters\([Hu et al\., 2022](https://arxiv.org/html/2610.12865#bib.bib44)\)\. Studies of language modeling report gradient\-dependent local curvature during training, motivating a relaxation of global smoothness\([Crawshaw et al\., 2022](https://arxiv.org/html/2610.12865#bib.bib23)\)\. In addition, empirical studies in optimization report heavy\-tailed gradient noise in BERT training and ALBERT and RoBERTa fine\-tuning\([Zhang et al\., 2020b](https://arxiv.org/html/2610.12865#bib.bib29);[Battash et al\., 2024](https://arxiv.org/html/2610.12865#bib.bib27);[Chezhegov et al\., 2024](https://arxiv.org/html/2610.12865#bib.bib28)\)\. These findings raise a natural question: does similar heavy\-tailed gradient noise arise during Bayesian fine\-tuning of LLMs with LoRA? If so, what mechanisms can enable LMC to accommodate this noise while retaining non\-asymptotic FI guarantees for non\-log\-concave targets whose potentials need not be globally smooth?

Our experiments on Bayesian LoRA fine\-tuning of LLMs provide an affirmative empirical answer to the first question, showing evidence of heavy\-tailed stochastic gradient noise and state\-dependent noise growth consistent with BG\-0\. For the second question, non\-convex optimization offers a natural starting point\. Normalized gradient methods achieve stationarity guarantees under generalized smoothness\([Chen et al\., 2023](https://arxiv.org/html/2610.12865#bib.bib30)\), while clipping, normalization, and momentum handle heavy\-tailed noise\([Cutkosky and Mehta, 2021](https://arxiv.org/html/2610.12865#bib.bib32)\)\. Anchoring and regularization provide guarantees under BG\-0\([Fazla et al\., 2026](https://arxiv.org/html/2610.12865#bib.bib36)\)\. These optimization results motivate seeking analogous FI guarantees for non\-log\-concave sampling\.

The difficulty is that modifying an optimization update to promote descent need not preserve the distribution sought by a sampler\. Even with exact gradients, clipping or normalizing the Langevin drift while leaving the Brownian noise unchanged can alter the invariant distribution\. A further difficulty is that existing FI guarantees depend on the expected squared error in the gradient estimates\. Rare but very large errors can make this quantity infinite, even when the estimates are unbiased and have a finite moment of order<p<21\\\!<\\\!p\\\!<\\\!2\. The standard analysis then yields an infinite error bound, providing no useful convergence guarantee\. Controlling these extreme errors without introducing excessive bias is therefore essential\. Generalized smoothness creates a separate obstacle because discretization error depends on gradient values along the random path between updates\. Bounding this error requires controlling higher\-order gradient moments that are not covered by the standard analysis\. Adaptive step sizes can limit these errors, but choosing them from noisy gradient estimates couples the estimation error with the duration over which it acts\. Establishing FI convergence therefore requires controlling both errors and their interaction without relying on log\-concavity, dissipativity, or a functional inequality\.

### 1\.1Contributions

We address these challenges through robust gradient estimation and adaptive discretization\. Geometric\-median centering and residual clipping control mean\-squared estimation error even when oracle variance is infinite, while adaptive steps limit discretization error where curvature grows\. Robust adaptive LMC \(RA\-LMC\) integrates these mechanisms to handle state\-dependent heavy\-tailed noise and generalized smoothness simultaneously\. Its analysis controls the dependence between estimation error and step duration through a noise\-aware step size\. For all three methods, we establish non\-asymptotic FI guarantees and explicit iteration and oracle complexities\. Our contributions are summarized below\.

- •State\-dependent heavy\-tailed oracle\.We propose BG\-pp, which generalizes both the BG\-0 condition and heavy\-tailed noise models with a state\-independent bound on thepp\-th moment\. Under global smoothness, we establish FI guarantees for robust LMC \(R\-LMC\), allowing both state dependence and infinite variance\.
- •Sampling under generalized smoothness\.We establish FI guarantees for adaptive LMC \(A\-LMC\) under symmetric generalized smoothness, assuming access to exact gradients\. The resulting complexity bounds recover the standard globally smooth LMC rate as a special case\.
- •Robust adaptive sampling\.We propose robust adaptive LMC \(RA\-LMC\) and control the dependence between gradient estimation errors and adaptive step sizes to establish FI guarantees under generalized smoothness and BG\-ppnoise\.
- •Empirical evaluation\.Synthetic experiments verify FI convergence of all three methods and demonstrate the benefits of each mechanism and their combination\. Bayesian LoRA fine\-tuning experiments provide empirical evidence supporting BG\-pporacle\. On LLM and VLM experiments, RA\-LMC matches or improves upon leading baselines in predictive accuracy and uncertainty quantification\. Code to reproduce our experiments is publicly available\.

## 2Preliminaries

Notation:We work on a probability space\(Ω,ℱ,ℙ\)\(\\Omega,\\mathcal\{F\},\\mathbb\{P\}\)\. Expectations with respect toℙ\\mathbb\{P\}are denoted by𝔼\\mathbb\{E\}\. We usex,y,z,xref∈ℝdx,y,z,x\_\{\\mathrm\{ref\}\}\\in\\mathbb\{R\}^\{d\}for deterministic points andXtX\_\{t\}for random states\. In particular,xrefx\_\{\\mathrm\{ref\}\}is a deterministic reference point chosen at initialization and used as the oracle anchor, whereasX0∼μ0X\_\{0\}\\sim\\mu\_\{0\}is the random initial state\. When there is no ambiguity, we suppress brackets and write𝔼​‖X‖2\\mathbb\{E\}\\\|X\\\|^\{2\}in place of𝔼⁡\[‖X‖2\]\\mathbb\{E\}\[\\\|X\\\|^\{2\}\]\.

###### Definition 2\.1\.

LetV∈C1​\(ℝd\)V\\in C^\{1\}\(\\mathbb\{R\}^\{d\}\)with0<Z:=∫ℝde−V⁡\(x\)​𝑑x<∞0<Z:=\\int\_\{\\mathbb\{R\}^\{d\}\}e^\{\-V\(x\)\}\\,\\mathrm\{d\}x<\\infty, and setπ⁡\(x\)=Z−1​e−V⁡\(x\)\\pi\(x\)=Z^\{\-1\}e^\{\-V\(x\)\}\. We also writeπ⁡\(d​x\)=π⁡\(x\)​d​x\\pi\(\\mathrm\{d\}x\)=\\pi\(x\)\\,\\mathrm\{d\}xfor the target measure\. Forμ≪π\\mu\\ll\\piandf=d​μ/d​πf=\\mathrm\{d\}\\mu/\\mathrm\{d\}\\pi, writeKL\(μ∥π\):=∫flogfdπ\\mathrm\{KL\}\(\\mu\\\|\\pi\):=\\int f\\log f\\,\\mathrm\{d\}\\piand

FI\(μ∥π\):=4∫∥∇f∥2dπ\.\\mathrm\{FI\}\(\\mu\\\|\\pi\):=4\\int\\\|\\nabla\\sqrt\{f\}\\\|^\{2\}\\,\\mathrm\{d\}\\pi\.\(1\)

Following prior LMC analyses\([Vempala and Wibisono, 2019](https://arxiv.org/html/2610.12865#bib.bib9);[Balasubramanian et al\., 2022](https://arxiv.org/html/2610.12865#bib.bib1);[Chewi et al\., 2025](https://arxiv.org/html/2610.12865#bib.bib58)\), we assumeK0:=KL\(μ0∥π\)=𝒪\(d\)K\_\{0\}:=\\mathrm\{KL\}\(\\mu\_\{0\}\\\|\\pi\)=\\mathcal\{O\}\(d\)for iteration and oracle complexity bounds, while retainingK0K\_\{0\}explicitly in finite\-time bounds and parameter choices\.

Langevin Sampling:Conventionally, theLangevin diffusion, the solution to the stochastic differential equation

d​Xt=−∇V​\(Xt\)​d​t\+2​d​Wt,\\mathrm\{d\}X\_\{t\}=\-\\nabla V\(X\_\{t\}\)\\,\\mathrm\{d\}t\+\\sqrt\{2\}\\,\\mathrm\{d\}W\_\{t\},\(2\)hasπ\\pias an invariant distribution whenever the diffusion is well posed; under standard additional conditions, this invariant distribution is unique and the diffusion converges to it ast→∞t\\to\\infty\. Here,\(Wt\)t≥0\(W\_\{t\}\)\_\{t\\geq 0\}denotes a standarddd\-dimensional Brownian motion\. Euler discretization with step sizeh\>0h\>0yields the standardLangevin Monte Carlo\(LMC\) algorithm

X\(k\+1\)​h:=Xk​h−h∇V\(Xk​h\)\+2​hZk,X\_\{\(k\+1\)h\}:=X\_\{kh\}\-h\\nabla V\(X\_\{kh\}\)\+\\sqrt\{2h\}Z\_\{k\},
whereZk∼𝒩⁡\(0,Id\)Z\_\{k\}\\\!\\sim\\\!\{\\mathcal\{N\}\}\(0,I\_\{d\}\)\. For the continuous\-time interpolation specified in Section[3](https://arxiv.org/html/2610.12865#S3), letμt:=Law⁡\(Xt\)\\mu\_\{t\}:=\\operatorname\{Law\}\(X\_\{t\}\)andμ¯T:=T−1​∫0Tμt​𝑑t\\bar\{\\mu\}\_\{T\}:=T^\{\-1\}\\int\_\{0\}^\{T\}\\mu\_\{t\}\\,\\mathrm\{d\}t\. To ensure the stability of discretization, previous works\([Balasubramanian et al\., 2022](https://arxiv.org/html/2610.12865#bib.bib1);[Chewi et al\., 2023](https://arxiv.org/html/2610.12865#bib.bib4);[Sahin et al\., 2026b](https://arxiv.org/html/2610.12865#bib.bib2)\)adopt the following assumption\.

###### Assumption 2\.2\(GlobalLL\-smoothness\)\.

The potentialVVis globallyLL\-smooth; that is, its gradient is globallyLL\-Lipschitz:‖∇V​\(x1\)−∇V​\(x2\)‖≤L​‖x1−x2‖\\\|\\nabla V\(x\_\{1\}\)\-\\nabla V\(x\_\{2\}\)\\\|\\leq L\\\|x\_\{1\}\-x\_\{2\}\\\|for allx1,x2∈ℝdx\_\{1\},x\_\{2\}\\in\\mathbb\{R\}^\{d\}and someL\>0L\>0\.

Under this assumption,[Balasubramanian et al\. \(2022\)](https://arxiv.org/html/2610.12865#bib.bib1)showed that𝒪⁡\(L2​d2/ε2\)\\mathcal\{O\}\(L^\{2\}d^\{2\}/\\varepsilon^\{2\}\)iterations are sufficient to achieveFI\(μ¯T∥π\)≤ε\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\varepsilon\. When exact gradients are unavailable, SG\-LMC averagesbbindependent stochastic\-oracle evaluations per update:

X\(k\+1\)​h=Xk​h−hb​∑j=1b𝒢⁡\(Xk​h,ξk,j\)\+2​h​Zk,X\_\{\(k\+1\)h\}=X\_\{kh\}\-\\frac\{h\}\{b\}\\sum\_\{j=1\}^\{b\}\{\\mathcal\{G\}\}\(X\_\{kh\},\\xi\_\{k,j\}\)\+\\sqrt\{2h\}\\,Z\_\{k\},whereZk∼𝒩⁡\(0,Id\)Z\_\{k\}\\sim\\mathcal\{N\}\(0,I\_\{d\}\)\. We assume𝔼ξ​\[𝒢⁡\(x,ξ\)\]=∇V​\(x\)\\mathbb\{E\}\_\{\\xi\}\[\\mathcal\{G\}\(x,\\xi\)\]=\\nabla V\(x\)and𝔼ξ​‖𝒢⁡\(x,ξ\)−∇V​\(x\)‖2≤σ2\\mathbb\{E\}\_\{\\xi\}\\\|\\mathcal\{G\}\(x,\\xi\)\-\\nabla V\(x\)\\\|^\{2\}\\leq\\sigma^\{2\}\. Choosingb=𝒪⁡\(σ2/ε\)b=\\mathcal\{O\}\(\\sigma^\{2\}/\\varepsilon\)in the FI bound of[Balasubramanian et al\. \(2022, Theorem 15\)](https://arxiv.org/html/2610.12865#bib.bib1)gives𝒪⁡\(L2​d2​σ2/ε3\)\\mathcal\{O\}\(L^\{2\}d^\{2\}\\sigma^\{2\}/\\varepsilon^\{3\}\)oracle calls; the step\-by\-step derivation is in Appendix[A\.6](https://arxiv.org/html/2610.12865#A1.SS6)\. The SL\-VRLD method of[Sahin et al\. \(2026b\)](https://arxiv.org/html/2610.12865#bib.bib2)attains the same total oracle order with constant expected calls per iteration under an additional mean\-square smoothness assumption on stochastic gradients\.

Symmetric Generalized Smoothness:Motivated by empirical evidence of gradient\-dependent local curvature in language modeling and ResNet\-20 image classification\([Zhang et al\., 2020a](https://arxiv.org/html/2610.12865#bib.bib22)\),[Chen et al\. \(2023\)](https://arxiv.org/html/2610.12865#bib.bib30)proposed the following symmetric generalization of globalLL\-smoothness\.

###### Assumption 2\.3\.

For someL0,L1≥0L\_\{0\},L\_\{1\}\\geq 0andγ∈\(0,1\]\\gamma\\in\(0,1\], the potentialV:ℝd→ℝV:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}satisfies

‖∇V​\(y\)−∇V​\(x\)‖≤‖y−x‖​\(L0\+L1​supz∈\[x,y\]‖∇V​\(z\)‖γ\)\\\|\\nabla V\(y\)\-\\nabla V\(x\)\\\|\\leq\\\|y\-x\\\|\(L\_\{0\}\+L\_\{1\}\\sup\_\{z\\in\[x,y\]\}\\\|\\nabla V\(z\)\\\|^\{\\gamma\}\)\\vskip\-10\.00002ptfor allx,y∈ℝdx,y\\in\\mathbb\{R\}^\{d\}\.

The term “symmetric” refers to the use of the supremum gradient norm along the segment joiningxxandyy, which treats the two endpoints symmetrically\. This allows local curvature to grow with the gradient magnitude and reduces to globalL0L\_\{0\}\-smoothness whenL1=0L\_\{1\}=0\. Compared with Assumption[2\.2](https://arxiv.org/html/2610.12865#S2.Thmtheorem2), this assumption covers examples such as superquadratic potentials, polynomial double\-well models, and generalized Gaussian distributions\.

BG\-0 and Heavy\-Tailed Oracles:Uniformly bounded variance restricts both the state dependence and tail behavior of gradient noise\. In Bayesian linear regression, a noise component linear in the parameter can cause quadratically growing variance\([Brosse et al\., 2018](https://arxiv.org/html/2610.12865#bib.bib24)\)\. Such state\-dependent bounds are studied in[Baker et al\. \(2019\)](https://arxiv.org/html/2610.12865#bib.bib25);[Putcha et al\. \(2023\)](https://arxiv.org/html/2610.12865#bib.bib26)\. Heavy\-tailed gradient noise is reported in BERT training\([Zhang et al\., 2020b](https://arxiv.org/html/2610.12865#bib.bib29)\), CLIP\([Battash et al\., 2024](https://arxiv.org/html/2610.12865#bib.bib27)\), and ALBERT and RoBERTa fine\-tuning\([Chezhegov et al\., 2024](https://arxiv.org/html/2610.12865#bib.bib28)\)\. These observations motivate two relaxations:

𝔼ξ​‖𝒢⁡\(x,ξ\)−∇V​\(x\)‖2\\displaystyle\\mathbb\{E\}\_\{\\xi\}\\\|\\mathcal\{G\}\(x,\\xi\)\-\\nabla V\(x\)\\\|^\{2\}≤B​‖x−xref‖2\+G,\\displaystyle\\leq B\\\|x\-x\_\{\\mathrm\{ref\}\}\\\|^\{2\}\+G,\(3\)𝔼ξ​‖𝒢⁡\(x,ξ\)−∇V​\(x\)‖p\\displaystyle\\mathbb\{E\}\_\{\\xi\}\\\|\\mathcal\{G\}\(x,\\xi\)\-\\nabla V\(x\)\\\|^\{p\}≤σp,∀x,xref∈ℝd,\\displaystyle\\leq\\sigma^\{p\},\\quad\\forall x,x\_\{\\mathrm\{ref\}\}\\in\{\\mathbb\{R\}\}^\{d\},wherep∈\(1,2\]p\\in\(1,2\],B,G≥0B,G\\geq 0, andσ\>0\\sigma\>0are constants\. The first inequality is the Blum–Gladyshev \(BG\-0\) condition\([Blum, 1954](https://arxiv.org/html/2610.12865#bib.bib54);[Gladyshev, 1965](https://arxiv.org/html/2610.12865#bib.bib55)\)and it permits the upper bound to grow quadratically with the distance from the reference pointxrefx\_\{\\mathrm\{ref\}\}\. The second inequality imposes a state\-independent finitepp\-th\-moment bound and permits infinite variance when1<p<21<p<2\.

## 3Methods

We develop our framework in three stages\. First, under globalLL\-smoothness, we propose a stochastic oracle that generalizes both BG\-0 and heavy\-tailed noise assumptions and construct a robust gradient estimator to control the resulting estimation error\. Second, with exact\-gradient access, we replace globalLL\-smoothness by Assumption[2\.3](https://arxiv.org/html/2610.12865#S2.Thmtheorem3)and develop an adaptive step\-size scheme to control discretization error\. Finally, we combine these ideas in RA\-LMC, which handles both relaxations simultaneously by addressing the coupling between stochastic estimation error and gradient\-dependent local curvature\.

We express all three methods through a common Euler interpolation\. At iterationkk, letFkF\_\{k\}be the Langevin force \(the negative gradient or its estimate\) andHk\>0H\_\{k\}\>0the step size, both chosen before the next Brownian increment\. Setτ0=0\\tau\_\{0\}=0andτk\+1=τk\+Hk\\tau\_\{k\+1\}=\\tau\_\{k\}\+H\_\{k\}\. Fort∈\[τk,τk\+1\]t\\in\[\\tau\_\{k\},\\tau\_\{k\+1\}\], define

Xt:=Xτk\+\(t−τk\)​Fk\+2​\(Wt−Wτk\)\.X\_\{t\}:=X\_\{\\tau\_\{k\}\}\+\(t\-\\tau\_\{k\}\)F\_\{k\}\+\\sqrt\{2\}\(W\_\{t\}\-W\_\{\\tau\_\{k\}\}\)\.\(4\)Standard LMC hasFk=−∇V​\(Xτk\)F\_\{k\}=\-\\nabla V\(X\_\{\\tau\_\{k\}\}\)andHk=hH\_\{k\}=h\. ForT\>0T\>0, letNT:=inf\{n∈ℕ0:τn≥T\}N\_\{T\}:=\\inf\\\{n\\in\\mathbb\{N\}\_\{0\}:\\tau\_\{n\}\\geq T\\\}be the number of updates needed to reachTT\. If updatekkusesqkq\_\{k\}gradient\-oracle evaluations, letQT:=∑k=0NT−1qkQ\_\{T\}:=\\sum\_\{k=0\}^\{N\_\{T\}\-1\}q\_\{k\}be the total oracle count through timeTT\. We specify the force and step size for each method below and establish non\-asymptotic FI guarantees for all three settings\.

### 3\.1Robust LMC under State\-Dependent Heavy\-Tailed Noise

Building on the BG\-0 and heavy\-tailed noise models in \([3](https://arxiv.org/html/2610.12865#S2.E3)\), we propose a generalized BG\-pporacle that permits both state\-dependent noise growth and potentially infinite variance\. This formulation recovers both models as special cases and allows us to address their combined challenges in non\-log\-concave sampling\. Our proposed oracle is given below\.

###### Assumption 3\.1\(BG\-pporacle\)\.

For1<p≤21<p\\leq 2andB,G≥0B,G\\geq 0,𝒢⁡\(x,ξ\)\\mathcal\{G\}\(x,\\xi\)satisfies𝔼ξ​\[𝒢⁡\(x,ξ\)\]=∇V​\(x\)\\mathbb\{E\}\_\{\\xi\}\[\\mathcal\{G\}\(x,\\xi\)\]=\\nabla V\(x\)and

𝔼ξ​‖𝒢⁡\(x,ξ\)−∇V​\(x\)‖p\\displaystyle\\mathbb\{E\}\_\{\\xi\}\\\|\\mathcal\{G\}\(x,\\xi\)\-\\nabla V\(x\)\\\|^\{p\}≤B​‖x−xref‖p\+G\.\\displaystyle\\leq B\\\|x\-x\_\{\\mathrm\{ref\}\}\\\|^\{p\}\+G\.

This assumption allows heavy\-tailed stochastic gradient noise whose magnitude can increase as the sampler moves away from the reference point\. The exponentppspecifies both the finite moment required of the noise and the power of its permitted growth with distance\. Settingp=2p=2recovers the BG\-0 condition, whereasB=0B=0recovers a state\-independent finite\-pp\-moment bound\. For1<p<21<p<2andB\>0B\>0, the model can accommodate infinite variance and state\-dependent noise simultaneously\.

To handle a BG\-pporacle with potentially infinite variance, we adapt the two\-stage centering and residual clipping approach of[Lugosi and Mendelson \(2019\)](https://arxiv.org/html/2610.12865#bib.bib56);[Whitehouse et al\. \(2024\)](https://arxiv.org/html/2610.12865#bib.bib57)to stochastic LMC\. Specifically, we use the geometric median of three independent oracle evaluations as a center, clip the residual of a fourth around it, and establish the mean\-squared error control needed for FI analysis\. At iterationkkand statexx, we draw independent oracle evaluations𝒢k,i,j:=𝒢⁡\(x,ξk,i,j\)\\mathcal\{G\}\_\{k,i,j\}:=\\mathcal\{G\}\(x,\\xi\_\{k,i,j\}\)and define

Ck,i\\displaystyle C\_\{k,i\}:=GeoMed⁡\(𝒢k,i,1,𝒢k,i,2,𝒢k,i,3\),\\displaystyle:=\\mathrm\{GeoMed\}\(\\mathcal\{G\}\_\{k,i,1\},\\mathcal\{G\}\_\{k,i,2\},\\mathcal\{G\}\_\{k,i,3\}\),\(5\)𝒢^k,i\\displaystyle\\widehat\{\\mathcal\{G\}\}\_\{k,i\}:=Ck,i\+clipλk​\(𝒢k,i,4−Ck,i\),\\displaystyle:=C\_\{k,i\}\+\\mathrm\{clip\}\_\{\\lambda\_\{k\}\}\(\\mathcal\{G\}\_\{k,i,4\}\-C\_\{k,i\}\),whereclipλk​\(v\):=v​min⁡\{1,λk/‖v‖\}\\mathrm\{clip\}\_\{\\lambda\_\{k\}\}\(v\):=v\\min\\\{1,\\lambda\_\{k\}/\\\|v\\\|\\\}forv≠0v\\neq 0, andclipλk​\(0\):=0\\mathrm\{clip\}\_\{\\lambda\_\{k\}\}\(0\):=0\. Here,Ck,iC\_\{k,i\}is the*geometric\-median center*: a large deviation from the true gradient requires at least two of its three independent oracle evaluations to deviate substantially, yielding a finite second moment forp\>1p\>1\. We clip the residual rather than the gradient itself to control extreme noise without directly truncating the Langevin drift, which could otherwise steer the sampler toward a different target distribution\. Averagingbbindependent estimates from \([5](https://arxiv.org/html/2610.12865#S3.E5)\), we define robust LMC \(R\-LMC\) through \([4](https://arxiv.org/html/2610.12865#S3.E4)\) by

Fk=−1b∑i=1b𝒢^k,iandHk=h\.F\_\{k\}=\-\\frac\{1\}\{b\}\\sum\_\{i=1\}^\{b\}\\widehat\{\\mathcal\{G\}\}\_\{k,i\}\\quad\\text\{and\}\\quad H\_\{k\}=h\.\(6\)Thusτk=k​h\\tau\_\{k\}=kh, and each update usesqk=4​bq\_\{k\}=4boracle evaluations\. Building on the FI framework of[Balasubramanian et al\. \(2022\)](https://arxiv.org/html/2610.12865#bib.bib1), we establish our first convergence guarantee under the BG\-pporacle\.

###### Theorem 3\.2\.

Suppose Assumptions[2\.2](https://arxiv.org/html/2610.12865#S2.Thmtheorem2)and[3\.1](https://arxiv.org/html/2610.12865#S3.Thmtheorem1)hold, andK0<∞K\_\{0\}<\\infty\. Letμt\\mu\_\{t\}be the law of the interpolation \([4](https://arxiv.org/html/2610.12865#S3.E4)\) with \([6](https://arxiv.org/html/2610.12865#S3.E6)\)\. For anyδ\>0\\delta\>0, integerN≥1N\\geq 1, and0<h≤\(8​L\)−10<h\\leq\(8L\)^\{\-1\}, there exist a batch sizebband clipping radiiλk\\lambda\_\{k\}such that, withT=N​hT=Nh, the time\-averaged lawμ¯T\\bar\{\\mu\}\_\{T\}satisfies

FI\(μ¯T∥π\)≤2​K0N​h\+323L2dh\+4912δ2\.\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\frac\{2K\_\{0\}\}\{Nh\}\+\\frac\{32\}\{3\}L^\{2\}dh\+\\frac\{49\}\{12\}\\delta^\{2\}\.IfK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\), then for sufficiently smallε\>0\\varepsilon\>0we can chooseN,h,bN,h,b, andλk\\lambda\_\{k\}so thatFI\(μ¯T∥π\)≤ε\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\varepsilonwithNT=𝒪⁡\(L2​d2/ε2\)N\_\{T\}=\\mathcal\{O\}\(L^\{2\}d^\{2\}/\\varepsilon^\{2\}\)in both cases below\. WhenB\>0B\>0, the oracle complexity is

QT=𝒪⁡\(B1/\(p−1\)​L2\+p/\[2​\(p−1\)\]​d2\+3​p/\[2​\(p−1\)\]ε2\+3​p/\[2​\(p−1\)\]\)\.Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{B^\{1/\(p\-1\)\}L^\{2\+p/\[2\(p\-1\)\]\}d^\{2\+3p/\[2\(p\-1\)\]\}\}\{\\varepsilon^\{2\+3p/\[2\(p\-1\)\]\}\}\\right\)\.WhenB=0B=0andG\>0G\>0, it improves to

QT=𝒪⁡\(L2​d2​G1/\(p−1\)ε2\+p/\[2​\(p−1\)\]\)\.Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{L^\{2\}d^\{2\}G^\{1/\(p\-1\)\}\}\{\\varepsilon^\{2\+p/\[2\(p\-1\)\]\}\}\\right\)\.

For fixedB,G\>0B,G\>0, the state\-dependent term determines the leading high\-accuracy oracle rate;GGremains in the finite bound\. The full statement and proof are provided in Theorem[A\.2](https://arxiv.org/html/2610.12865#A1.Thmtheorem2)and Appendix[A\.4](https://arxiv.org/html/2610.12865#A1.SS4)\.

### 3\.2Adaptive LMC under Generalized Smoothness

We next develop an adaptive step\-size mechanism for controlling the LMC discretization error when globalLL\-smoothness is replaced by Assumption[2\.3](https://arxiv.org/html/2610.12865#S2.Thmtheorem3)\. Under globalLL\-smoothness, the gradient varies at a uniform rate throughout the state space, allowing a single step size to control the discretization error along the entire trajectory\. Under generalized smoothness, however, the gradient can change more rapidly as its norm increases\. A fixed step size may therefore be safe where the gradient changes slowly but too aggressive where it changes rapidly, causing large discretization errors or instability\. This motivates using shorter steps in regions of large gradients while allowing longer steps in more regular regions\. We define adaptive LMC \(A\-LMC\) through \([4](https://arxiv.org/html/2610.12865#S3.E4)\) by

Fk=−∇V​\(Xτk\)andHk=h1\+ω​‖Fk‖γ,F\_\{k\}=\-\\nabla V\(X\_\{\\tau\_\{k\}\}\)\\quad\\text\{and\}\\quad H\_\{k\}=\\frac\{h\}\{1\+\\omega\\\|F\_\{k\}\\\|^\{\\gamma\}\},\(7\)whereω≥0\\omega\\geq 0\. Here,γ\\gammadetermines how rapidly the step size decreases with the gradient norm\. Matching this exponent to Assumption[2\.3](https://arxiv.org/html/2610.12865#S2.Thmtheorem3)offsets the corresponding growth in local curvature, whileω\\omegacontrols the overall strength of the adaptation\. Becauseτk\+1−τk=Hk\\tau\_\{k\+1\}\-\\tau\_\{k\}=H\_\{k\}records elapsed time along the interpolation, theh→0h\\to 0limit of A\-LMC is the original Langevin diffusion \([2](https://arxiv.org/html/2610.12865#S2.E2)\), which preservesπ\\pi\. Indexing the same updates byk​hkhinstead yields a time\-rescaled diffusion with a different invariant law\([Leroy et al\., 2024](https://arxiv.org/html/2610.12865#bib.bib39)\), so our guarantee concerns the time\-averaged lawμ¯T\\bar\{\\mu\}\_\{T\}\.

Extending the FI framework of[Balasubramanian et al\. \(2022\)](https://arxiv.org/html/2610.12865#bib.bib1)to A\-LMC requires controlling how the gradient changes during each adaptive step\. Under globalLL\-smoothness, this change is bounded byLLtimes the displacement\. Under Assumption[2\.3](https://arxiv.org/html/2610.12865#S2.Thmtheorem3), the bound depends on gradient values along the random interpolation path, producing higher\-order terms that displacement alone cannot control\. We control these terms with estimates of gradient changes along each adaptive interval and a bound on the required gradient moments in terms of FI\. Combined with the adaptive step size, these estimates absorb the state\-dependent terms into FI and leave an explicit discretization remainder\. This yields the following theorem\.

###### Theorem 3\.3\.

Suppose Assumption[2\.3](https://arxiv.org/html/2610.12865#S2.Thmtheorem3)holds with0<γ≤10<\\gamma\\leq 1andK0<∞K\_\{0\}<\\infty\. Setω=L1/\(1\+L0\)\\omega=L\_\{1\}/\(1\+L\_\{0\}\)and letμt\\mu\_\{t\}be the law of the interpolation \([4](https://arxiv.org/html/2610.12865#S3.E4)\) with \([7](https://arxiv.org/html/2610.12865#S3.E7)\)\. For any horizonT\>0T\>0and sufficiently smallh\>0h\>0,NT<∞N\_\{T\}<\\inftyalmost surely, andμ¯T\\bar\{\\mu\}\_\{T\}satisfies

FI\(μ¯T∥π\)≤4​K03​T\+83Rγ\(h,1\+Cγ\+K0T\),\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\frac\{4K\_\{0\}\}\{3T\}\+\\frac\{8\}\{3\}R\_\{\\gamma\}\\\!\\left\(h,1\+C\_\{\\gamma\}\+\\frac\{K\_\{0\}\}\{T\}\\right\),whereCγC\_\{\\gamma\}and the discretization remainderRγR\_\{\\gamma\}are defined in Theorem[A\.4](https://arxiv.org/html/2610.12865#A1.Thmtheorem4)\. IfK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\), then for sufficiently smallε\>0\\varepsilon\>0we can chooseTTandhhso thatFI\(μ¯T∥π\)≤ε\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\varepsilonwithNT=QTN\_\{T\}=Q\_\{T\}\. LetΛ:=max⁡\{1,L0,L1\}\\Lambda:=\\max\\\{1,L\_\{0\},L\_\{1\}\\\}\. WhenL1\>0L\_\{1\}\>0and0<γ≤10<\\gamma\\leq 1, the expected oracle complexity is

𝔼​QT=𝒪⁡\(Λ4/\(2−γ\)​d4/\(2−γ\)ε2\)\.\\mathbb\{E\}Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{\\Lambda^\{4/\(2\-\\gamma\)\}d^\{4/\(2\-\\gamma\)\}\}\{\\varepsilon^\{2\}\}\\right\)\.WhenL1=0L\_\{1\}=0, the oracle count is deterministic and satisfies

QT=𝒪⁡\(L02​d2ε2\)\.Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{L\_\{0\}^\{2\}d^\{2\}\}\{\\varepsilon^\{2\}\}\\right\)\.

The complete statement and proof are provided in Theorem[A\.4](https://arxiv.org/html/2610.12865#A1.Thmtheorem4)and Appendix[A\.7](https://arxiv.org/html/2610.12865#A1.SS7)\.

### 3\.3Robust Adaptive LMC \(RA\-LMC\)

For a non\-log\-concave target satisfying Assumption[2\.3](https://arxiv.org/html/2610.12865#S2.Thmtheorem3)and a stochastic oracle satisfying Assumption[3\.1](https://arxiv.org/html/2610.12865#S3.Thmtheorem1), a natural approach is to combine the robust estimator \([5](https://arxiv.org/html/2610.12865#S3.E5)\) with the adaptive step size \([7](https://arxiv.org/html/2610.12865#S3.E7)\)\. This combination, however, is not straightforward\. The adaptive step\-size rule in \([7](https://arxiv.org/html/2610.12865#S3.E7)\) assumes access to the exact gradient\. Replacing it with the robust estimate makes the estimation error and step size dependent, so the same inaccurate estimate can affect both the drift and the duration over which its error is applied\. Consequently, the conditional mean\-squared error bound for \([5](https://arxiv.org/html/2610.12865#S3.E5)\) does not directly control the error accumulated over elapsed time, and the two preceding analyses cannot be combined through a simple substitution\.

Figure 1:Robust estimation and adaptive steps improve FI convergence under heavy\-tailed noise and generalized smoothness, respectively, while RA\-LMC improves convergence when both challenges arise by integrating robust estimation with a noise\-aware step size\. SG, R, and A denote stochastic gradient averaging, proposed robust estimator and adaptive steps, respectively\. Panels \(a\)–\(c\) use a Gaussian mixture with a BG\-pporacle of infinite variance, a radial quartic target with exact gradients, and a radial cubic target with the same oracle, respectively\. Each method’s curve shows the median across 10 seeds, with its shaded region indicating the interquartile range\.Figure 2:Panel \(a\) shows minibatch gradient error norms measured at step 2,500 of Qwen3\-4B LoRA fine\-tuning on OpenBookQA\. Panel \(b\) shows synthetic norms generated from a one\-sided Lévy\-stable reference construction with an infinite second moment\. The similarity between the histograms suggests heavy\-tailed behavior in the stochastic gradient errors\.Figure 3:Panels \(a\) and \(b\) show the empirical first moment of minibatch gradient\-error norms versus distance from the final LoRA iterate during fine\-tuning of Qwen3\-4B on OpenBookQA and Qwen3\-VL\-4B\-Instruct on SLAKE\. Each point averages error norms from 32 independent minibatches of size four\. The increase in the first moment with distance motivates the state\-dependent term in the BG\-pporacle\.To address this dependence, we retain the robust estimator in \([5](https://arxiv.org/html/2610.12865#S3.E5)\) and define the localLpL^\{p\}\-noise scale at statexxby

sp​\(x\):=\[B​‖x−xref‖p\+G\]1/p,s\_\{p\}\(x\):=\\bigl\[B\\\|x\-x\_\{\\mathrm\{ref\}\}\\\|^\{p\}\+G\\bigr\]^\{1/p\},which bounds the conditionalLpL^\{p\}norm of the oracle error\. Using both this scale and the robust gradient estimate, we define robust adaptive LMC \(RA\-LMC\) through \([4](https://arxiv.org/html/2610.12865#S3.E4)\) by

Fk=−1b∑i=1b𝒢^k,i,Hk=h1\+ω​‖Fk‖γ\+κ​sp​\(Xτk\)2,F\_\{k\}=\-\\frac\{1\}\{b\}\\sum\_\{i=1\}^\{b\}\\widehat\{\\mathcal\{G\}\}\_\{k,i\},\\quad H\_\{k\}=\\frac\{h\}\{1\+\\omega\\\|F\_\{k\}\\\|^\{\\gamma\}\+\\kappa s\_\{p\}\(X\_\{\\tau\_\{k\}\}\)^\{2\}\},\(8\)whereκ,ω≥0\\kappa,\\omega\\geq 0\. The term‖Fk‖γ\\\|F\_\{k\}\\\|^\{\\gamma\}adapts the step size to the gradient\-dependent curvature allowed by generalized smoothness, whilesp​\(Xτk\)2s\_\{p\}\(X\_\{\\tau\_\{k\}\}\)^\{2\}further shortens the step in regions of high oracle noise and controls the coupling between the estimation error and its duration\. Building on Theorem[3\.3](https://arxiv.org/html/2610.12865#S3.Thmtheorem3), we control the interaction between gradient estimation and discretization errors under generalized smoothness\. The following theorem gives a non\-asymptotic FI guarantee for the time\-averaged law and the corresponding iteration and stochastic\-oracle complexities\.

###### Theorem 3\.4\.

Suppose Assumptions[2\.3](https://arxiv.org/html/2610.12865#S2.Thmtheorem3)and[3\.1](https://arxiv.org/html/2610.12865#S3.Thmtheorem1)hold with0<γ≤10<\\gamma\\leq 1and1<p≤21<p\\leq 2, andK0<∞K\_\{0\}<\\infty\. IfB\>0B\>0, assume additionally that𝔼​‖X0−xref‖2≤M2,0<∞\\mathbb\{E\}\\left\\lVert X\_\{0\}\-x\_\{\\mathrm\{ref\}\}\\right\\rVert^\{2\}\\leq M\_\{2,0\}<\\infty\. Setω=L1/\(1\+L0\)\\omega=L\_\{1\}/\(1\+L\_\{0\}\)and letμt\\mu\_\{t\}be the law of the interpolation \([4](https://arxiv.org/html/2610.12865#S3.E4)\) with \([8](https://arxiv.org/html/2610.12865#S3.E8)\)\. For sufficiently smallε\>0\\varepsilon\>0, we can chooseT,h,b,κT,h,b,\\kappa, andλk\\lambda\_\{k\}so thatNT<∞N\_\{T\}<\\inftyalmost surely andμ¯T\\bar\{\\mu\}\_\{T\}satisfiesFI\(μ¯T∥π\)≤ε\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\varepsilonwithQT=4​b​NTQ\_\{T\}=4bN\_\{T\}\. LetΛ:=max⁡\{1,L0,L1\}\\Lambda:=\\max\\\{1,L\_\{0\},L\_\{1\}\\\}\. IfK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)andL1\>0L\_\{1\}\>0, then the expected iteration complexity is

𝔼​NT=𝒪⁡\(Λ\(8−2​γ\)/\(2−γ\)​d\(6−γ\)/\(2−γ\)ε2\)\.\\mathbb\{E\}N\_\{T\}=\\mathcal\{O\}\\left\(\\frac\{\\Lambda^\{\(8\-2\\gamma\)/\(2\-\\gamma\)\}d^\{\(6\-\\gamma\)/\(2\-\\gamma\)\}\}\{\\varepsilon^\{2\}\}\\right\)\.\(9\)WhenB\>0B\>0, the expected oracle complexity is

𝔼​QT\\displaystyle\\mathbb\{E\}Q\_\{T\}=𝒪⁡\(B1/\(p−1\)​Λ\(8−2​γ\)​\(p−1\)\+2​p\(p−1\)​\(2−γ\)​d\(3​p−2\)​\(6−γ\)2​\(p−1\)​\(2−γ\)ε2\+3​p/\[2​\(p−1\)\]\)\.\\displaystyle=\\mathcal\{O\}\\\!\\left\(\\frac\{B^\{1/\(p\-1\)\}\\Lambda^\{\\frac\{\(8\-2\\gamma\)\(p\-1\)\+2p\}\{\(p\-1\)\(2\-\\gamma\)\}\}d^\{\\frac\{\(3p\-2\)\(6\-\\gamma\)\}\{2\(p\-1\)\(2\-\\gamma\)\}\}\}\{\\varepsilon^\{2\+3p/\[2\(p\-1\)\]\}\}\\right\)\.WhenB=0B=0, the expected oracle complexity is

𝔼​QT\\displaystyle\\mathbb\{E\}Q\_\{T\}=𝒪⁡\(\(1\+G\)1/\(p−1\)​Λ\(8−2​γ\)​\(p−1\)\+p\(p−1\)​\(2−γ\)​d1\+8​\(p−1\)\+γ​p2​\(p−1\)​\(2−γ\)ε2\+p/\[2​\(p−1\)\]\)\.\\displaystyle\\\!=\\\!\\mathcal\{O\}\\\!\\left\(\\\!\\frac\{\(1\+G\)^\{1/\(p\-1\)\}\\Lambda^\{\\frac\{\(8\-2\\gamma\)\(p\-1\)\+p\}\{\(p\-1\)\(2\-\\gamma\)\}\}\\\!d^\{1\+\\frac\{8\(p\-1\)\+\\gamma p\}\{2\(p\-1\)\(2\-\\gamma\)\}\}\}\{\\varepsilon^\{2\+p/\[2\(p\-1\)\]\}\}\\\!\\right\)\.

The full statement, including parameter choices and theFI\\mathrm\{FI\}bound, is given in Theorem[A\.6](https://arxiv.org/html/2610.12865#A1.Thmtheorem6), with complexity specializations in Corollary[A\.7](https://arxiv.org/html/2610.12865#A1.Thmtheorem7)and proofs in Appendix[A\.9](https://arxiv.org/html/2610.12865#A1.SS9)\.

## 4Experiments

### 4\.1Synthetic Experiments

We conduct synthetic experiments to validate the FI convergence of our three methods and assess the benefit of each mechanism\. We consider three non\-log\-concave sampling problems inℝ2\{\\mathbb\{R\}\}^\{2\}, testing R\-LMC under a state\-dependent heavy\-tailed oracle, A\-LMC under generalized smoothness with exact gradients, and RA\-LMC under both relaxations\. Figure[1](https://arxiv.org/html/2610.12865#S3.F1)reports estimatedFI\(μ¯T∥π\)\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)versus timeTT, showing medians across ten random seeds with interquartile bands\. Appendix[B\.1](https://arxiv.org/html/2610.12865#A2.SS1)provides the experimental settings and FI estimation details\.

Figure[1](https://arxiv.org/html/2610.12865#S3.F1)\(a\) shows results for R\-LMC on a Gaussian\-mixture target with a globally smooth potential and a state\-dependent Student oracle having a finitepp\-th moment but infinite variance\. We compare R\-LMC, SG\-LMC, and exact\-gradient LMC using the same fixed step size\. Both stochastic methods use 16 oracle queries per update\. R\-LMC closely follows the FI decrease of exact\-gradient LMC, whereas SG\-LMC diverges, illustrating the benefit of robust estimation\.

Figure[1](https://arxiv.org/html/2610.12865#S3.F1)\(b\) shows results for A\-LMC on a radial quartic potential satisfying Assumption[2\.3](https://arxiv.org/html/2610.12865#S2.Thmtheorem3)withγ=2/3\\gamma=2/3\. Using exact gradients, we compare fixed\-step LMC with A\-LMC\. Both methods initially reduce FI, but A\-LMC reaches lower FI over the observed horizon, supporting the use of adaptive steps to control discretization error under unbounded curvature\.

Figure[1](https://arxiv.org/html/2610.12865#S3.F1)\(c\) shows results for RA\-LMC on a radial cubic potential satisfying Assumption[2\.3](https://arxiv.org/html/2610.12865#S2.Thmtheorem3)withγ=1/2\\gamma=1/2\. We compare six methods, using the same oracle as in panel \(a\) and 16 queries per update for each stochastic method\. Fixed\-step SG\-LMC and R\-LMC encounter numerical failures, while A\-SG\-LMC plateaus at higher FI\. RA\-LMC closely tracks exact\-gradient A\-LMC and has lower estimated FI than A\-SG\-LMC and fixed\-step LMC\. These results highlight the benefit of robust estimation with adaptive steps when state\-dependent heavy\-tailed noise and growing curvature occur together\.

Table 1:OpenBookQA and SLAKE test results \(mean±\\pmstd\. dev\.\)\. Accuracy \(ACC, in percent\) is higher\-is\-better; negative log\-likelihood \(NLL\), Brier score, and expected calibration error \(ECE\) are lower\-is\-better\.Bold:the best\.Underline:the second best\.
### 4\.2Bayesian Fine\-Tuning Experiments

We conduct LLM and VLM fine\-tuning experiments to assess the empirical relevance of the BG\-pporacle model and the practical benefits of combining robust gradient estimation with adaptive updates\. The step\-size rule analyzed above requires smoothness and local noise\-scale quantities that are difficult to specify for large models\. In practice, we retain geometric\-median centering and residual clipping, use AdamW\([Loshchilov and Hutter, 2019](https://arxiv.org/html/2610.12865#bib.bib68)\)for adaptive updates, and add Gaussian noise after each update\. We consider the following two datasets\.

OpenBookQA\.This dataset provides a benchmark for reasoning in language models\. Its questions require applying elementary science facts and commonsense knowledge to everyday situations\. Each question has four answer choices, with one correct answer\([Mihaylov et al\., 2018](https://arxiv.org/html/2610.12865#bib.bib66)\)\. It contains 4,957 training, 500 validation, and 500 test examples\.

SLAKE\.This dataset provides a benchmark for medical visual question answering in vision\-language models\. It pairs X\-ray, CT, and MRI images with questions about anatomical structures and abnormalities\([Liu et al\., 2021](https://arxiv.org/html/2610.12865#bib.bib67)\)\. We use its yes/no subset, treating each image–question pair as a binary classification example\. This subset contains 1,681 training, 358 validation, and 355 test examples\.

#### 4\.2\.1Empirical Evidence for BG\-ppOracle

We examine the empirical relevance of the BG\-pporacle by studying minibatch\-gradient errors during Bayesian fine\-tuning\. We consider LMC with minibatch\-averaged gradients and AdamW\-based adaptive updates rather than fixed\-step LMC\. This allows us to investigate whether heavy\-tailed, state\-dependent errors occur when updates adapt to changing gradient scales, as motivated by generalized smoothness\. Their presence would motivate our robust estimator as a complement to step\-size adaptation\.

We fine\-tune Qwen3\-4B and Qwen3\-VL\-4B\-Instruct on OpenBookQA and SLAKE, respectively, for 5,000 steps with minibatch size four\. Following[Zhang et al\. \(2020b\)](https://arxiv.org/html/2610.12865#bib.bib29), we measure the Euclidean norm of the difference between each minibatch gradient and the average gradient over the full training set\. Figure[2](https://arxiv.org/html/2610.12865#S3.F2)\(a\) shows the histogram of these error norms for OpenBookQA at step 2,500, while Figure[2](https://arxiv.org/html/2610.12865#S3.F2)\(b\) shows synthetic norms generated from a one\-sided Lévy\-stable reference construction\. The similarity between the histograms suggests heavy\-tailed behavior in stochastic gradient errors during LoRA fine\-tuning with LMC\. Appendix[B\.2\.2](https://arxiv.org/html/2610.12865#A2.SS2.SSS2)provides the details and corresponding results at additional checkpoints for both datasets\.

To examine the state\-dependent component of the BG\-pporacle, we select 50 checkpoints from each fine\-tuning trajectory, uniformly spaced in distance from the final LoRA iterate\. At each checkpoint, we average the error norms over 32 independent minibatches and repeat this calculation 8 times\. Figure[3](https://arxiv.org/html/2610.12865#S3.F3)plots these first\-moment estimates against the Euclidean distance from the final iterate\. The observed increase in error magnitude with distance supports modeling the noise scale as state dependent\. Appendix[B\.2\.3](https://arxiv.org/html/2610.12865#A2.SS2.SSS3)provides the sampling and estimation details\.

#### 4\.2\.2Comparison with Baselines

We compare RA\-LMC against SG\-LMC and nine baselines from Bayesian Adaptation Gym\([Samplawski et al\., 2026](https://arxiv.org/html/2610.12865#bib.bib53)\)\. We use the benchmark’s reference implementations for these nine methods and repeat each fine\-tuning experiment with four random seeds\. For each seed, we average class probabilities from ten predictions per test example and compute accuracy, negative log\-likelihood \(NLL\), Brier score, and expected calibration error \(ECE\)\. Table[1](https://arxiv.org/html/2610.12865#S4.T1)reports the mean and standard deviation across seeds\.

RA\-LMC outperforms SG\-LMC on all four metrics for both datasets\. Compared with the benchmark baselines, it has the highest accuracy and lowest Brier score on both datasets, as well as the lowest NLL on OpenBookQA\. It remains competitive on the other metrics\. These results support the practical benefits of combining our robust estimator with adaptive updates for Bayesian LoRA fine\-tuning\.

## 5Conclusion

We extended the FI framework or non\-log\-concave Langevin sampling beyond globally Lipschitz gradients and uniformly bounded noise variance\. The proposed BG\-pporacle and our three methods support non\-asymptotic guarantees for time\-averaged laws, with explicit iteration and oracle complexities and without assuming any dissipativity or functional inqualities\. RA\-LMC controls the interaction between robust estimation and adaptive step sizes, enabling guarantees under both relaxations\. Synthetic experiments support these mechanisms\. In Bayesian LoRA fine\-tuning, we found evidence for BG\-pporacle and showed that RA\-LMC, using our robust estimator and adaptive step sizes, matches or outperforms leading baselines in accuracy and uncertainty quantification\.

### AI Use Statement

In this work, we used generative AI tools for assist in writing of proofs, implement methods, support qualitative and thematic data analysis, and formulate mathematical claims\. We have not used generative AI tools for generate synthetic datasets, help develop theoretical models or conceptual frameworks, provide critical ingredients for proving mathematical claims, propose or refine hypothesis, design or provide feedback on research methodology or experiments, interpret results, and the remaining required\-disclosure tasks are not applicable to this work\. Additionally, we used generative AI tools for create or edit software code, summarize or analyze existing literature, discover research topics or identify gaps, and edit a research paper to improve readability\. We have reviewed all AI\-assisted work and tested AI\-generated experiment code to run as intended, and all conversations with AI are verified before being applied to the paper\. We take responsibility for the final content of this work, including text, claims, or artifacts produced with the aid of generative AI\.

### Acknowledgements

This work was partially supported by National Insititutes of Health grant NIH/NHLBI R01\-HL153430 \(PI: Sharif\), and supported in part by NSF DMS\-2502560 and CNS\-2313109\.

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Robust Langevin Sampling under State\-Dependent Heavy\-Tailed Noise with Applications to LLM Fine\-Tuning: Supplementary Materials

## Appendix Table of Contents

## Appendix AFull Theorem Statements and Proofs

### A\.1Information and interpolation conventions

##### Information functionals\.

As in Definition[2\.1](https://arxiv.org/html/2610.12865#S2.Thmtheorem1), we useπ⁡\(x\)\\pi\(x\)for the target density andπ⁡\(d​x\)=π⁡\(x\)​d​x\\pi\(\\mathrm\{d\}x\)=\\pi\(x\)\\,\\mathrm\{d\}xfor its measure\. Ifμ≪π\\mu\\ll\\piandf:=d​μ/d​πf:=\\mathrm\{d\}\\mu/\\mathrm\{d\}\\pi, the KL convention is0​log⁡0:=00\\log 0:=0; ifμ≪̸π\\mu\\not\\ll\\pi, we setKL\(μ∥π\):=\+∞\\mathrm\{KL\}\(\\mu\\\|\\pi\):=\+\\infty\. The Fisher information in \([1](https://arxiv.org/html/2610.12865#S2.E1)\) uses the weak gradient off\\sqrt\{f\}:FI\(μ∥π\):=\+∞\\mathrm\{FI\}\(\\mu\\\|\\pi\):=\+\\inftyunlessμ≪π\\mu\\ll\\pi,f∈Wloc1,2​\(ℝd\)\\sqrt\{f\}\\in W^\{1,2\}\_\{\\mathrm\{loc\}\}\(\\mathbb\{R\}^\{d\}\), and∫ℝd‖∇f‖2​𝑑π<∞\\int\_\{\\mathbb\{R\}^\{d\}\}\\\|\\nabla\\sqrt\{f\}\\\|^\{2\}\\,\\mathrm\{d\}\\pi<\\infty\. Here the weak gradient is distributional with respect to Lebesgue measure\. Ifμ\\muhas a smooth positive Lebesgue densityρ\\rho, then equivalently

FI\(μ∥π\)=∫ℝd∥∇logρ\(x\)\+∇V\(x\)∥2ρ\(x\)dx\.\\mathrm\{FI\}\(\\mu\\\|\\pi\)=\\int\_\{\\mathbb\{R\}^\{d\}\}\\\|\\nabla\\log\\rho\(x\)\+\\nabla V\(x\)\\\|^\{2\}\\rho\(x\)\\,\\mathrm\{d\}x\.
###### Definition A\.1\(Interpolation and algorithmic laws\)\.

Letτ0=0\\tau\_\{0\}=0\. At eventkk, let a step sizeHk\>0H\_\{k\}\>0and a Langevin forceFkF\_\{k\}be selected using the current and past states and any oracle samples drawn at that event, before the Brownian increment on the next interval is generated, and setτk\+1:=τk\+Hk\\tau\_\{k\+1\}:=\\tau\_\{k\}\+H\_\{k\}\. We use the continuous\-time interpolation

Xt:=Xτk\+\(t−τk\)​Fk\+2​\(Wt−Wτk\),X\_\{t\}:=X\_\{\\tau\_\{k\}\}\+\(t\-\\tau\_\{k\}\)F\_\{k\}\+\\sqrt\{2\}\\bigl\(W\_\{t\}\-W\_\{\\tau\_\{k\}\}\\bigr\),fort∈\[τk,τk\+1\)t\\in\[\\tau\_\{k\},\\tau\_\{k\+1\}\)\. Letτ∞:=limk→∞τk\\tau\_\{\\infty\}:=\\lim\_\{k\\to\\infty\}\\tau\_\{k\}\. The interpolation is defined pathwise on\[0,τ∞\)\[0,\\tau\_\{\\infty\}\); wheneverℙ⁡\(τ∞\>t\)=1\\mathbb\{P\}\(\\tau\_\{\\infty\}\>t\)=1, letμt:=Law⁡\(Xt\)\\mu\_\{t\}:=\\operatorname\{Law\}\(X\_\{t\}\), where the law includes both Brownian and stochastic oracle randomness\. ForT\>0T\>0, define

NT\\displaystyle N\_\{T\}:=inf\{n∈ℕ0:τn≥T\},\\displaystyle:=\\inf\\\{n\\in\\mathbb\{N\}\_\{0\}:\\tau\_\{n\}\\geq T\\\},μ¯T​\(A\)\\displaystyle\\bar\{\\mu\}\_\{T\}\(A\):=1T​∫0Tμt​\(A\)​dt,\\displaystyle:=\\frac\{1\}\{T\}\\int\_\{0\}^\{T\}\\mu\_\{t\}\(A\)\\,\\mathrm\{d\}t,whereℕ0:=\{0,1,…\}\\mathbb\{N\}\_\{0\}:=\\\{0,1,\\ldots\\\}andinf∅:=∞\\inf\\emptyset:=\\infty, and where the second identity holds for every Borel setAAwheneverNT<∞N\_\{T\}<\\inftyalmost surely\. Thus,μ¯T\\bar\{\\mu\}\_\{T\}is a deterministic mixture law\. Equivalently, ifU∼Unif⁡\(\[0,T\]\)U\\sim\\mathrm\{Unif\}\(\[0,T\]\)is independent of the process, thenXU∼μ¯TX\_\{U\}\\sim\\bar\{\\mu\}\_\{T\}\. If eventkkusesqkq\_\{k\}individual gradient oracle evaluations, its oracle count through timeTTis

QT:=∑k=0NT−1qk\.Q\_\{T\}:=\\sum\_\{k=0\}^\{N\_\{T\}\-1\}q\_\{k\}\.For a fixed step size and a horizonT=N​hT=Nh, we haveτk=k​h\\tau\_\{k\}=khandNT=NN\_\{T\}=N\.

##### Sampling convention\.

Throughout the stochastic methods, conditional on the past and current state, all oracle evaluations drawn at an event are fresh and mutually independent, and they are independent of the Brownian increment on the following interval\.

### A\.2Notation, constants, and deterministic schedules

This appendix gives the complete statements and proofs of the three main results\. The definitions in this subsection make the theorem statements explicit\. Each proof recalls the relevant formulas where they enter the argument, together with a brief explanation of their role\. For real numbersa,ba,b, we writea∧b:=min⁡\{a,b\}a\\wedge b:=\\min\\\{a,b\\\}anda∨b:=max⁡\{a,b\}a\\vee b:=\\max\\\{a,b\\\}\. All laws, event times, interpolations, time\-averaged laws, and oracle counts follow Definition[A\.1](https://arxiv.org/html/2610.12865#A1.Thmtheorem1)\. We write

FI¯T:=1T∫0TFI\(μt∥π\)dt\.\\overline\{\\mathrm\{FI\}\}\_\{T\}:=\\frac\{1\}\{T\}\\int\_\{0\}^\{T\}\\mathrm\{FI\}\(\\mu\_\{t\}\\\|\\pi\)\\,\\mathrm\{d\}t\.Convexity of relative Fisher information givesFI\(μ¯T∥π\)≤FI¯T\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\overline\{\\mathrm\{FI\}\}\_\{T\}\. ThusFI¯T\\overline\{\\mathrm\{FI\}\}\_\{T\}averages the information of the individual laws, whereasFI\(μ¯T∥π\)\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)is the information of their time average; these quantities need not coincide\. The initialization quantity common to all three results isK0:=KL\(μ0∥π\)K\_\{0\}:=\\mathrm\{KL\}\(\\mu\_\{0\}\\\|\\pi\)\. For the stochastic methods withB\>0B\>0, the proofs additionally require a known finite bound𝔼​‖X0−xref‖2≤M2,0\\mathbb\{E\}\\left\\lVert X\_\{0\}\-x\_\{\\mathrm\{ref\}\}\\right\\rVert^\{2\}\\leq M\_\{2,0\}\. WhenB=0B=0, no second moment bound on the initial state is needed, and we setM2,0:=0M\_\{2,0\}:=0\. Finite\-time bounds retain the initialization quantities explicitly\. To report iteration and oracle complexities, we adopt the conventionK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)and showM2,0M\_\{2,0\}explicitly whenB\>0B\>0\.

For the BG\-pporacle, we define its localLpL^\{p\}\-noise scale as

sp​\(x\):=\[B​‖x−xref‖p\+G\]1/p\.s\_\{p\}\(x\):=\\bigl\[B\\left\\lVert x\-x\_\{\\mathrm\{ref\}\}\\right\\rVert^\{p\}\+G\\bigr\]^\{1/p\}\.The robust estimator error bound uses the constant

Cp:=22/p−1​\[22​p−2​\(1\+2​3​4p\)2\+32​p​31/pp−1\+2p​\(1\+2​3​4p\)\]\.C\_\{p\}:=2^\{2/p\-1\}\\Bigg\[2^\{2p\-2\}\\bigl\(1\+2\\sqrt\{3\}\\,4^\{p\}\\bigr\)^\{2\}\+\\frac\{32p\\,3^\{1/p\}\}\{p\-1\}\+2^\{p\}\\bigl\(1\+2\\sqrt\{3\}\\,4^\{p\}\\bigr\)\\Bigg\]\.\(10\)This constant depends only onppand is obtained in Lemma[A\.9](https://arxiv.org/html/2610.12865#A1.Thmtheorem9)by bounding the squared bias and variance of the robust estimator\. Throughout,GeoMed\\operatorname\{GeoMed\}denotes a fixed measurable, translation\-equivariant selection from the minimizers of the sum of Euclidean distances\. The proof uses only its defining minimization property\. Writingu=‖x−xref‖2u=\\left\\lVert x\-x\_\{\\mathrm\{ref\}\}\\right\\rVert^\{2\}, the square of the localLpL^\{p\}noise scale becomes

sp​\(x\)2=\[B​up/2\+G\]2/p\.s\_\{p\}\(x\)^\{2\}=\\bigl\[Bu^\{p/2\}\+G\\bigr\]^\{2/p\}\.As a function of the squared distanceu≥0u\\geq 0, this expression is nonnegative, nondecreasing, and concave for1<p≤21<p\\leq 2\. Indeed, away from its zero set, its second derivative equals

B⁡\(p2−1\)​G​up/2−2​\(B​up/2\+G\)2/p−2≤0\.B\\left\(\\frac\{p\}\{2\}\-1\\right\)Gu^\{p/2\-2\}\(Bu^\{p/2\}\+G\)^\{2/p\-2\}\\leq 0\.Continuity covers the remaining points\. Consequently, Jensen’s inequality gives

𝔼​sp​\(X\)2≤\[B​\(𝔼​‖X−xref‖2\)p/2\+G\]2/p\.\\mathbb\{E\}s\_\{p\}\(X\)^\{2\}\\leq\\left\[B\\bigl\(\\mathbb\{E\}\\left\\lVert X\-x\_\{\\mathrm\{ref\}\}\\right\\rVert^\{2\}\\bigr\)^\{p/2\}\+G\\right\]^\{2/p\}\.Thus, a second\-moment bound on the trajectory controls its average squared localLpL^\{p\}noise scale in Theorems[A\.2](https://arxiv.org/html/2610.12865#A1.Thmtheorem2)and[A\.6](https://arxiv.org/html/2610.12865#A1.Thmtheorem6)\.

We next collect the generalized smoothness coefficients shared by the last two results\. Set

ω:=L11\+L0,andCγ:=4​d​L0\+\(2−γ\)​γγ/\(2−γ\)​\(2​d​L1\)2/\(2−γ\)\.\\omega:=\\frac\{L\_\{1\}\}\{1\+L\_\{0\}\},\\quad\\text\{and\}\\quad C\_\{\\gamma\}:=4dL\_\{0\}\+\(2\-\\gamma\)\\gamma^\{\\gamma/\(2\-\\gamma\)\}\(2dL\_\{1\}\)^\{2/\(2\-\\gamma\)\}\.\(11\)The constantCγC\_\{\\gamma\}appears in the gradient second\-moment bound𝔼μ‖∇V‖2≤2FI\(μ∥π\)\+Cγ\\mathbb\{E\}\_\{\\mu\}\\left\\lVert\\nabla V\\right\\rVert^\{2\}\\leq 2\\mathrm\{FI\}\(\\mu\\\|\\pi\)\+C\_\{\\gamma\}\. The coefficientsAγA\_\{\\gamma\},BγB\_\{\\gamma\}, andQγQ\_\{\\gamma\}below control the discretization error for updates using exact gradients\. WhenL1\>0L\_\{1\}\>0and0<γ<10<\\gamma<1, let

Aγ​\(h\)\\displaystyle A\_\{\\gamma\}\(h\)=3​L02​\(1\+2γ2/\(1−γ\)\)2​d​h\+32​γ\+1​2\(2​γ2\+γ\+2\)/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​L12/\(1−γ\)​\[\(d\+21−γ\)​h\]1/\(1−γ\),\\displaystyle=3L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}dh\+3^\{2\\gamma\+1\}2^\{\(2\\gamma^\{2\}\+\\gamma\+2\)/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}L\_\{1\}^\{2/\(1\-\\gamma\)\}\\left\[\\left\(d\+\\frac\{2\}\{1\-\\gamma\}\\right\)h\\right\]^\{1/\(1\-\\gamma\)\},Bγ​\(h\)\\displaystyle B\_\{\\gamma\}\(h\)=\[L02​\(1\+2γ2/\(1−γ\)\)2\+32​γ​22​γ2/\(1−γ\)​\(1\+L0\)2\]​h2\\displaystyle=\\left\[L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}\+3^\{2\\gamma\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}\(1\+L\_\{0\}\)^\{2\}\\right\]h^\{2\}\+32​γ\+1​2\(2​γ2\+γ\+1\)/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​\[\(1\+L0\)​h\]2/\(1−γ\),\\displaystyle\\quad\+3^\{2\\gamma\+1\}2^\{\(2\\gamma^\{2\}\+\\gamma\+1\)/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}\[\(1\+L\_\{0\}\)h\]^\{2/\(1\-\\gamma\)\},Qγ​\(h\)\\displaystyle Q\_\{\\gamma\}\(h\)=32​γ\+1​22​γ2/\(1−γ\)​L1​\(1\+L0\)​d​h\.\\displaystyle=3^\{2\\gamma\+1\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}L\_\{1\}\(1\+L\_\{0\}\)dh\.\(12\)AtL1\>0L\_\{1\}\>0andγ=1\\gamma=1, define

E1ex​\(h\)\\displaystyle E\_\{1\}^\{\\mathrm\{ex\}\}\(h\):=exp⁡\{2​\(1\+L0\)​h\+2​2​L1​d​h\+8​L12​h\},\\displaystyle:=\\exp\\\{2\(1\+L\_\{0\}\)h\+2\\sqrt\{2\}L\_\{1\}\\sqrt\{dh\}\+8L\_\{1\}^\{2\}h\\\},A1​\(h\)\\displaystyle A\_\{1\}\(h\):=4​E1ex​\(h\)​L02​\(d\+1\)​h,\\displaystyle:=4E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)L\_\{0\}^\{2\}\(d\+1\)h,B1​\(h\)\\displaystyle B\_\{1\}\(h\):=43​E1ex​\(h\)​\{L02\+\(1\+L0\)2\}​h2,\\displaystyle:=\\tfrac\{4\}\{3\}E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)\\\{L\_\{0\}^\{2\}\+\(1\+L\_\{0\}\)^\{2\}\\\}h^\{2\},Q1​\(h\)\\displaystyle Q\_\{1\}\(h\):=4​E1ex​\(h\)​L1​\(1\+L0\)​\(d\+1\)​h\.\\displaystyle:=4E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)L\_\{1\}\(1\+L\_\{0\}\)\(d\+1\)h\.\(13\)WhenL1=0L\_\{1\}=0, the sharper coefficients, valid for every0<γ≤10<\\gamma\\leq 1, are

Aγ​\(h\)=d​L02​h,Bγ​\(h\)=13​L02​h2,Qγ​\(h\)=0\.A\_\{\\gamma\}\(h\)=dL\_\{0\}^\{2\}h,\\qquad B\_\{\\gamma\}\(h\)=\\tfrac\{1\}\{3\}L\_\{0\}^\{2\}h^\{2\},\\qquad Q\_\{\\gamma\}\(h\)=0\.\(14\)The following quantities define the coefficient multiplying the discretization error and the terms independent of that error, respectively\.

Δγ​\(h\)\\displaystyle\\Delta\_\{\\gamma\}\(h\):=8​Bγ​\(h\)\+8γ/2​Qγ​\(h\),\\displaystyle:=8B\_\{\\gamma\}\(h\)\+8^\{\\gamma/2\}Q\_\{\\gamma\}\(h\),Rγ​\(h,M\)\\displaystyle R\_\{\\gamma\}\(h,M\):=Aγ​\(h\)\+8​Bγ​\(h\)​M\+8γ/2​Qγ​\(h\)​Mγ/2\.\\displaystyle:=A\_\{\\gamma\}\(h\)\+8B\_\{\\gamma\}\(h\)M\+8^\{\\gamma/2\}Q\_\{\\gamma\}\(h\)M^\{\\gamma/2\}\.\(15\)Thus,RγR\_\{\\gamma\}contains the error remaining after applying the moment bounds, whileΔγ\\Delta\_\{\\gamma\}is the coefficient of the error that must be absorbed\.

We use a tilde to distinguish the coefficients for updates using gradient estimates\. In the discretization bound of Lemma[A\.20](https://arxiv.org/html/2610.12865#A1.Thmtheorem20),A~γ\\widetilde\{A\}\_\{\\gamma\}collects terms independent of the gradient norms, whileU~γ\\widetilde\{U\}\_\{\\gamma\}andV~γ\\widetilde\{V\}\_\{\\gamma\}multiply the squared norms of the exact gradient and the applied force, respectively\. For the stochastic methods, the force is the negative gradient estimate and has the same norm\. IfL1=0L\_\{1\}=0, set

A~γ​\(h\)=L02​d​h,U~γ​\(h\)=0,V~γ​\(h\)=13​L02​h2\.\\widetilde\{A\}\_\{\\gamma\}\(h\)=L\_\{0\}^\{2\}dh,\\qquad\\widetilde\{U\}\_\{\\gamma\}\(h\)=0,\\qquad\\widetilde\{V\}\_\{\\gamma\}\(h\)=\\tfrac\{1\}\{3\}L\_\{0\}^\{2\}h^\{2\}\.\(16\)IfL1\>0L\_\{1\}\>0and0<γ<10<\\gamma<1, set

A~γ​\(h\)\\displaystyle\\widetilde\{A\}\_\{\\gamma\}\(h\)=3​\[L02​\(1\+2γ2/\(1−γ\)\)2\+32​γ​22​γ2/\(1−γ\)​L12\]​d​h\\displaystyle=3\\left\[L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}\+3^\{2\\gamma\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}L\_\{1\}^\{2\}\\right\]dh\+32​γ\+1​2\(2​γ2\+γ\+2\)/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​L12/\(1−γ\)​\[\(d\+21−γ\)​h\]1/\(1−γ\),\\displaystyle\\quad\+3^\{2\\gamma\+1\}2^\{\(2\\gamma^\{2\}\+\\gamma\+2\)/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}L\_\{1\}^\{2/\(1\-\\gamma\)\}\\left\[\\left\(d\+\\frac\{2\}\{1\-\\gamma\}\\right\)h\\right\]^\{1/\(1\-\\gamma\)\},U~γ​\(h\)\\displaystyle\\widetilde\{U\}\_\{\\gamma\}\(h\)=32​γ​22​γ2/\(1−γ\)​\[γ​\(1\+L0\)2​h2\+3​L12​d​h\],\\displaystyle=3^\{2\\gamma\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}\\left\[\\gamma\(1\+L\_\{0\}\)^\{2\}h^\{2\}\+3L\_\{1\}^\{2\}dh\\right\],V~γ​\(h\)\\displaystyle\\widetilde\{V\}\_\{\\gamma\}\(h\)=\[L02​\(1\+2γ2/\(1−γ\)\)2\+\(1−γ\)​32​γ​22​γ2/\(1−γ\)​\(1\+L0\)2\]​h2\\displaystyle=\\left\[L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}\+\(1\-\\gamma\)3^\{2\\gamma\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}\(1\+L\_\{0\}\)^\{2\}\\right\]h^\{2\}\+32​γ\+1​2\(2​γ2\+γ\+1\)/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​\[\(1\+L0\)​h\]2/\(1−γ\)\.\\displaystyle\\quad\+3^\{2\\gamma\+1\}2^\{\(2\\gamma^\{2\}\+\\gamma\+1\)/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}\[\(1\+L\_\{0\}\)h\]^\{2/\(1\-\\gamma\)\}\.\(17\)AtL1\>0L\_\{1\}\>0andγ=1\\gamma=1, put

A~1​\(h\)\\displaystyle\\widetilde\{A\}\_\{1\}\(h\)=4​E1ex​\(h\)​L02​\(d\+1\)​h,\\displaystyle=4E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)L\_\{0\}^\{2\}\(d\+1\)h,U~1​\(h\)\\displaystyle\\widetilde\{U\}\_\{1\}\(h\)=E1ex​\(h\)​\{43​\(1\+L0\)2​h2\+4​L12​\(d\+1\)​h\},\\displaystyle=E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)\\\{\\tfrac\{4\}\{3\}\(1\+L\_\{0\}\)^\{2\}h^\{2\}\+4L\_\{1\}^\{2\}\(d\+1\)h\\\},V~1​\(h\)\\displaystyle\\widetilde\{V\}\_\{1\}\(h\)=43​E1ex​\(h\)​L02​h2\.\\displaystyle=\\tfrac\{4\}\{3\}E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)L\_\{0\}^\{2\}h^\{2\}\.\(18\)All coefficients defined above are deterministic, continuous, and nondecreasing inhh, and vanish ath=0h=0\. The caseL1=0L\_\{1\}=0uses the sharper bounds available under global smoothness\. The caseγ=1\\gamma=1uses the exponential gradient variation bound instead of the power bound for0<γ<10<\\gamma<1\.

### A\.3Complete convergence guarantees

###### Theorem A\.2\(Fixed\-batch robust LMC\)\.

Suppose Assumptions[2\.2](https://arxiv.org/html/2610.12865#S2.Thmtheorem2)and[3\.1](https://arxiv.org/html/2610.12865#S3.Thmtheorem1)hold\. AssumeK0<∞K\_\{0\}<\\inftyand, ifB\>0B\>0, a known bound𝔼​‖X0−xref‖2≤M2,0<∞\\mathbb\{E\}\\left\\lVert X\_\{0\}\-x\_\{\\mathrm\{ref\}\}\\right\\rVert^\{2\}\\leq M\_\{2,0\}<\\infty\. Fixδ\>0\\delta\>0, an integerN≥1N\\geq 1, and0<h≤1/\(8​L\)0<h\\leq 1/\(8L\), and writeT=N​hT=Nh\. Define

RT,δ:=\(3​M2,0\+645​K0​T\+33215​L​d​T2\+3​d​T\+883​T2​δ2\)1/2R\_\{T,\\delta\}:=\\left\(3M\_\{2,0\}\+\\frac\{64\}\{5\}K\_\{0\}T\+\\frac\{332\}\{15\}LdT^\{2\}\+3dT\+\\frac\{88\}\{3\}T^\{2\}\\delta^\{2\}\\right\)^\{1/2\}and use at every step

b⁡\(T,δ\)\\displaystyle b\(T,\\delta\):=max\{1,⌈\(2​Cp​\[B​RT,δp\+G\]2/pδ2\)p/\[2​\(p−1\)\]⌉\}andλk:=sp\(Xk​h\)b\(T,δ\)1/p\.\\displaystyle:=\\max\\left\\\{1,\\left\\lceil\\left\(\\frac\{2C\_\{p\}\[BR\_\{T,\\delta\}^\{p\}\+G\]^\{2/p\}\}\{\\delta^\{2\}\}\\right\)^\{p/\[2\(p\-1\)\]\}\\right\\rceil\\right\\\}\\quad\\text\{and\}\\quad\\lambda\_\{k\}:=s\_\{p\}\(X\_\{kh\}\)b\(T,\\delta\)^\{1/p\}\.Then the updates in \([6](https://arxiv.org/html/2610.12865#S3.E6)\) satisfy

FI\(μ¯T∥π\)≤FI¯T\\displaystyle\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\overline\{\\mathrm\{FI\}\}\_\{T\}≤2​K0T\+323​L2​d​h\+4912​δ2\.\\displaystyle\\leq\\frac\{2K\_\{0\}\}\{T\}\+\\frac\{32\}\{3\}L^\{2\}dh\+\\frac\{49\}\{12\}\\delta^\{2\}\.
For anyε\>0\\varepsilon\>0, chooseδ,h\\delta,h, andTTso thatδ2=ε16\\delta^\{2\}=\\frac\{\\varepsilon\}\{16\},h=min⁡\{18​L,ε32​L2​d\}h=\\min\\left\\\{\\frac\{1\}\{8L\},\\frac\{\\varepsilon\}\{32L^\{2\}d\}\\right\\\}andT=h​max⁡\{1,⌈6​K0ε​h⌉\}T=h\\max\\left\\\{1,\\left\\lceil\\frac\{6K\_\{0\}\}\{\\varepsilon h\}\\right\\rceil\\right\\\}\. Setb=b⁡\(T,δ\)b=b\(T,\\delta\)\. The resulting law satisfiesFI\(μ¯T∥π\)≤ε\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\varepsilonand

NT\\displaystyle N\_\{T\}≤1\+6​K0ε​max⁡\{8​L,32​L2​dε\},QT=4​b​NT≤4​NT​\[1\+\(2​Cp​\[B​RT,δp\+G\]2/pδ2\)p/\[2​\(p−1\)\]\]\.\\displaystyle\\leq 1\+\\frac\{6K\_\{0\}\}\{\\varepsilon\}\\max\\left\\\{8L,\\frac\{32L^\{2\}d\}\{\\varepsilon\}\\right\\\},\\quad Q\_\{T\}=4bN\_\{T\}\\leq 4N\_\{T\}\\left\[1\+\\left\(\\frac\{2C\_\{p\}\[BR\_\{T,\\delta\}^\{p\}\+G\]^\{2/p\}\}\{\\delta^\{2\}\}\\right\)^\{p/\[2\(p\-1\)\]\}\\right\]\.For fixed problem parameters withK0,L\>0K\_\{0\},L\>0andB\>0B\>0, asε↓0\\varepsilon\\downarrow 0,

NT\\displaystyle N\_\{T\}=𝒪⁡\(K0​L2​dε2\),QT=𝒪⁡\(B1/\(p−1\)​K01\+p/\(p−1\)​L2\+p/\[2​\(p−1\)\]​d1\+p/\[2​\(p−1\)\]ε2\+3​p/\[2​\(p−1\)\]\)\.\\displaystyle=\\mathcal\{O\}\\\!\\left\(\\frac\{K\_\{0\}L^\{2\}d\}\{\\varepsilon^\{2\}\}\\right\),\\quad Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{B^\{1/\(p\-1\)\}K\_\{0\}^\{1\+p/\(p\-1\)\}L^\{2\+p/\[2\(p\-1\)\]\}d^\{1\+p/\[2\(p\-1\)\]\}\}\{\\varepsilon^\{2\+3p/\[2\(p\-1\)\]\}\}\\right\)\.WhenB=0B=0andG\>0G\>0, the oracle rate is instead

QT=𝒪⁡\(K0​L2​d​G1/\(p−1\)ε2\+p/\[2​\(p−1\)\]\)\.Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{K\_\{0\}L^\{2\}d\\,G^\{1/\(p\-1\)\}\}\{\\varepsilon^\{2\+p/\[2\(p\-1\)\]\}\}\\right\)\.

###### Corollary A\.3\(Special cases of fixed\-batch robust LMC\)\.

Under the assumptions and accuracy schedule of Theorem[A\.2](https://arxiv.org/html/2610.12865#A1.Thmtheorem2), supposeK0\>0K\_\{0\}\>0andK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)\. Then

NT=𝒪⁡\(L2​d2ε2\)\.N\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{L^\{2\}d^\{2\}\}\{\\varepsilon^\{2\}\}\\right\)\.With the remaining problem parameters fixed, asε↓0\\varepsilon\\downarrow 0, the corresponding oracle complexities specialize as follows\.

1. 1\.Ifp=2p=2,B=0B=0, andG\>0G\>0, then QT=𝒪⁡\(L2​d2​Gε3\)\.Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{L^\{2\}d^\{2\}G\}\{\\varepsilon^\{3\}\}\\right\)\.
2. 2\.Under BG\-0 noise, corresponding top=2p=2andB\>0B\>0, QT=𝒪⁡\(B​L3​d5ε5\)\.Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{BL^\{3\}d^\{5\}\}\{\\varepsilon^\{5\}\}\\right\)\.
3. 3\.IfB=G=0B=G=0, the robust construction generates the standard exact\-gradient LMC iterates\. Removing its three redundant oracle calls per iteration leaves the iterates unchanged and gives QT=NT=𝒪⁡\(L2​d2ε2\)\.Q\_\{T\}=N\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{L^\{2\}d^\{2\}\}\{\\varepsilon^\{2\}\}\\right\)\.

###### Theorem A\.4\(Exact\-gradient adaptive\-step LMC\)\.

LetV∈C1​\(ℝd\)V\\in C^\{1\}\(\\mathbb\{R\}^\{d\}\), let0<Z<∞0<Z<\\infty, and supposeK0<∞K\_\{0\}<\\infty\. Under Assumption[2\.3](https://arxiv.org/html/2610.12865#S2.Thmtheorem3), run the adaptive LMC in \([7](https://arxiv.org/html/2610.12865#S3.E7)\), truncate only its last occupied interval at a prescribed horizonT\>0T\>0\. LetMT:=1\+Cγ\+K0/TM\_\{T\}:=1\+C\_\{\\gamma\}\+K\_\{0\}/Tand fix a base step satisfying0<h≤\[1\+2​Δγ​\(1\)\]−10<h\\leq\[1\+2\\Delta\_\{\\gamma\}\(1\)\]^\{\-1\}\. ThenNT<∞N\_\{T\}<\\inftyalmost surely, and

FI\(μ¯T∥π\)≤FI¯T≤4​K03​T\+83Rγ\(h,MT\)\.\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\overline\{\\mathrm\{FI\}\}\_\{T\}\\leq\\frac\{4K\_\{0\}\}\{3T\}\+\\frac\{8\}\{3\}R\_\{\\gamma\}\(h,M\_\{T\}\)\.\(19\)Moreover,QT=NTQ\_\{T\}=N\_\{T\}, and

𝔼​NT≤1\+Th​\[1\+ω​\{8​\(MT\+2​Rγ​\(h,MT\)\)\}γ/2\]\.\\mathbb\{E\}N\_\{T\}\\leq 1\+\\frac\{T\}\{h\}\\left\[1\+\\omega\\\{8\(M\_\{T\}\+2R\_\{\\gamma\}\(h,M\_\{T\}\)\)\\\}^\{\\gamma/2\}\\right\]\.\(20\)
For0<ε≤10<\\varepsilon\\leq 1andK0\>0K\_\{0\}\>0, choose

T:=4​K0ε,h:=max⁡\{u∈\[0,\(1\+2​Δγ​\(1\)\)−1\]:Rγ​\(u,MT\)≤3​ε16\}\.T:=\\frac\{4K\_\{0\}\}\{\\varepsilon\},\\qquad h:=\\max\\left\\\{u\\in\[0,\(1\+2\\Delta\_\{\\gamma\}\(1\)\)^\{\-1\}\]:R\_\{\\gamma\}\(u,M\_\{T\}\)\\leq\\frac\{3\\varepsilon\}\{16\}\\right\\\}\.\(21\)This choice is well defined and yieldsh\>0h\>0\. It givesFI\(μ¯T∥π\)≤ε\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\varepsilonand

𝔼​QT=𝔼​NT≤1\+4​K0ε​h​\(1\+4​ω​MTγ/2\)\.\\mathbb\{E\}Q\_\{T\}=\\mathbb\{E\}N\_\{T\}\\leq 1\+\\frac\{4K\_\{0\}\}\{\\varepsilon h\}\\left\(1\+4\\omega M\_\{T\}^\{\\gamma/2\}\\right\)\.\(22\)IfL1\>0L\_\{1\}\>0, letΛ:=max⁡\{1,L0,L1\}\\Lambda:=\\max\\\{1,L\_\{0\},L\_\{1\}\\\}\. With the remaining problem parameters fixed, asε↓0\\varepsilon\\downarrow 0,

𝔼​NT=𝔼​QT=𝒪⁡\(K0​Λ4/\(2−γ\)​d\(2\+γ\)/\(2−γ\)ε2\)\.\\mathbb\{E\}N\_\{T\}=\\mathbb\{E\}Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{K\_\{0\}\\Lambda^\{4/\(2\-\\gamma\)\}d^\{\(2\+\\gamma\)/\(2\-\\gamma\)\}\}\{\\varepsilon^\{2\}\}\\right\)\.\(23\)

###### Corollary A\.5\(Complexity specializations of adaptive\-step LMC\)\.

Under the assumptions and accuracy schedule of Theorem[A\.4](https://arxiv.org/html/2610.12865#A1.Thmtheorem4), supposeK0\>0K\_\{0\}\>0andK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)\. Then the following statements hold asε↓0\\varepsilon\\downarrow 0\.

1. 1\.IfL1\>0L\_\{1\}\>0andΛ:=max⁡\{1,L0,L1\}\\Lambda:=\\max\\\{1,L\_\{0\},L\_\{1\}\\\}, then 𝔼​NT=𝔼​QT=𝒪⁡\(Λ4/\(2−γ\)​d4/\(2−γ\)ε2\)\.\\mathbb\{E\}N\_\{T\}=\\mathbb\{E\}Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{\\Lambda^\{4/\(2\-\\gamma\)\}d^\{4/\(2\-\\gamma\)\}\}\{\\varepsilon^\{2\}\}\\right\)\.
2. 2\.IfL1=0L\_\{1\}=0, the step size is constant and the oracle count is deterministic\. For all sufficiently smallε\\varepsilon, QT=NT≤1\+128​K0​d​L023​ε2Q\_\{T\}=N\_\{T\}\\leq 1\+\\frac\{128K\_\{0\}dL\_\{0\}^\{2\}\}\{3\\varepsilon^\{2\}\}\(24\)and henceQT=NT=𝒪⁡\(d2​L02/ε2\)Q\_\{T\}=N\_\{T\}=\\mathcal\{O\}\(d^\{2\}L\_\{0\}^\{2\}/\\varepsilon^\{2\}\)underK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)\.

###### Theorem A\.6\(Robust adaptive step LMC\)\.

LetV∈C1​\(ℝd\)V\\in C^\{1\}\(\\mathbb\{R\}^\{d\}\), let0<Z<∞0<Z<\\infty, and supposeK0<∞K\_\{0\}<\\infty\. Suppose Assumptions[2\.3](https://arxiv.org/html/2610.12865#S2.Thmtheorem3)and[3\.1](https://arxiv.org/html/2610.12865#S3.Thmtheorem1)hold with0<γ≤10<\\gamma\\leq 1and1<p≤21<p\\leq 2\. IfB\>0B\>0, assume𝔼​‖X0−xref‖2≤M2,0<∞\\mathbb\{E\}\\left\\lVert X\_\{0\}\-x\_\{\\mathrm\{ref\}\}\\right\\rVert^\{2\}\\leq M\_\{2,0\}<\\infty\. WhenB=0B=0, setM2,0=0M\_\{2,0\}=0\. Define

𝒟γ​\(u\):=A~γ​\(u\)\+\{U~γ​\(u\)\+2​V~γ​\(u\)\}​\(1\+Cγ\)\.\\mathcal\{D\}\_\{\\gamma\}\(u\):=\\widetilde\{A\}\_\{\\gamma\}\(u\)\+\\\{\\widetilde\{U\}\_\{\\gamma\}\(u\)\+2\\widetilde\{V\}\_\{\\gamma\}\(u\)\\\}\(1\+C\_\{\\gamma\}\)\.For0<ε≤10<\\varepsilon\\leq 1, chooseT:=8​\(K0\+1\)εT:=\\frac\{8\(K\_\{0\}\+1\)\}\{\\varepsilon\},MT:=1\+Cγ\+K0/TM\_\{T\}:=1\+C\_\{\\gamma\}\+K\_\{0\}/T, andh:=max⁡\{u∈\[0,1\]:𝒟γ​\(u\)≤ε/64\}\.h:=\\max\\\{u\\in\[0,1\]:\\mathcal\{D\}\_\{\\gamma\}\(u\)\\leq\\varepsilon/64\\\}\.Set

κ:=\(1\+s¯T2\)−1,andb:=⌈\[260​Cp​κ−1​\(1\+ω​MTγ/2\)ε\]p/\[2​\(p−1\)\]⌉,\\kappa:=\(1\+\\bar\{s\}\_\{T\}^\{2\}\)^\{\-1\},\\quad\\text\{and\}\\quad b:=\\left\\lceil\\left\[\\frac\{260C\_\{p\}\\kappa^\{\-1\}\(1\+\\omega M\_\{T\}^\{\\gamma/2\}\)\}\{\\varepsilon\}\\right\]^\{p/\[2\(p\-1\)\]\}\\right\\rceil,wheres¯T2:=\[B​\(6​M2,0\+8​d​T\+80​T2​MT\)p/2\+G\]2/p\\bar\{s\}\_\{T\}^\{2\}:=\\left\[B\\left\(6M\_\{2,0\}\+8dT\+80T^\{2\}M\_\{T\}\\right\)^\{p/2\}\+G\\right\]^\{2/p\}andω:=L11\+L0\.\\omega:=\\frac\{L\_\{1\}\}\{1\+L\_\{0\}\}\.Run \([8](https://arxiv.org/html/2610.12865#S3.E8)\) usingbbindependent quadruples per iteration and clipping radiusλk=sp​\(Xτk\)​b1/p\\lambda\_\{k\}=s\_\{p\}\(X\_\{\\tau\_\{k\}\}\)b^\{1/p\}, truncating the last occupied interval atTT\. Thenh\>0h\>0,NT<∞N\_\{T\}<\\inftyalmost surely, and

FI\(μ¯T∥π\)≤FI¯T≤2​K0T\+9𝒟γ\(h\)\+ε2≤ε\.\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\overline\{\\mathrm\{FI\}\}\_\{T\}\\leq\\frac\{2K\_\{0\}\}\{T\}\+9\\mathcal\{D\}\_\{\\gamma\}\(h\)\+\\frac\{\\varepsilon\}\{2\}\\leq\\varepsilon\.Moreover,QT=4​b​NTQ\_\{T\}=4bN\_\{T\}pathwise, with

𝔼​NT≤2\+24​Th​\(1\+ω​MTγ/2\),𝔼​QT=4​b​𝔼​NT\.\\mathbb\{E\}N\_\{T\}\\leq 2\+\\frac\{24T\}\{h\}\(1\+\\omega M\_\{T\}^\{\\gamma/2\}\),\\qquad\\mathbb\{E\}Q\_\{T\}=4b\\mathbb\{E\}N\_\{T\}\.The asymptotic complexity specializations are given in Corollary[A\.7](https://arxiv.org/html/2610.12865#A1.Thmtheorem7)\.

###### Corollary A\.7\(Complexity specializations of robust adaptive step LMC\)\.

Under the assumptions and accuracy schedule of Theorem[A\.6](https://arxiv.org/html/2610.12865#A1.Thmtheorem6), supposeK0\>0K\_\{0\}\>0andK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)\. With the remaining problem parameters fixed, the following statements hold asε↓0\\varepsilon\\downarrow 0\.

1. 1\.IfL1\>0L\_\{1\}\>0andΛ:=max⁡\{1,L0,L1\}\\Lambda:=\\max\\\{1,L\_\{0\},L\_\{1\}\\\}, then 𝔼​NT=𝒪⁡\(Λ\(8−2​γ\)/\(2−γ\)​d\(6−γ\)/\(2−γ\)ε2\)\.\\mathbb\{E\}N\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{\\Lambda^\{\(8\-2\\gamma\)/\(2\-\\gamma\)\}d^\{\(6\-\\gamma\)/\(2\-\\gamma\)\}\}\{\\varepsilon^\{2\}\}\\right\)\.WhenB\>0B\>0, 𝔼​QT\\displaystyle\\mathbb\{E\}Q\_\{T\}=𝒪⁡\(B1/\(p−1\)​Λ\(8−2​γ\)​\(p−1\)\+2​p\(p−1\)​\(2−γ\)​d\(3​p−2\)​\(6−γ\)2​\(p−1\)​\(2−γ\)ε2\+3​p/\[2​\(p−1\)\]\),\\displaystyle=\\mathcal\{O\}\\\!\\left\(\\frac\{B^\{1/\(p\-1\)\}\\Lambda^\{\\frac\{\(8\-2\\gamma\)\(p\-1\)\+2p\}\{\(p\-1\)\(2\-\\gamma\)\}\}d^\{\\frac\{\(3p\-2\)\(6\-\\gamma\)\}\{2\(p\-1\)\(2\-\\gamma\)\}\}\}\{\\varepsilon^\{2\+3p/\[2\(p\-1\)\]\}\}\\right\),whereas, whenB=0B=0, 𝔼​QT\\displaystyle\\mathbb\{E\}Q\_\{T\}=𝒪⁡\(\(1\+G\)1/\(p−1\)​Λ\(8−2​γ\)​\(p−1\)\+p\(p−1\)​\(2−γ\)​d1\+8​\(p−1\)\+γ​p2​\(p−1\)​\(2−γ\)ε2\+p/\[2​\(p−1\)\]\)\.\\displaystyle=\\mathcal\{O\}\\\!\\left\(\\frac\{\(1\+G\)^\{1/\(p\-1\)\}\\Lambda^\{\\frac\{\(8\-2\\gamma\)\(p\-1\)\+p\}\{\(p\-1\)\(2\-\\gamma\)\}\}d^\{1\+\\frac\{8\(p\-1\)\+\\gamma p\}\{2\(p\-1\)\(2\-\\gamma\)\}\}\}\{\\varepsilon^\{2\+p/\[2\(p\-1\)\]\}\}\\right\)\.In particular, forp=2p=2, the oracle complexity has accuracy dependenceε−5\\varepsilon^\{\-5\}under BG\-0 noise \(B\>0B\>0\) andε−3\\varepsilon^\{\-3\}under state\-independent finite\-variance noise \(B=0B=0andG\>0G\>0\)\. With the constant inK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)fixed, the hidden numerical constants depend only onγ\\gammafor the iteration bound and onp,γp,\\gammafor the oracle bounds\. Atγ=1\\gamma=1, the same displayed powers follow by using the corresponding coefficients directly\.
2. 2\.IfL1=0L\_\{1\}=0, the noise\-aware iteration count can remain random, but 𝔼​NT=𝒪⁡\(\(K0\+1\)​d​L02ε2\)=𝒪⁡\(d2​L02ε2\)\.\\mathbb\{E\}N\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{\(K\_\{0\}\+1\)dL\_\{0\}^\{2\}\}\{\\varepsilon^\{2\}\}\\right\)=\\mathcal\{O\}\\\!\\left\(\\frac\{d^\{2\}L\_\{0\}^\{2\}\}\{\\varepsilon^\{2\}\}\\right\)\.

### A\.4Proof of Theorem[A\.2](https://arxiv.org/html/2610.12865#A1.Thmtheorem2)

The proof has three components\. First, in Subsection[A\.4\.1](https://arxiv.org/html/2610.12865#A1.SS4.SSS1), we show that geometric\-median centering followed by residual clipping converts a conditionalpp\-th\-moment oracle bound into a conditional mean\-squared error bound\. Second, Subsection[A\.4\.2](https://arxiv.org/html/2610.12865#A1.SS4.SSS2)uses a one\-step KL\-dissipation argument to convert the mean\-squared drift error into a bound on time\-averaged relative Fisher information\. Finally, Subsection[A\.4\.3](https://arxiv.org/html/2610.12865#A1.SS4.SSS3)controls the average squared distance of the chain fromxrefx\_\{\\mathrm\{ref\}\}\. This estimate controls the dependence of the gradient\-estimation error on the state\-dependent noise scale while retaining a fixed batch size\. Subsection[A\.4\.4](https://arxiv.org/html/2610.12865#A1.SS4.SSS4)then verifies the accuracy schedule and derives the iteration and oracle complexities\. Throughout this proof,Hk=hH\_\{k\}=h,τk=k​h\\tau\_\{k\}=kh, and

𝒢^k:=1b​∑i=1b𝒢^k,i=−Fk\.\\widehat\{\\mathcal\{G\}\}\_\{k\}:=\\frac\{1\}\{b\}\\sum\_\{i=1\}^\{b\}\\widehat\{\\mathcal\{G\}\}\_\{k,i\}=\-F\_\{k\}\.Thus𝒢^k\\widehat\{\\mathcal\{G\}\}\_\{k\}estimates the gradient, whereasFkF\_\{k\}is the Langevin force in \([4](https://arxiv.org/html/2610.12865#S3.E4)\)\. All expectations below include both the oracle and Brownian randomness unless a conditioning event or state is specified\.

#### A\.4\.1Mean\-squared error of the robust estimator

We begin with the geometric median of three oracle evaluations\. It provides a center with a finite second moment even when one oracle call has only a finitepp\-th moment\. A large error in this center requires at least two of the three independent oracle calls to be large, which squares the tail probability\.

The good\-points implication below is the three\-point specialization of[Minsker \(2015, Lemma 2\.1\)](https://arxiv.org/html/2610.12865#bib.bib59); its use of independent pilot estimators is also related to the amplification principle in their Theorem 3\.1\. We retain the short calculation because it supplies the particular tail integral and constants needed here\.

###### Lemma A\.8\(Tail amplification by the geometric\-median center\)\.

LetY1,Y2,Y3Y\_\{1\},Y\_\{2\},Y\_\{3\}be independent copies of a random vectorYYwith meanθ\\thetaand

𝔼​‖Y−θ‖p≤sp,1<p≤2\.\\mathbb\{E\}\\\|Y\-\\theta\\\|^\{p\}\\leq s^\{p\},\\qquad 1<p\\leq 2\.LetC=GeoMed⁡\(Y1,Y2,Y3\)C=\\mathrm\{GeoMed\}\(Y\_\{1\},Y\_\{2\},Y\_\{3\}\)\. For everyt\>0t\>0and0<a<2​p0<a<2p,

ℙ⁡\(‖C−θ‖\>t\)\\displaystyle\\mathbb\{P\}\(\\\|C\-\\theta\\\|\>t\)≤min⁡\{1,3​\(4​s/t\)2​p\},\\displaystyle\\leq\\min\\\{1,3\(4s/t\)^\{2p\}\\\},\(25\)𝔼​‖C−θ‖a\\displaystyle\\mathbb\{E\}\\\|C\-\\theta\\\|^\{a\}≤2​p2​p−a​4a​3a/\(2​p\)​sa\.\\displaystyle\\leq\\frac\{2p\}\{2p\-a\}\\,4^\{a\}3^\{a/\(2p\)\}s^\{a\}\.\(26\)

###### Proof\.

Suppose two of the three observations, sayY1Y\_\{1\}andY2Y\_\{2\}, lie within distanceuuofθ\\theta\. SinceCCminimizes the sum of distances,

∑j=13‖C−Yj‖≤∑j=13‖θ−Yj‖\.\\sum\_\{j=1\}^\{3\}\\\|C\-Y\_\{j\}\\\|\\leq\\sum\_\{j=1\}^\{3\}\\\|\\theta\-Y\_\{j\}\\\|\.The triangle inequality bounds the left\-hand side from below by

2​‖C−θ‖−2​u\+‖θ−Y3‖−‖C−θ‖,2\\\|C\-\\theta\\\|\-2u\+\\\|\\theta\-Y\_\{3\}\\\|\-\\\|C\-\\theta\\\|,whereas the right\-hand side is at most2​u\+‖θ−Y3‖2u\+\\\|\\theta\-Y\_\{3\}\\\|\. Hence‖C−θ‖≤4​u\\\|C\-\\theta\\\|\\leq 4u\. Consequently,‖C−θ‖\>t\\\|C\-\\theta\\\|\>trequires at least two of the events\{∥Yj−θ∥\>t/4\}\\\{\\\|Y\_\{j\}\-\\theta\\\|\>t/4\\\}to occur\. A union bound over the three pairs, independence, and Markov’s inequality give

ℙ⁡\(‖C−θ‖\>t\)≤3​ℙ​\(‖Y−θ‖\>t/4\)2≤3⋅42​p​\(s/t\)2​p\.\\mathbb\{P\}\(\\\|C\-\\theta\\\|\>t\)\\leq 3\\,\\mathbb\{P\}\(\\\|Y\-\\theta\\\|\>t/4\)^\{2\}\\leq 3\\cdot 4^\{2p\}\(s/t\)^\{2p\}\.Every probability is at most one, so the preceding estimate proves \([25](https://arxiv.org/html/2610.12865#A1.E25)\)\.

Whens\>0s\>0, sett∗=4 31/\(2​p\)​st\_\{\*\}=4\\,3^\{1/\(2p\)\}s, where the two bounds in \([25](https://arxiv.org/html/2610.12865#A1.E25)\) coincide\. Tail integration yields

𝔼​‖C−θ‖a\\displaystyle\\mathbb\{E\}\\\|C\-\\theta\\\|^\{a\}≤a​∫0t∗ta−1​𝑑t\+3​a​\(4​s\)2​p​∫t∗∞ta−1−2​p​𝑑t\\displaystyle\\leq a\\int\_\{0\}^\{t\_\{\*\}\}t^\{a\-1\}\\,\\mathrm\{d\}t\+3a\(4s\)^\{2p\}\\int\_\{t\_\{\*\}\}^\{\\infty\}t^\{a\-1\-2p\}\\,\\mathrm\{d\}t=t∗a​\(1\+a2​p−a\)=2​p2​p−a​4a​3a/\(2​p\)​sa,\\displaystyle=t\_\{\*\}^\{a\}\\left\(1\+\\frac\{a\}\{2p\-a\}\\right\)=\\frac\{2p\}\{2p\-a\}\\,4^\{a\}3^\{a/\(2p\)\}s^\{a\},which is \([26](https://arxiv.org/html/2610.12865#A1.E26)\)\. Ifs=0s=0, thenY=θY=\\thetaalmost surely and both conclusions are immediate\. ∎

We call each four\-call estimate𝒢^k,i\\widehat\{\\mathcal\{G\}\}\_\{k,i\}in \([5](https://arxiv.org/html/2610.12865#S3.E5)\) a*robust replicate*\. Three oracle evaluations form its geometric\-median center, and the residual of a fourth evaluation is clipped around this center\. The next lemma bounds the mean\-squared error of the average ofbbindependent robust replicates\. Clipping limits the variance of each replicate but introduces bias\. Increasing the clipping radius reduces this bias but weakens the variance control, while averaging independent replicates reduces the variance by a factor of1/b1/b\. We choose the clipping radius to balance the squared bias and the variance of the averaged estimator\. Recall the final estimator error coefficient

Cp:=22/p−1​\[22​p−2​\(1\+2​3​4p\)2\+32​p​31/pp−1​2p​\(1\+2​3​4p\)\]\.\\displaystyle C\_\{p\}:=2^\{2/p\-1\}\\Bigg\[2^\{2p\-2\}\\bigl\(1\+2\\sqrt\{3\}\\,4^\{p\}\\bigr\)^\{2\}\+\\frac\{32p\\,3^\{1/p\}\}\{p\-1\}2^\{p\}\\bigl\(1\+2\\sqrt\{3\}\\,4^\{p\}\\bigr\)\\Bigg\]\.
###### Lemma A\.9\(Conditional mean\-squared error of the robust estimator\)\.

Fix a deterministic statexxand lets=sp​\(x\)=\[B​‖x−xref‖p\+G\]1/ps=s\_\{p\}\(x\)=\[B\\\|x\-x\_\{\\mathrm\{ref\}\}\\\|^\{p\}\+G\]^\{1/p\}\. For an integerb≥1b\\geq 1, formbbindependent four\-call replicates as in \([5](https://arxiv.org/html/2610.12865#S3.E5)\) and average them to obtain𝒢^b,λ​\(x\)\\widehat\{\\mathcal\{G\}\}\_\{b,\\lambda\}\(x\)\. Ifλ≥s\>0\\lambda\\geq s\>0, then

𝔼⁡\[‖𝒢^b,λ​\(x\)−∇V​\(x\)‖2\]\\displaystyle\\mathbb\{E\}\\\!\\left\[\\\|\\widehat\{\\mathcal\{G\}\}\_\{b,\\lambda\}\(x\)\-\\nabla V\(x\)\\\|^\{2\}\\right\]≤22​p−2​\(1\+2​3​4p\)2​s2​p​λ2−2​p\+1b​\[32​p​31/pp−1\+2p​\(1\+2​3​4p\)\]​sp​λ2−p\.\\displaystyle\\leq 2^\{2p\-2\}\\bigl\(1\+2\\sqrt\{3\}\\,4^\{p\}\\bigr\)^\{2\}s^\{2p\}\\lambda^\{2\-2p\}\+\\frac\{1\}\{b\}\\left\[\\frac\{32p\\,3^\{1/p\}\}\{p\-1\}\+2^\{p\}\\bigl\(1\+2\\sqrt\{3\}\\,4^\{p\}\\bigr\)\\right\]s^\{p\}\\lambda^\{2\-p\}\.\(27\)The first term bounds the squared bias, and the second bounds the variance after averaging\. In particular, forλ=s​b1/p\\lambda=sb^\{1/p\},

𝔼\[∥𝒢^b,λ\(x\)−∇V\(x\)∥2\]≤Cpsp\(x\)2b−2\(p−1\)/p\.\\mathbb\{E\}\\\!\\left\[\\\|\\widehat\{\\mathcal\{G\}\}\_\{b,\\lambda\}\(x\)\-\\nabla V\(x\)\\\|^\{2\}\\right\]\\leq C\_\{p\}s\_\{p\}\(x\)^\{2\}b^\{\-2\(p\-1\)/p\}\.Ifs=0s=0, the oracle and estimator are exact almost surely, so the bound remains valid withλ=0\\lambda=0\.

###### Proof\.

The assertion is immediate whens=0s=0, so assumes\>0s\>0\. Suppress the replicate index and writeYj=𝒢⁡\(x,ξj\)Y\_\{j\}=\\mathcal\{G\}\(x,\\xi\_\{j\}\),θ=∇V​\(x\)\\theta=\\nabla V\(x\),

C=GeoMed⁡\(Y1,Y2,Y3\),Y^=C\+clipλ​\(Y4−C\)\.C=\\mathrm\{GeoMed\}\(Y\_\{1\},Y\_\{2\},Y\_\{3\}\),\\qquad\\widehat\{Y\}=C\+\\mathrm\{clip\}\_\{\\lambda\}\(Y\_\{4\}\-C\)\.LetA=C−θA=C\-\\theta,U=Y4−θU=Y\_\{4\}\-\\theta, andR=Y4−C=U−AR=Y\_\{4\}\-C=U\-A\. Lemma[A\.8](https://arxiv.org/html/2610.12865#A1.Thmtheorem8)gives

𝔼​‖A‖p≤2​3​4p​sp,𝔼​‖A‖2≤16​p​31/pp−1​s2\.\\mathbb\{E\}\\\|A\\\|^\{p\}\\leq 2\\sqrt\{3\}\\,4^\{p\}s^\{p\},\\qquad\\mathbb\{E\}\\\|A\\\|^\{2\}\\leq\\frac\{16p\\,3^\{1/p\}\}\{p\-1\}s^\{2\}\.Since‖u\+v‖p≤2p−1​\(‖u‖p\+‖v‖p\)\\\|u\+v\\\|^\{p\}\\leq 2^\{p\-1\}\(\\\|u\\\|^\{p\}\+\\\|v\\\|^\{p\}\),

𝔼​‖R‖p≤2p−1​\(1\+2​3​4p\)​sp\.\\mathbb\{E\}\\\|R\\\|^\{p\}\\leq 2^\{p\-1\}\\bigl\(1\+2\\sqrt\{3\}\\,4^\{p\}\\bigr\)s^\{p\}\.
Define the discarded residualTλ​\(R\):=R−clipλ​\(R\)T\_\{\\lambda\}\(R\):=R\-\\mathrm\{clip\}\_\{\\lambda\}\(R\)\. BecauseY^=Y4−Tλ​\(R\)\\widehat\{Y\}=Y\_\{4\}\-T\_\{\\lambda\}\(R\)and𝔼​Y4=θ\\mathbb\{E\}Y\_\{4\}=\\theta,

‖𝔼​Y^−θ‖\\displaystyle\\\|\\mathbb\{E\}\\widehat\{Y\}\-\\theta\\\|≤𝔼​‖Tλ​\(R\)‖≤λ1−p​𝔼​‖R‖p\\displaystyle\\leq\\mathbb\{E\}\\\|T\_\{\\lambda\}\(R\)\\\|\\leq\\lambda^\{1\-p\}\\mathbb\{E\}\\\|R\\\|^\{p\}≤2p−1​\(1\+2​3​4p\)​sp​λ1−p\.\\displaystyle\\leq 2^\{p\-1\}\\bigl\(1\+2\\sqrt\{3\}\\,4^\{p\}\\bigr\)s^\{p\}\\lambda^\{1\-p\}\.\(28\)Here we used\(z−λ\)\+≤z𝟏\{z\>λ\}≤λ1−pzp\(z\-\\lambda\)\_\{\+\}\\leq z\\mathbf\{1\}\_\{\\\{z\>\\lambda\\\}\}\\leq\\lambda^\{1\-p\}z^\{p\}forz≥0z\\geq 0\. Similarly,min⁡\{z,λ\}2≤λ2−p​zp\\min\\\{z,\\lambda\\\}^\{2\}\\leq\\lambda^\{2\-p\}z^\{p\}, and therefore

𝔼​‖Y^−θ‖2\\displaystyle\\mathbb\{E\}\\\|\\widehat\{Y\}\-\\theta\\\|^\{2\}≤2​𝔼​‖A‖2\+2​𝔼​‖clipλ​\(R\)‖2\\displaystyle\\leq 2\\mathbb\{E\}\\\|A\\\|^\{2\}\+2\\mathbb\{E\}\\\|\\mathrm\{clip\}\_\{\\lambda\}\(R\)\\\|^\{2\}≤32​p​31/pp−1​s2\+2p​\(1\+2​3​4p\)​sp​λ2−p\\displaystyle\\leq\\frac\{32p\\,3^\{1/p\}\}\{p\-1\}s^\{2\}\+2^\{p\}\\bigl\(1\+2\\sqrt\{3\}\\,4^\{p\}\\bigr\)s^\{p\}\\lambda^\{2\-p\}≤\[32​p​31/pp−1\+2p​\(1\+2​3​4p\)\]​sp​λ2−p\.\\displaystyle\\leq\\left\[\\frac\{32p\\,3^\{1/p\}\}\{p\-1\}\+2^\{p\}\\bigl\(1\+2\\sqrt\{3\}\\,4^\{p\}\\bigr\)\\right\]s^\{p\}\\lambda^\{2\-p\}\.\(29\)The last inequality usesλ≥s\\lambda\\geq sandp≤2p\\leq 2\.

LetY^1,…,Y^b\\widehat\{Y\}\_\{1\},\\ldots,\\widehat\{Y\}\_\{b\}be the independent replicates\. The exact bias–variance decomposition gives

𝔼​‖1b​∑j=1bY^j−θ‖2=‖𝔼​Y^−θ‖2\+1b​𝔼​‖Y^−𝔼​Y^‖2\.\\mathbb\{E\}\\left\\\|\\frac\{1\}\{b\}\\sum\_\{j=1\}^\{b\}\\widehat\{Y\}\_\{j\}\-\\theta\\right\\\|^\{2\}=\\\|\\mathbb\{E\}\\widehat\{Y\}\-\\theta\\\|^\{2\}\+\\frac\{1\}\{b\}\\mathbb\{E\}\\\|\\widehat\{Y\}\-\\mathbb\{E\}\\widehat\{Y\}\\\|^\{2\}\.Using \([28](https://arxiv.org/html/2610.12865#A1.E28)\) and \([29](https://arxiv.org/html/2610.12865#A1.E29)\) proves \([27](https://arxiv.org/html/2610.12865#A1.E27)\)\. Withλ=s​b1/p\\lambda=sb^\{1/p\}, both terms are proportional tos2b−2\(p−1\)/ps^\{2\}b^\{\-2\(p\-1\)/p\}, so

𝔼​‖𝒢^b,λ​\(x\)−θ‖2\\displaystyle\\mathbb\{E\}\\\|\\widehat\{\\mathcal\{G\}\}\_\{b,\\lambda\}\(x\)\-\\theta\\\|^\{2\}≤21−2/pCps2b−2\(p−1\)/p\\displaystyle\\leq 2^\{1\-2/p\}C\_\{p\}s^\{2\}b^\{\-2\(p\-1\)/p\}≤Cps2b−2\(p−1\)/p\.\\displaystyle\\leq C\_\{p\}s^\{2\}b^\{\-2\(p\-1\)/p\}\.After the substitution, factoring outs2b−2\(p−1\)/ps^\{2\}b^\{\-2\(p\-1\)/p\}in \([27](https://arxiv.org/html/2610.12865#A1.E27)\) leaves the coefficient21−2/p​Cp2^\{1\-2/p\}C\_\{p\}, whereCpC\_\{p\}is defined above\. The final inequality uses21−2/p≤12^\{1\-2/p\}\\leq 1for1<p≤21<p\\leq 2\. ∎

The fixed\-batch argument uses only a second\-moment trajectory bound\. The following elementary observation is what allows that bound to control the state\-dependent local noise scale\.

###### Lemma A\.10\(Concavity of the squared localLpL^\{p\}noise scale\)\.

For1<p≤21<p\\leq 2, the squared local noise scale

sp​\(x\)2=\[B​‖x−xref‖p\+G\]2/ps\_\{p\}\(x\)^\{2\}=\\bigl\[B\\left\\lVert x\-x\_\{\\mathrm\{ref\}\}\\right\\rVert^\{p\}\+G\\bigr\]^\{2/p\}\(30\)is a nonnegative, nondecreasing, and concave function of the squared distance‖x−xref‖2\\left\\lVert x\-x\_\{\\mathrm\{ref\}\}\\right\\rVert^\{2\}\. Moreover,

sp​\(x\)2≤\(B\+G\)2/p​\(1\+‖x−xref‖2\)\.s\_\{p\}\(x\)^\{2\}\\leq\(B\+G\)^\{2/p\}\\bigl\(1\+\\left\\lVert x\-x\_\{\\mathrm\{ref\}\}\\right\\rVert^\{2\}\\bigr\)\.\(31\)

###### Proof\.

IfB=0B=0orG=0G=0, the squared local noise scale is respectively constant or linear in the squared distance, and the conclusions follow directly\. Suppose now thatB,G\>0B,G\>0\. At a positive squared distancerr, direct differentiation gives

d2d​r2​\[B​rp/2\+G\]2/p=B⁡\(p2−1\)​G​rp/2−2​\(B​rp/2\+G\)2/p−2≤0,\\displaystyle\\frac\{\\mathrm\{d\}^\{2\}\}\{\\mathrm\{d\}r^\{2\}\}\[Br^\{p/2\}\+G\]^\{2/p\}=B\\left\(\\frac\{p\}\{2\}\-1\\right\)Gr^\{p/2\-2\}\(Br^\{p/2\}\+G\)^\{2/p\-2\}\\leq 0,becausep≤2p\\leq 2\. Continuity extends concavity tor=0r=0\. Monotonicity is clear from the first derivative\.

For0≤r≤10\\leq r\\leq 1, monotonicity gives the bound in \([31](https://arxiv.org/html/2610.12865#A1.E31)\)\. Forr\>1r\>1, concavity at1=r−1​r\+\(1−r−1\)​01=r^\{\-1\}r\+\(1\-r^\{\-1\}\)0and nonnegativity at zero give

\[B\+G\]2/p≥r−1​\[B​rp/2\+G\]2/p\.\[B\+G\]^\{2/p\}\\geq r^\{\-1\}\[Br^\{p/2\}\+G\]^\{2/p\}\.This is stronger than \([31](https://arxiv.org/html/2610.12865#A1.E31)\) forr\>1r\>1\. ∎

#### A\.4\.2From drift error to relative Fisher information

We restate the following two results from[Balasubramanian et al\. \(2022\)](https://arxiv.org/html/2610.12865#bib.bib1)in our notation and use them without repeating their proofs\. The first gives KL dissipation for an interpolation that uses a possibly random frozen gradient vector\.

###### Lemma A\.11\(\([Balasubramanian et al\., 2022](https://arxiv.org/html/2610.12865#bib.bib1), Lemma 23\)\)\.

Let\(X,g\)\(X,g\)be jointly distributed random vectors with finite second moments, and let\(Wt\)t≥0\(W\_\{t\}\)\_\{t\\geq 0\}be a Brownian motion independent of\(X,g\)\(X,g\)\. Define

Yt=X−t​g\+2​Wt,νt=Law⁡\(Yt\)\.Y\_\{t\}=X\-tg\+\\sqrt\{2\}W\_\{t\},\\qquad\\nu\_\{t\}=\\operatorname\{Law\}\(Y\_\{t\}\)\.Then, fort\>0t\>0,

dd​tKL\(νt∥π\)\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\mathrm\{KL\}\(\\nu\_\{t\}\\\|\\pi\)≤−34FI\(νt∥π\)\+𝔼∥∇V\(Yt\)−𝔼\[g∣Yt\]∥2\\displaystyle\\leq\-\\frac\{3\}\{4\}\\mathrm\{FI\}\(\\nu\_\{t\}\\\|\\pi\)\+\\mathbb\{E\}\\bigl\\\|\\nabla V\(Y\_\{t\}\)\-\\mathbb\{E\}\[g\\mid Y\_\{t\}\]\\bigr\\\|^\{2\}≤−34FI\(νt∥π\)\+𝔼∥∇V\(Yt\)−g∥2\.\\displaystyle\\leq\-\\frac\{3\}\{4\}\\mathrm\{FI\}\(\\nu\_\{t\}\\\|\\pi\)\+\\mathbb\{E\}\\\|\\nabla V\(Y\_\{t\}\)\-g\\\|^\{2\}\.\(32\)

The second result controls the gradient second moment by relative Fisher information\. It is the point at which dimension enters the globally smooth analysis\.

###### Lemma A\.12\(\([Balasubramanian et al\., 2022](https://arxiv.org/html/2610.12865#bib.bib1), Lemma 24\)\)\.

Under Assumption[2\.2](https://arxiv.org/html/2610.12865#S2.Thmtheorem2), every probability measureν\\nusatisfies

𝔼ν∥∇V∥2≤FI\(ν∥π\)\+2Ld\.\\mathbb\{E\}\_\{\\nu\}\\\|\\nabla V\\\|^\{2\}\\leq\\mathrm\{FI\}\(\\nu\\\|\\pi\)\+2Ld\.\(33\)

The preceding two lemmas yield the following transfer principle\. It isolates the Langevin discretization error from the mean\-squared error of the robust drift estimates\. HereK0:=KL\(μ0∥π\)K\_\{0\}:=\\mathrm\{KL\}\(\\mu\_\{0\}\\\|\\pi\)is the initial KL divergence\.

###### Proposition A\.13\(Mean\-squared drift error implies Fisher control\)\.

Suppose Assumption[2\.2](https://arxiv.org/html/2610.12865#S2.Thmtheorem2)holds,K0<∞K\_\{0\}<\\infty, and𝔼​‖X0−xref‖2<∞\\mathbb\{E\}\\\|X\_\{0\}\-x\_\{\\mathrm\{ref\}\}\\\|^\{2\}<\\infty\. Let

ek:=𝔼​‖𝒢^k−∇V​\(Xk​h\)‖2,e¯N:=1N​∑k=0N−1ek,e\_\{k\}:=\\mathbb\{E\}\\\|\\widehat\{\\mathcal\{G\}\}\_\{k\}\-\\nabla V\(X\_\{kh\}\)\\\|^\{2\},\\qquad\\bar\{e\}\_\{N\}:=\\frac\{1\}\{N\}\\sum\_\{k=0\}^\{N\-1\}e\_\{k\},and supposeek<∞e\_\{k\}<\\inftyfork=0,…,N−1k=0,\\ldots,N\-1\. If0<h≤\(8​L\)−10<h\\leq\(8L\)^\{\-1\}, then

1N​h∫0N​hFI\(μt∥π\)dt\\displaystyle\\frac\{1\}\{Nh\}\\int\_\{0\}^\{Nh\}\\mathrm\{FI\}\(\\mu\_\{t\}\\\|\\pi\)\\,\\mathrm\{d\}t≤2​K0N​h\+8​L2​d​h\+643​L3​d​h2\+\(4\+163​L2​h2\)​e¯N\.\\displaystyle\\leq\\frac\{2K\_\{0\}\}\{Nh\}\+8L^\{2\}dh\+\\frac\{64\}\{3\}L^\{3\}dh^\{2\}\+\\left\(4\+\\frac\{16\}\{3\}L^\{2\}h^\{2\}\\right\)\\bar\{e\}\_\{N\}\.\(34\)

###### Proof\.

Fixkkandu∈\[0,h\]u\\in\[0,h\], and set

Δk,u:=𝔼∥Xk​h\+u−Xk​h∥2,Ik​h\+u:=FI\(μk​h\+u∥π\)\.\\Delta\_\{k,u\}:=\\mathbb\{E\}\\\|X\_\{kh\+u\}\-X\_\{kh\}\\\|^\{2\},\\qquad I\_\{kh\+u\}:=\\mathrm\{FI\}\(\\mu\_\{kh\+u\}\\\|\\pi\)\.Independence and centering of the Brownian increment give

Δk,u=u2​𝔼​‖𝒢^k‖2\+2​d​u\.\\Delta\_\{k,u\}=u^\{2\}\\mathbb\{E\}\\\|\\widehat\{\\mathcal\{G\}\}\_\{k\}\\\|^\{2\}\+2du\.\(35\)Twice applying‖a\+b‖2≤2​‖a‖2\+2​‖b‖2\\\|a\+b\\\|^\{2\}\\leq 2\\\|a\\\|^\{2\}\+2\\\|b\\\|^\{2\}and using globalLL\-smoothness yields

𝔼​‖𝒢^k‖2≤4​𝔼​‖∇V​\(Xk​h\+u\)‖2\+4​L2​Δk,u\+2​ek\.\\mathbb\{E\}\\\|\\widehat\{\\mathcal\{G\}\}\_\{k\}\\\|^\{2\}\\leq 4\\mathbb\{E\}\\\|\\nabla V\(X\_\{kh\+u\}\)\\\|^\{2\}\+4L^\{2\}\\Delta\_\{k,u\}\+2e\_\{k\}\.Sinceu≤h≤\(8​L\)−1u\\leq h\\leq\(8L\)^\{\-1\},\(1−4​L2​u2\)−1≤16/15<4/3\(1\-4L^\{2\}u^\{2\}\)^\{\-1\}\\leq 16/15<4/3\. Solving \([35](https://arxiv.org/html/2610.12865#A1.E35)\) forΔk,u\\Delta\_\{k,u\}therefore gives

Δk,u≤163​u2​𝔼​‖∇V​\(Xk​h\+u\)‖2\+83​u2​ek\+83​d​u\.\\Delta\_\{k,u\}\\leq\\frac\{16\}\{3\}u^\{2\}\\mathbb\{E\}\\\|\\nabla V\(X\_\{kh\+u\}\)\\\|^\{2\}\+\\frac\{8\}\{3\}u^\{2\}e\_\{k\}\+\\frac\{8\}\{3\}du\.\(36\)Apply Lemma[A\.12](https://arxiv.org/html/2610.12865#A1.Thmtheorem12)withν=μk​h\+u\\nu=\\mu\_\{kh\+u\}\. SinceIk​h\+u=FI\(μk​h\+u∥π\)I\_\{kh\+u\}=\\mathrm\{FI\}\(\\mu\_\{kh\+u\}\\\|\\pi\), it gives

𝔼​‖∇V​\(Xk​h\+u\)‖2≤Ik​h\+u\+2​L​d\.\\mathbb\{E\}\\\|\\nabla V\(X\_\{kh\+u\}\)\\\|^\{2\}\\leq I\_\{kh\+u\}\+2Ld\.\(37\)Moreover, globalLL\-smoothness and the definition ofeke\_\{k\}imply

𝔼​‖∇V​\(Xk​h\+u\)−𝒢^k‖2≤2​L2​Δk,u\+2​ek\.\\mathbb\{E\}\\\|\\nabla V\(X\_\{kh\+u\}\)\-\\widehat\{\\mathcal\{G\}\}\_\{k\}\\\|^\{2\}\\leq 2L^\{2\}\\Delta\_\{k,u\}\+2e\_\{k\}\.Substituting \([36](https://arxiv.org/html/2610.12865#A1.E36)\) and \([37](https://arxiv.org/html/2610.12865#A1.E37)\) into this inequality gives

𝔼​‖∇V​\(Xk​h\+u\)−𝒢^k‖2\\displaystyle\\mathbb\{E\}\\\|\\nabla V\(X\_\{kh\+u\}\)\-\\widehat\{\\mathcal\{G\}\}\_\{k\}\\\|^\{2\}≤323​L2​u2​Ik​h\+u\+643​L3​d​u2\+163​L2​d​u\+\(2\+163​L2​u2\)​ek\\displaystyle\\leq\\frac\{32\}\{3\}L^\{2\}u^\{2\}I\_\{kh\+u\}\+\\frac\{64\}\{3\}L^\{3\}du^\{2\}\+\\frac\{16\}\{3\}L^\{2\}du\+\\left\(2\+\\frac\{16\}\{3\}L^\{2\}u^\{2\}\\right\)e\_\{k\}≤16​L2​u2​Ik​h\+u\+32​L3​d​u2\+8​L2​d​u\+\(2\+8​L2​u2\)​ek\.\\displaystyle\\leq 16L^\{2\}u^\{2\}I\_\{kh\+u\}\+32L^\{3\}du^\{2\}\+8L^\{2\}du\+\(2\+8L^\{2\}u^\{2\}\)e\_\{k\}\.\(38\)
We next apply Lemma[A\.11](https://arxiv.org/html/2610.12865#A1.Thmtheorem11)withX=Xk​hX=X\_\{kh\},g=𝒢^kg=\\widehat\{\\mathcal\{G\}\}\_\{k\}, andWu=Wk​h\+u−Wk​hW\_\{u\}=W\_\{kh\+u\}\-W\_\{kh\}\. The fresh Brownian motion above is independent of this pair by the standing sampling convention\. Under this substitution,

Yu=Xk​h−u​𝒢^k\+2​\(Wk​h\+u−Wk​h\)=Xk​h\+u,νu=μk​h\+u\.Y\_\{u\}=X\_\{kh\}\-u\\widehat\{\\mathcal\{G\}\}\_\{k\}\+\\sqrt\{2\}\(W\_\{kh\+u\}\-W\_\{kh\}\)=X\_\{kh\+u\},\\qquad\\nu\_\{u\}=\\mu\_\{kh\+u\}\.Consequently, the second inequality in \([32](https://arxiv.org/html/2610.12865#A1.E32)\) gives, foru∈\(0,h\]u\\in\(0,h\],

dd​uKL\(μk​h\+u∥π\)≤−34Ik​h\+u\+𝔼∥∇V\(Xk​h\+u\)−𝒢^k∥2\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}u\}\\mathrm\{KL\}\(\\mu\_\{kh\+u\}\\\|\\pi\)\\leq\-\\frac\{3\}\{4\}I\_\{kh\+u\}\+\\mathbb\{E\}\\\|\\nabla V\(X\_\{kh\+u\}\)\-\\widehat\{\\mathcal\{G\}\}\_\{k\}\\\|^\{2\}\.\(39\)Thus, we apply the lemma to the joint law rather than conditioning on the realized state, which would replaceμk​h\\mu\_\{kh\}by a point mass\.

Substitute \([38](https://arxiv.org/html/2610.12865#A1.E38)\) into \([39](https://arxiv.org/html/2610.12865#A1.E39)\)\. Since16​L2​u2≤16​L2​h2≤1/416L^\{2\}u^\{2\}\\leq 16L^\{2\}h^\{2\}\\leq 1/4, the resulting Fisher information term is absorbed into the coefficient3/43/4\. Integrating overu∈\[0,h\]u\\in\[0,h\]gives

KL\(μ\(k\+1\)​h∥π\)−KL\(μk​h∥π\)≤−12∫k​h\(k\+1\)​hFI\(μt∥π\)dt\+323L3dh3\+4L2dh2\+\(2h\+83L2h3\)ek\.\\mathrm\{KL\}\(\\mu\_\{\(k\+1\)h\}\\\|\\pi\)\-\\mathrm\{KL\}\(\\mu\_\{kh\}\\\|\\pi\)\\leq\-\\frac\{1\}\{2\}\\int\_\{kh\}^\{\(k\+1\)h\}\\mathrm\{FI\}\(\\mu\_\{t\}\\\|\\pi\)\\,\\mathrm\{d\}t\+\\frac\{32\}\{3\}L^\{3\}dh^\{3\}\+4L^\{2\}dh^\{2\}\+\\left\(2h\+\\frac\{8\}\{3\}L^\{2\}h^\{3\}\\right\)e\_\{k\}\.\(40\)Summing \([40](https://arxiv.org/html/2610.12865#A1.E40)\) overkk, discarding the nonnegative terminal KL divergence, and dividing byN​h/2Nh/2proves \([34](https://arxiv.org/html/2610.12865#A1.E34)\)\. ∎

#### A\.4\.3Trajectory moment control and completion of the finite bound

We first considerB\>0B\>0, using the known initial second\-moment bound𝔼​‖X0−xref‖2≤M2,0\\mathbb\{E\}\\\|X\_\{0\}\-x\_\{\\mathrm\{ref\}\}\\\|^\{2\}\\leq M\_\{2,0\}in Theorem[A\.2](https://arxiv.org/html/2610.12865#A1.Thmtheorem2)\. The caseB=0B=0is treated separately below\. WriteT=N​hT=Nhand let

M¯N:=1N​∑k=0N−1𝔼​‖Xk​h−xref‖2\.\\overline\{M\}\_\{N\}:=\\frac\{1\}\{N\}\\sum\_\{k=0\}^\{N\-1\}\\mathbb\{E\}\\\|X\_\{kh\}\-x\_\{\\mathrm\{ref\}\}\\\|^\{2\}\.The prescribed clipping radius isλk=sp​\(Xk​h\)​b1/p\\lambda\_\{k\}=s\_\{p\}\(X\_\{kh\}\)b^\{1/p\}, wheresp​\(x\)=\[B​‖x−xref‖p\+G\]1/ps\_\{p\}\(x\)=\[B\\\|x\-x\_\{\\mathrm\{ref\}\}\\\|^\{p\}\+G\]^\{1/p\}\. Lemma[A\.9](https://arxiv.org/html/2610.12865#A1.Thmtheorem9), conditioning first on the current state, and two applications of Jensen’s inequality using Lemma[A\.10](https://arxiv.org/html/2610.12865#A1.Thmtheorem10)give

ek\\displaystyle e\_\{k\}≤Cpb−2\(p−1\)/p𝔼\[\(B∥Xk​h−xref∥p\+G\)2/p\]\\displaystyle\\leq C\_\{p\}b^\{\-2\(p\-1\)/p\}\\mathbb\{E\}\\\!\\left\[\(B\\\|X\_\{kh\}\-x\_\{\\mathrm\{ref\}\}\\\|^\{p\}\+G\)^\{2/p\}\\right\]≤Cpb−2\(p−1\)/p\[B\(𝔼∥Xk​h−xref∥2\)p/2\+G\]2/p,\\displaystyle\\leq C\_\{p\}b^\{\-2\(p\-1\)/p\}\[B\(\\mathbb\{E\}\\\|X\_\{kh\}\-x\_\{\\mathrm\{ref\}\}\\\|^\{2\}\)^\{p/2\}\+G\]^\{2/p\},\(41\)e¯N\\displaystyle\\bar\{e\}\_\{N\}≤Cpb−2\(p−1\)/p\[BM¯Np/2\+G\]2/p\.\\displaystyle\\leq C\_\{p\}b^\{\-2\(p\-1\)/p\}\[B\\overline\{M\}\_\{N\}^\{p/2\}\+G\]^\{2/p\}\.\(42\)All quantities in \([41](https://arxiv.org/html/2610.12865#A1.E41)\) are finite\. Indeed, the initial second\-moment bound starts an induction\. IfXk​hX\_\{kh\}has a finite second moment, Lemma[A\.9](https://arxiv.org/html/2610.12865#A1.Thmtheorem9)and \([31](https://arxiv.org/html/2610.12865#A1.E31)\) imply thatek=𝔼​‖𝒢^k−∇V​\(Xk​h\)‖2<∞e\_\{k\}=\\mathbb\{E\}\\\|\\widehat\{\\mathcal\{G\}\}\_\{k\}\-\\nabla V\(X\_\{kh\}\)\\\|^\{2\}<\\infty\. GlobalLL\-smoothness then gives𝔼​‖∇V​\(Xk​h\)‖2<∞\\mathbb\{E\}\\\|\\nabla V\(X\_\{kh\}\)\\\|^\{2\}<\\infty, so𝔼​‖𝒢^k‖2<∞\\mathbb\{E\}\\\|\\widehat\{\\mathcal\{G\}\}\_\{k\}\\\|^\{2\}<\\infty\. The update \([6](https://arxiv.org/html/2610.12865#S3.E6)\) therefore gives a finite second moment forX\(k\+1\)​hX\_\{\(k\+1\)h\}\.

Proposition[A\.13](https://arxiv.org/html/2610.12865#A1.Thmtheorem13)andL​h≤1/8Lh\\leq 1/8imply

∫0TFI\(μt∥π\)dt≤2K0\+43LdT\+4912Te¯N\.\\int\_\{0\}^\{T\}\\mathrm\{FI\}\(\\mu\_\{t\}\\\|\\pi\)\\,\\mathrm\{d\}t\\leq 2K\_\{0\}\+\\frac\{4\}\{3\}LdT\+\\frac\{49\}\{12\}T\\bar\{e\}\_\{N\}\.\(43\)To see the constants, after multiplying \([34](https://arxiv.org/html/2610.12865#A1.E34)\) byTT, the two discretization terms are at mostL​d​TLdTandL​d​T/3LdT/3, while4\+\(16/3\)​L2​h2≤49/124\+\(16/3\)L^\{2\}h^\{2\}\\leq 49/12\.

We next control the energy of the applied robust drifts\. The estimate used to derive \([36](https://arxiv.org/html/2610.12865#A1.E36)\), together with \([35](https://arxiv.org/html/2610.12865#A1.E35)\), gives

\(1−4​L2​u2\)​𝔼​‖𝒢^k‖2≤4​𝔼​‖∇V​\(Xk​h\+u\)‖2\+8​L2​d​u\+2​ek\.\(1\-4L^\{2\}u^\{2\}\)\\mathbb\{E\}\\\|\\widehat\{\\mathcal\{G\}\}\_\{k\}\\\|^\{2\}\\leq 4\\mathbb\{E\}\\\|\\nabla V\(X\_\{kh\+u\}\)\\\|^\{2\}\+8L^\{2\}du\+2e\_\{k\}\.Use\(1−4​L2​u2\)−1≤16/15\(1\-4L^\{2\}u^\{2\}\)^\{\-1\}\\leq 16/15, integrate overu∈\[0,h\]u\\in\[0,h\], and apply Lemma[A\.12](https://arxiv.org/html/2610.12865#A1.Thmtheorem12)to obtain

h𝔼∥𝒢^k∥2≤6415∫k​h\(k\+1\)​hFI\(μt∥π\)dt\+12815Ldh\+6415L2dh2\+3215hek\.h\\mathbb\{E\}\\\|\\widehat\{\\mathcal\{G\}\}\_\{k\}\\\|^\{2\}\\leq\\frac\{64\}\{15\}\\int\_\{kh\}^\{\(k\+1\)h\}\\mathrm\{FI\}\(\\mu\_\{t\}\\\|\\pi\)\\,\\mathrm\{d\}t\+\\frac\{128\}\{15\}Ldh\+\\frac\{64\}\{15\}L^\{2\}dh^\{2\}\+\\frac\{32\}\{15\}he\_\{k\}\.Summing this inequality, substituting \([43](https://arxiv.org/html/2610.12865#A1.E43)\), and usingL​h≤1/8Lh\\leq 1/8gives

h​∑k=0N−1𝔼​‖𝒢^k‖2≤12815​K0\+66445​L​d​T\+1769​T​e¯N\.h\\sum\_\{k=0\}^\{N\-1\}\\mathbb\{E\}\\\|\\widehat\{\\mathcal\{G\}\}\_\{k\}\\\|^\{2\}\\leq\\frac\{128\}\{15\}K\_\{0\}\+\\frac\{664\}\{45\}LdT\+\\frac\{176\}\{9\}T\\bar\{e\}\_\{N\}\.\(44\)
Fortk=k​ht\_\{k\}=kh, the summed recursion is

Xk​h−xref=X0−xref−h​∑j=0k−1𝒢^j\+2​\(Wk​h−W0\)\.X\_\{kh\}\-x\_\{\\mathrm\{ref\}\}=X\_\{0\}\-x\_\{\\mathrm\{ref\}\}\-h\\sum\_\{j=0\}^\{k\-1\}\\widehat\{\\mathcal\{G\}\}\_\{j\}\+\\sqrt\{2\}\(W\_\{kh\}\-W\_\{0\}\)\.The three term square inequality, Cauchy–Schwarz for the drift sum, and𝔼​‖Wk​h−W0‖2=d​tk\\mathbb\{E\}\\\|W\_\{kh\}\-W\_\{0\}\\\|^\{2\}=dt\_\{k\}yield

𝔼​‖Xk​h−xref‖2≤3​M2,0\+3​tk​h​∑j=0N−1𝔼​‖𝒢^j‖2\+6​d​tk\.\\mathbb\{E\}\\\|X\_\{kh\}\-x\_\{\\mathrm\{ref\}\}\\\|^\{2\}\\leq 3M\_\{2,0\}\+3t\_\{k\}h\\sum\_\{j=0\}^\{N\-1\}\\mathbb\{E\}\\\|\\widehat\{\\mathcal\{G\}\}\_\{j\}\\\|^\{2\}\+6dt\_\{k\}\.SinceN−1​∑k=0N−1tk=\(N−1\)​h/2≤T/2N^\{\-1\}\\sum\_\{k=0\}^\{N\-1\}t\_\{k\}=\(N\-1\)h/2\\leq T/2, averaging overkkand using \([44](https://arxiv.org/html/2610.12865#A1.E44)\) proves

M¯N≤3​M2,0\+645​K0​T\+33215​L​d​T2\+3​d​T\+883​T2​e¯N\.\\overline\{M\}\_\{N\}\\leq 3M\_\{2,0\}\+\\frac\{64\}\{5\}K\_\{0\}T\+\\frac\{332\}\{15\}LdT^\{2\}\+3dT\+\\frac\{88\}\{3\}T^\{2\}\\bar\{e\}\_\{N\}\.\(45\)
It remains to close \([42](https://arxiv.org/html/2610.12865#A1.E42)\) and \([45](https://arxiv.org/html/2610.12865#A1.E45)\) without assuming the desired error bound\. Denote the right\-hand side of \([45](https://arxiv.org/html/2610.12865#A1.E45)\) bySNS\_\{N\}\. Then

e¯N≤Cpb−2\(p−1\)/p\[BSNp/2\+G\]2/p\.\\bar\{e\}\_\{N\}\\leq C\_\{p\}b^\{\-2\(p\-1\)/p\}\[BS\_\{N\}^\{p/2\}\+G\]^\{2/p\}\.\(46\)At an arbitrary error levelδ2\\delta^\{2\}, the corresponding value ofSNS\_\{N\}is exactly the deterministic radius squared from Theorem[A\.2](https://arxiv.org/html/2610.12865#A1.Thmtheorem2), which we recall here:

RT,δ2=3​M2,0\+645​K0​T\+33215​L​d​T2\+3​d​T\+883​T2​δ2\.R\_\{T,\\delta\}^\{2\}=3M\_\{2,0\}\+\\frac\{64\}\{5\}K\_\{0\}T\+\\frac\{332\}\{15\}LdT^\{2\}\+3dT\+\\frac\{88\}\{3\}T^\{2\}\\delta^\{2\}\.The prescribed fixed batch size is

b=b⁡\(T,δ\):=max⁡\{1,⌈\(2​Cp​\[B​RT,δp\+G\]2/pδ2\)p/\[2​\(p−1\)\]⌉\}\.b=b\(T,\\delta\):=\\max\\left\\\{1,\\left\\lceil\\left\(\\frac\{2C\_\{p\}\[BR\_\{T,\\delta\}^\{p\}\+G\]^\{2/p\}\}\{\\delta^\{2\}\}\\right\)^\{p/\[2\(p\-1\)\]\}\\right\\rceil\\right\\\}\.This choice ensures

Cpb−2\(p−1\)/p\[B\(RT,δ2\)p/2\+G\]2/p≤δ22\.C\_\{p\}b^\{\-2\(p\-1\)/p\}\[B\(R\_\{T,\\delta\}^\{2\}\)^\{p/2\}\+G\]^\{2/p\}\\leq\\frac\{\\delta^\{2\}\}\{2\}\.\(47\)
Suppose for contradiction thate¯N≥δ2\>0\\bar\{e\}\_\{N\}\\geq\\delta^\{2\}\>0, and setϑ=δ2/e¯N∈\(0,1\]\\vartheta=\\delta^\{2\}/\\bar\{e\}\_\{N\}\\in\(0,1\]\. ComparingSNS\_\{N\}withRT,δ2R\_\{T,\\delta\}^\{2\}givesRT,δ2≥ϑ​SNR\_\{T,\\delta\}^\{2\}\\geq\\vartheta S\_\{N\}\. By the monotonicity and concavity in squared distance established in Lemma[A\.10](https://arxiv.org/html/2610.12865#A1.Thmtheorem10),

\[B​\(RT,δ2\)p/2\+G\]2/p≥\[B​\(ϑ​SN\)p/2\+G\]2/p≥ϑ​\[B​SNp/2\+G\]2/p\.\[B\(R\_\{T,\\delta\}^\{2\}\)^\{p/2\}\+G\]^\{2/p\}\\geq\[B\(\\vartheta S\_\{N\}\)^\{p/2\}\+G\]^\{2/p\}\\geq\\vartheta\[BS\_\{N\}^\{p/2\}\+G\]^\{2/p\}\.Combining this inequality with \([46](https://arxiv.org/html/2610.12865#A1.E46)\) and \([47](https://arxiv.org/html/2610.12865#A1.E47)\) yields

e¯N≤ϑ−1​δ22=e¯N2,\\bar\{e\}\_\{N\}\\leq\\vartheta^\{\-1\}\\frac\{\\delta^\{2\}\}\{2\}=\\frac\{\\bar\{e\}\_\{N\}\}\{2\},a contradiction\. Thereforee¯N<δ2\\bar\{e\}\_\{N\}<\\delta^\{2\}\. The factor two in the batch choice avoids a boundary case in this non\-strict concavity argument\.

Substitutinge¯N<δ2\\bar\{e\}\_\{N\}<\\delta^\{2\}into \([34](https://arxiv.org/html/2610.12865#A1.E34)\) and using convexity of relative Fisher information under mixtures gives

FI\(μ¯N​h∥π\)\\displaystyle\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{Nh\}\\\|\\pi\)≤1N​h∫0N​hFI\(μt∥π\)dt\\displaystyle\\leq\\frac\{1\}\{Nh\}\\int\_\{0\}^\{Nh\}\\mathrm\{FI\}\(\\mu\_\{t\}\\\|\\pi\)\\,\\mathrm\{d\}t≤2​K0N​h\+8​L2​d​h\+643​L3​d​h2\+\(4\+163​L2​h2\)​δ2\\displaystyle\\leq\\frac\{2K\_\{0\}\}\{Nh\}\+8L^\{2\}dh\+\\frac\{64\}\{3\}L^\{3\}dh^\{2\}\+\\left\(4\+\\frac\{16\}\{3\}L^\{2\}h^\{2\}\\right\)\\delta^\{2\}≤2​K0N​h\+323​L2​d​h\+4912​δ2,\\displaystyle\\leq\\frac\{2K\_\{0\}\}\{Nh\}\+\\frac\{32\}\{3\}L^\{2\}dh\+\\frac\{49\}\{12\}\\delta^\{2\},\(48\)where the last line again usesL​h≤1/8Lh\\leq 1/8\. This proves the finite\-time Fisher\-information bound in Theorem[A\.2](https://arxiv.org/html/2610.12865#A1.Thmtheorem2)\.

##### Removing the initial second\-moment condition whenB=0B=0\.

In the state\-independent case, the local\-noise scale is the constantG1/pG^\{1/p\}, and the batch size reduces to

b⁡\(T,δ\)=max⁡\{1,⌈\(2​Cp​G2/pδ2\)p/\[2​\(p−1\)\]⌉\}\.b\(T,\\delta\)=\\max\\left\\\{1,\\left\\lceil\\left\(\\frac\{2C\_\{p\}G^\{2/p\}\}\{\\delta^\{2\}\}\\right\)^\{p/\[2\(p\-1\)\]\}\\right\\rceil\\right\\\}\.Lemma[A\.9](https://arxiv.org/html/2610.12865#A1.Thmtheorem9)gives the uniform boundek≤CpG2/pb−2\(p−1\)/p≤δ2/2e\_\{k\}\\leq C\_\{p\}G^\{2/p\}b^\{\-2\(p\-1\)/p\}\\leq\\delta^\{2\}/2under the stated batch choice\. Thus, whenever the initial second moment is finite, Proposition[A\.13](https://arxiv.org/html/2610.12865#A1.Thmtheorem13)directly proves \([48](https://arxiv.org/html/2610.12865#A1.E48)\)\. No trajectory moment argument is needed in this case\.

We now remove the finite initial second\-moment condition\. Assuming onlyK0<∞K\_\{0\}<\\infty, condition the entire construction onER:=\{∥X0−xref∥≤R\}E\_\{R\}:=\\\{\\\|X\_\{0\}\-x\_\{\\mathrm\{ref\}\}\\\|\\leq R\\\}, for sufficiently largeRRthatpR:=ℙ⁡\(ER\)\>0p\_\{R\}:=\\mathbb\{P\}\(E\_\{R\}\)\>0\. The conditioned process has bounded initial states and uses the same oracle kernels and fresh Brownian increments\. Consequently, its grid\-point second moments are finite, and the same uniform error bound holds independently ofRR\. Iff0=d​μ0/d​πf\_\{0\}=\\mathrm\{d\}\\mu\_\{0\}/\\mathrm\{d\}\\pi, its initial KL divergence is

K0,R=1pR∫\{∥x−xref∥≤R\}logf0dμ0−logpR⟶K0\.K\_\{0,R\}=\\frac\{1\}\{p\_\{R\}\}\\int\_\{\\\{\\\|x\-x\_\{\\mathrm\{ref\}\}\\\|\\leq R\\\}\}\\log f\_\{0\}\\,\\mathrm\{d\}\\mu\_\{0\}\-\\log p\_\{R\}\\longrightarrow K\_\{0\}\.For eacht≤N​ht\\leq Nh, the conditional lawμt,R:=Law⁡\(Xt∣ER\)\\mu\_\{t,R\}:=\\operatorname\{Law\}\(X\_\{t\}\\mid E\_\{R\}\)converges toμt\\mu\_\{t\}in total variation aspR↑1p\_\{R\}\\uparrow 1\. Lower semicontinuity of relative FI and Fatou’s lemma therefore pass the integrated FI bound to the original process\. Convexity then gives \([48](https://arxiv.org/html/2610.12865#A1.E48)\) forμ¯N​h\\bar\{\\mu\}\_\{Nh\}\. This argument includes exact gradients \(B=G=0B=G=0\) and does not require continuity of the oracle kernel in the state\.

ForB\>0B\>0, the state\-dependent error is controlled by \([42](https://arxiv.org/html/2610.12865#A1.E42)\), so the known initial second\-moment bound remains an assumption of this proof\. Finite KL alone does not imply it for general non\-log\-concave targets\.

#### A\.4\.4Accuracy schedule and oracle complexity

We finish by deriving the stated iteration and stochastic\-oracle complexities\. Set

δ2=ε16,h=min⁡\{18​L,ε32​L2​d\},T=h​max⁡\{1,⌈6​K0ε​h⌉\}\.\\delta^\{2\}=\\frac\{\\varepsilon\}\{16\},\\qquad h=\\min\\left\\\{\\frac\{1\}\{8L\},\\frac\{\\varepsilon\}\{32L^\{2\}d\}\\right\\\},\\qquad T=h\\max\\left\\\{1,\\left\\lceil\\frac\{6K\_\{0\}\}\{\\varepsilon h\}\\right\\rceil\\right\\\}\.\(49\)Since the method uses fixed steps of lengthhh,NT=T/hN\_\{T\}=T/hequals the maximum in \([49](https://arxiv.org/html/2610.12865#A1.E49)\)\. The three terms in the last line of \([48](https://arxiv.org/html/2610.12865#A1.E48)\) are bounded, respectively, byε/3\\varepsilon/3,ε/3\\varepsilon/3, and49​ε/192<ε/349\\varepsilon/192<\\varepsilon/3\. HenceFI\(μ¯T∥π\)≤ε\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\varepsilon\. Sincemax⁡\{1,⌈z⌉\}≤1\+z\\max\\\{1,\\lceil z\\rceil\\\}\\leq 1\+zforz≥0z\\geq 0,

NT≤1\+6​K0ε​max⁡\{8​L,32​L2​dε\}\.N\_\{T\}\\leq 1\+\\frac\{6K\_\{0\}\}\{\\varepsilon\}\\max\\left\\\{8L,\\frac\{32L^\{2\}d\}\{\\varepsilon\}\\right\\\}\.\(50\)
AbbreviateR=RT,δR=R\_\{T,\\delta\}\. WhenB=0B=0, we setM2,0=0M\_\{2,0\}=0in this deterministic schedule\. This convention asserts no second\-moment bound, and the batch size depends only onGG\. Therefore

QT=4​b​NT≤4​NT​\[1\+\(2​Cp​\[B​Rp\+G\]2/pδ2\)p/\[2​\(p−1\)\]\]\.Q\_\{T\}=4bN\_\{T\}\\leq 4N\_\{T\}\\left\[1\+\\left\(\\frac\{2C\_\{p\}\[BR^\{p\}\+G\]^\{2/p\}\}\{\\delta^\{2\}\}\\right\)^\{p/\[2\(p\-1\)\]\}\\right\]\.\(51\)This is the claimed finite stochastic oracle bound\.

For completeness, we make every accuracy exponent explicit\. SupposeK0\>0K\_\{0\}\>0and

0<ε≤min⁡\{4​L​d,32​K0​L2​d\}\.0<\\varepsilon\\leq\\min\\\{4Ld,\\sqrt\{32K\_\{0\}L^\{2\}d\}\\\}\.Thenh=ε/\(32​L2​d\)≤K0/εh=\\varepsilon/\(32L^\{2\}d\)\\leq K\_\{0\}/\\varepsilon, and

T≤h\+6​K0ε≤7​K0ε\.T\\leq h\+\\frac\{6K\_\{0\}\}\{\\varepsilon\}\\leq\\frac\{7K\_\{0\}\}\{\\varepsilon\}\.\(52\)Substituting \([52](https://arxiv.org/html/2610.12865#A1.E52)\) andδ2=ε/16\\delta^\{2\}=\\varepsilon/16into the definition ofRRgives

R2≤R^2:=3​M2,0\+1626815​L​d​K02ε2\+\(5383/30\)​K02\+21​d​K0ε\.\\displaystyle R^\{2\}\\leq\\widehat\{R\}^\{2\}:=3M\_\{2,0\}\+\\frac\{16268\}\{15\}\\frac\{LdK\_\{0\}^\{2\}\}\{\\varepsilon^\{2\}\}\+\\frac\{\(5383/30\)K\_\{0\}^\{2\}\+21dK\_\{0\}\}\{\\varepsilon\}\.\(53\)The arithmetic follows by bounding each occurrence ofTTby7​K0/ε7K\_\{0\}/\\varepsiloninRT,δ2R\_\{T,\\delta\}^\{2\}and collecting theε−2\\varepsilon^\{\-2\}andε−1\\varepsilon^\{\-1\}terms\. Equations \([50](https://arxiv.org/html/2610.12865#A1.E50)\), \([51](https://arxiv.org/html/2610.12865#A1.E51)\), and monotonicity of the localLpL^\{p\}noise scale imply

QT=𝒪⁡\[\(1\+K0​L2​dε2\)​\{1\+\[B​R^p\+G\]1/\(p−1\)εp/\[2​\(p−1\)\]\}\],\\displaystyle Q\_\{T\}=\\mathcal\{O\}\\\!\\left\[\\left\(1\+\\frac\{K\_\{0\}L^\{2\}d\}\{\\varepsilon^\{2\}\}\\right\)\\left\\\{1\+\\frac\{\[B\\widehat\{R\}^\{p\}\+G\]^\{1/\(p\-1\)\}\}\{\\varepsilon^\{p/\[2\(p\-1\)\]\}\}\\right\\\}\\right\],where the hidden numerical constant depends only onpp\.

For fixed positiveK0,LK\_\{0\},Land fixed remaining problem parameters,R^≤64​K0​L​d/ε\\widehat\{R\}\\leq 64K\_\{0\}\\sqrt\{Ld\}/\\varepsilonfor all sufficiently smallε\\varepsilon\. Indeed, the leading coefficient16268/1516268/15in \([53](https://arxiv.org/html/2610.12865#A1.E53)\) is smaller than64264^\{2\}, and all other terms have lower order inε−1\\varepsilon^\{\-1\}\. Consequently, whenB\>0B\>0,

QT=𝒪⁡\(B1/\(p−1\)​K01\+p/\(p−1\)​L2\+p/\[2​\(p−1\)\]​d1\+p/\[2​\(p−1\)\]ε2\+3​p/\[2​\(p−1\)\]\)\.Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{B^\{1/\(p\-1\)\}K\_\{0\}^\{1\+p/\(p\-1\)\}L^\{2\+p/\[2\(p\-1\)\]\}d^\{1\+p/\[2\(p\-1\)\]\}\}\{\\varepsilon^\{2\+3p/\[2\(p\-1\)\]\}\}\\right\)\.\(54\)Under the conventionK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)from Section[2](https://arxiv.org/html/2610.12865#S2), this becomes

QT=𝒪⁡\(B1/\(p−1\)​L2\+p/\[2​\(p−1\)\]​d2\+3​p/\[2​\(p−1\)\]ε2\+3​p/\[2​\(p−1\)\]\),NT=𝒪⁡\(L2​d2ε2\)\.Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{B^\{1/\(p\-1\)\}L^\{2\+p/\[2\(p\-1\)\]\}d^\{2\+3p/\[2\(p\-1\)\]\}\}\{\\varepsilon^\{2\+3p/\[2\(p\-1\)\]\}\}\\right\),\\qquad N\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{L^\{2\}d^\{2\}\}\{\\varepsilon^\{2\}\}\\right\)\.
##### State\-independent finite\-ppnoise\.

IfB=0B=0, the radius drops out of the batch size, so it is important to specialize the finite bound directly rather than use \([54](https://arxiv.org/html/2610.12865#A1.E54)\)\. Equations \([50](https://arxiv.org/html/2610.12865#A1.E50)\) and \([51](https://arxiv.org/html/2610.12865#A1.E51)\) give

QT=𝒪⁡\(K0​L2​d​G1/\(p−1\)ε2\+p/\[2​\(p−1\)\]\),Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{K\_\{0\}L^\{2\}d\\,G^\{1/\(p\-1\)\}\}\{\\varepsilon^\{2\+p/\[2\(p\-1\)\]\}\}\\right\),\(55\)up to the lower\-order iteration term\. UnderK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\), replaceK0​dK\_\{0\}din \([55](https://arxiv.org/html/2610.12865#A1.E55)\) byd2d^\{2\}\.

This completes the proof of Theorem[A\.2](https://arxiv.org/html/2610.12865#A1.Thmtheorem2)\.

### A\.5Proof of Corollary[A\.3](https://arxiv.org/html/2610.12865#A1.Thmtheorem3)

The iteration bound follows directly from \([50](https://arxiv.org/html/2610.12865#A1.E50)\) underK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)\.

##### State\-independent second\-moment noise\.

Takingp=2p=2andB=0B=0in \([55](https://arxiv.org/html/2610.12865#A1.E55)\) yields

QT=𝒪⁡\(K0​L2​d​Gε3\)=𝒪⁡\(L2​d2​Gε3\)when​K0=𝒪⁡\(d\)\.Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{K\_\{0\}L^\{2\}dG\}\{\\varepsilon^\{3\}\}\\right\)=\\mathcal\{O\}\\\!\\left\(\\frac\{L^\{2\}d^\{2\}G\}\{\\varepsilon^\{3\}\}\\right\)\\quad\\text\{when \}K\_\{0\}=\\mathcal\{O\}\(d\)\.BecauseG\>0G\>0is fixed, this term absorbs the lower\-order iteration complexity asε↓0\\varepsilon\\downarrow 0\.

##### BG\-0 noise\.

Takingp=2p=2in \([54](https://arxiv.org/html/2610.12865#A1.E54)\) gives

QT=𝒪⁡\(B​K03​L3​d2ε5\)=𝒪⁡\(B​L3​d5ε5\)when​K0=𝒪⁡\(d\)\.Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{BK\_\{0\}^\{3\}L^\{3\}d^\{2\}\}\{\\varepsilon^\{5\}\}\\right\)=\\mathcal\{O\}\\\!\\left\(\\frac\{BL^\{3\}d^\{5\}\}\{\\varepsilon^\{5\}\}\\right\)\\quad\\text\{when \}K\_\{0\}=\\mathcal\{O\}\(d\)\.

##### Exact gradients\.

IfB=G=0B=G=0, the conditional moment bound in Assumption[3\.1](https://arxiv.org/html/2610.12865#S3.Thmtheorem1)is zero\. Thus, conditionally on each current state, every oracle call equals the gradient at that state almost surely\. Sinceb=1b=1, the four calls in the single robust replicate are identical, so the construction generates the standard exact\-gradient LMC iterates\. Replacing these four identical calls by one exact\-gradient evaluation leaves the iterates unchanged\. ThusQT=NTQ\_\{T\}=N\_\{T\}, and \([50](https://arxiv.org/html/2610.12865#A1.E50)\) gives

QT=NT=𝒪⁡\(K0​L2​dε2\)=𝒪⁡\(L2​d2ε2\)when​K0=𝒪⁡\(d\)\.Q\_\{T\}=N\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{K\_\{0\}L^\{2\}d\}\{\\varepsilon^\{2\}\}\\right\)=\\mathcal\{O\}\\\!\\left\(\\frac\{L^\{2\}d^\{2\}\}\{\\varepsilon^\{2\}\}\\right\)\\quad\\text\{when \}K\_\{0\}=\\mathcal\{O\}\(d\)\.
This proves Corollary[A\.3](https://arxiv.org/html/2610.12865#A1.Thmtheorem3)\.

### A\.6Minibatched SG\-LMC under uniform variance

For comparison with thep=2p=2,B=0B=0specialization of R\-LMC, we make the batch\-size choice in the SG\-LMC Fisher information bound explicit\. The next lemma specializes[Balasubramanian et al\. \(2022, Theorem 15\)](https://arxiv.org/html/2610.12865#bib.bib1)to unbiased minibatches\.

###### Lemma A\.14\(SG\-LMC Fisher\-information and oracle bounds\)\.

SupposeV∈C2​\(ℝd\)V\\in C^\{2\}\(\\mathbb\{R\}^\{d\}\)satisfies Assumption[2\.2](https://arxiv.org/html/2610.12865#S2.Thmtheorem2),K0:=KL\(μ0∥π\)<∞K\_\{0\}:=\\mathrm\{KL\}\(\\mu\_\{0\}\\\|\\pi\)<\\infty, and the stochastic oracle satisfies, for everyxx,

𝔼ξ​𝒢​\(x,ξ\)=∇V​\(x\),𝔼ξ​‖𝒢⁡\(x,ξ\)−∇V​\(x\)‖2≤σ2\.\\mathbb\{E\}\_\{\\xi\}\\mathcal\{G\}\(x,\\xi\)=\\nabla V\(x\),\\qquad\\mathbb\{E\}\_\{\\xi\}\\\|\\mathcal\{G\}\(x,\\xi\)\-\\nabla V\(x\)\\\|^\{2\}\\leq\\sigma^\{2\}\.Run \([4](https://arxiv.org/html/2610.12865#S3.E4)\) withHk=hH\_\{k\}=handFk=−1b∑j=1b𝒢\(Xk​h,ξk,j\),F\_\{k\}=\-\\frac\{1\}\{b\}\\sum\_\{j=1\}^\{b\}\\mathcal\{G\}\(X\_\{kh\},\\xi\_\{k,j\}\),where thebboracle calls are conditionally independent at each step, and independent of the next Brownian increment\. For integersN,b≥1N,b\\geq 1and0<h<\(14​L\)−10<h<\(14L\)^\{\-1\}, setT=N​hT=Nh\. ThenQT=N​bQ\_\{T\}=Nband

FI\(μ¯T∥π\)≤FI¯T≤2​K0T\+16L2dh\+8​σ2b\.\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\overline\{\\mathrm\{FI\}\}\_\{T\}\\leq\\frac\{2K\_\{0\}\}\{T\}\+16L^\{2\}dh\+\\frac\{8\\sigma^\{2\}\}\{b\}\.\(56\)Moreover, for everyε\>0\\varepsilon\>0there are choices ofh,b,Th,b,Tsuch thatFI\(μ¯T∥π\)≤ε\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\varepsilon\. For these choices, setNT:=T/hN\_\{T\}:=T/handQT:=b​NTQ\_\{T\}:=bN\_\{T\}\. IfK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)andσ2\>0\\sigma^\{2\}\>0is fixed, then, for sufficiently smallε\\varepsilon,

NT=𝒪⁡\(L2​d2ε2\),QT=𝒪⁡\(L2​d2​σ2ε3\)\.N\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{L^\{2\}d^\{2\}\}\{\\varepsilon^\{2\}\}\\right\),\\qquad Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{L^\{2\}d^\{2\}\\sigma^\{2\}\}\{\\varepsilon^\{3\}\}\\right\)\.

###### Proof\.

Fix a statexxand writeηj=𝒢⁡\(x,ξk,j\)−∇V​\(x\)\\eta\_\{j\}=\\mathcal\{G\}\(x,\\xi\_\{k,j\}\)\-\\nabla V\(x\)\. Independence and𝔼ξ​ηj=0\\mathbb\{E\}\_\{\\xi\}\\eta\_\{j\}=0make the cross terms vanish, so

𝔼ξ​‖1b​∑j=1bηj‖2=1b2​∑j=1b𝔼ξ​‖ηj‖2≤σ2b\.\\mathbb\{E\}\_\{\\xi\}\\left\\\|\\frac\{1\}\{b\}\\sum\_\{j=1\}^\{b\}\\eta\_\{j\}\\right\\\|^\{2\}=\\frac\{1\}\{b^\{2\}\}\\sum\_\{j=1\}^\{b\}\\mathbb\{E\}\_\{\\xi\}\\\|\\eta\_\{j\}\\\|^\{2\}\\leq\\frac\{\\sigma^\{2\}\}\{b\}\.Thus the batch average is unbiased and has variance at mostσ2/b\\sigma^\{2\}/bat every state\. Its expected gradient is∇V\\nabla V, which isLL\-Lipschitz by Assumption[2\.2](https://arxiv.org/html/2610.12865#S2.Thmtheorem2)\. Assumption 14 of[Balasubramanian et al\. \(2022\)](https://arxiv.org/html/2610.12865#bib.bib1)therefore holds with theirV^=V\\widehat\{V\}=V,L^=L\\widehat\{L\}=L, squared bias zero, and varianceσ2/b\\sigma^\{2\}/b\. Their Theorem 15 gives the time\-averaged FI inequality in \([56](https://arxiv.org/html/2610.12865#A1.E56)\); convexity of FI gives the bound forμ¯T\\bar\{\\mu\}\_\{T\}\. Each of theNNsteps usesbboracle calls, soQT=N​bQ\_\{T\}=Nb\.

To establish the complexity claim, we bound each term of \([56](https://arxiv.org/html/2610.12865#A1.E56)\) byε/3\\varepsilon/3\. It suffices to require

N≥6​K0ε​h,h≤ε48​L2​d,8​σ2b≤ε3⟺b≥24​σ2ε\.N\\geq\\frac\{6K\_\{0\}\}\{\\varepsilon h\},\\qquad h\\leq\\frac\{\\varepsilon\}\{48L^\{2\}d\},\\qquad\\frac\{8\\sigma^\{2\}\}\{b\}\\leq\\frac\{\\varepsilon\}\{3\}\\ \\Longleftrightarrow\\ b\\geq\\frac\{24\\sigma^\{2\}\}\{\\varepsilon\}\.The batch condition does not involveLLordd\. Choose the integer batch size and a step size satisfying the lemma’s restriction by setting

h:=min⁡\{128​L,ε48​L2​d\},b:=min⁡\{128​L,ε48​L2​d\},NT:=⌈6​\(K0\+1\)ε​h⌉,T:=NT​h,QT:=NT​b\.h:=\\min\\left\\\{\\frac\{1\}\{28L\},\\frac\{\\varepsilon\}\{48L^\{2\}d\}\\right\\\},\\quad b:=\\min\\left\\\{\\frac\{1\}\{28L\},\\frac\{\\varepsilon\}\{48L^\{2\}d\}\\right\\\},\\quad N\_\{T\}:=\\left\\lceil\\frac\{6\(K\_\{0\}\+1\)\}\{\\varepsilon h\}\\right\\rceil,\\quad T:=N\_\{T\}h,\\quad Q\_\{T\}:=N\_\{T\}b\.\(57\)These choices makeh<\(14​L\)−1h<\(14L\)^\{\-1\}and control the three terms of \([56](https://arxiv.org/html/2610.12865#A1.E56)\) separately:

2​K0NT​h≤ε​K03​\(K0\+1\)≤ε3,16​L2​d​h≤ε3,8​σ2b≤ε3\.\\displaystyle\\frac\{2K\_\{0\}\}\{N\_\{T\}h\}\\leq\\frac\{\\varepsilon K\_\{0\}\}\{3\(K\_\{0\}\+1\)\}\\leq\\frac\{\\varepsilon\}\{3\},\\quad 16L^\{2\}dh\\leq\\frac\{\\varepsilon\}\{3\},\\quad\\frac\{8\\sigma^\{2\}\}\{b\}\\leq\\frac\{\\varepsilon\}\{3\}\.Substituting these bounds into \([56](https://arxiv.org/html/2610.12865#A1.E56)\) proves the FI claim\. The last line also shows why it suffices to takebbproportional toσ2/ε\\sigma^\{2\}/\\varepsilon, without an additional factor ofL​dLd\.

Forε≤\(12/7\)​L​d\\varepsilon\\leq\(12/7\)Ld, the second branch ofhhis active\. The ceiling bounds then give, step by step,

NT≤1\+288​\(K0\+1\)​L2​dε2,b≤1\+24​σ2ε,N\_\{T\}\\leq 1\+\\frac\{288\(K\_\{0\}\+1\)L^\{2\}d\}\{\\varepsilon^\{2\}\},\\qquad b\\leq 1\+\\frac\{24\\sigma^\{2\}\}\{\\varepsilon\},and hence

QT=NT​b≤\(1\+288​\(K0\+1\)​L2​dε2\)​\(1\+24​σ2ε\)\.Q\_\{T\}=N\_\{T\}b\\leq\\left\(1\+\\frac\{288\(K\_\{0\}\+1\)L^\{2\}d\}\{\\varepsilon^\{2\}\}\\right\)\\left\(1\+\\frac\{24\\sigma^\{2\}\}\{\\varepsilon\}\\right\)\.For fixedσ2\>0\\sigma^\{2\}\>0and sufficiently smallε\\varepsilon, the leading term is𝒪⁡\(\(K0\+1\)​L2​d​σ2/ε3\)\\mathcal\{O\}\(\(K\_\{0\}\+1\)L^\{2\}d\\sigma^\{2\}/\\varepsilon^\{3\}\)\. UsingK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)gives the displayed iteration and oracle rates\. ∎

The SG\-LMC calculation in[Sahin et al\. \(2026b, Appendix B\.3\)](https://arxiv.org/html/2610.12865#bib.bib2)usesb=⌈σ2​NT⌉b=\\lceil\\sigma^\{2\}\\sqrt\{N\_\{T\}\}\\rceil\. SubstitutingNT=𝒪⁡\(L2​d2/ε2\)N\_\{T\}=\\mathcal\{O\}\(L^\{2\}d^\{2\}/\\varepsilon^\{2\}\)into that schedule givesb=𝒪⁡\(L​d​σ2/ε\)b=\\mathcal\{O\}\(Ld\\sigma^\{2\}/\\varepsilon\)and the reportedQT=𝒪⁡\(L3​d3​σ2/ε3\)Q\_\{T\}=\\mathcal\{O\}\(L^\{3\}d^\{3\}\\sigma^\{2\}/\\varepsilon^\{3\}\)whenK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)andσ2\>0\\sigma^\{2\}\>0is fixed\. The accuracy\-based choice \([57](https://arxiv.org/html/2610.12865#A1.E57)\) follows from the same FI bound and avoids the extraL​dLdfactor in that schedule\. The SL\-VRLD method of[Sahin et al\. \(2026b\)](https://arxiv.org/html/2610.12865#bib.bib2)achieves the same total oracle order with𝒪⁡\(1\)\{\\mathcal\{O\}\}\(1\)expected oracle calls per update under an additional mean\-square Lipschitz condition on stochastic gradients\. The selected SG\-LMC schedule above uses a batch of sizeb=𝒪⁡\(σ2/ε\)b=\{\\mathcal\{O\}\}\(\\sigma^\{2\}/\\varepsilon\)at every update, so its oracle work per update grows asε\\varepsilondecreases\.

### A\.7Proof of Theorem[A\.4](https://arxiv.org/html/2610.12865#A1.Thmtheorem4)

This subsection proves the exact gradient adaptive\-step result\. We first set up the adaptive intervals and continuous\-time interpolation and explain the roles of the constants used in the finite horizon bound\. Subsection[A\.7\.1](https://arxiv.org/html/2610.12865#A1.SS7.SSS1)establishes endpoint control under symmetric generalized smoothness, and Subsection[A\.7\.2](https://arxiv.org/html/2610.12865#A1.SS7.SSS2)gives a gradient second moment bound\. Subsection[A\.7\.3](https://arxiv.org/html/2610.12865#A1.SS7.SSS3)proves KL dissipation for an adapted approximate drift, while Subsection[A\.7\.5](https://arxiv.org/html/2610.12865#A1.SS7.SSS5)bounds the discretization error caused by holding the gradient fixed over each adaptive interval\. Subsection[A\.7\.6](https://arxiv.org/html/2610.12865#A1.SS7.SSS6)then absorbs the resulting error term and controls the random number of adaptive intervals\. Finally, Subsection[A\.7\.7](https://arxiv.org/html/2610.12865#A1.SS7.SSS7)verifies the accuracy schedule and derives the iteration and oracle complexities\. No finite initial second moment assumption is needed here, we require onlyK0=KL\(μ0∥π\)<∞K\_\{0\}=\\mathrm\{KL\}\(\\mu\_\{0\}\\\|\\pi\)<\\infty\.

Throughout this subsection,K0:=KL\(μ0∥π\)K\_\{0\}:=\\mathrm\{KL\}\(\\mu\_\{0\}\\\|\\pi\)and0<γ≤10<\\gamma\\leq 1\. For the adaptive\-step method in \([7](https://arxiv.org/html/2610.12865#S3.E7)\), letℱk\\mathcal\{F\}\_\{k\}be the history atτk\\tau\_\{k\}, before the next Brownian increment\. For each interval withτk<T\\tau\_\{k\}<T, setℓk:=min⁡\{Hk,T−τk\}\\ell\_\{k\}:=\\min\\\{H\_\{k\},T\-\\tau\_\{k\}\\\}\. Thusℓk=Hk\\ell\_\{k\}=H\_\{k\}on every complete interval, and only the last occupied interval can be shortened\. For0≤s≤ℓk0\\leq s\\leq\\ell\_\{k\}, the interpolation in Definition[A\.1](https://arxiv.org/html/2610.12865#A1.Thmtheorem1), withFk=−∇V​\(Xτk\)F\_\{k\}=\-\\nabla V\(X\_\{\\tau\_\{k\}\}\), is

Xτk\+s=Xτk−s∇V\(Xτk\)\+2\(Wτk\+s−Wτk\)\.X\_\{\\tau\_\{k\}\+s\}=X\_\{\\tau\_\{k\}\}\-s\\nabla V\(X\_\{\\tau\_\{k\}\}\)\+\\sqrt\{2\}\\bigl\(W\_\{\\tau\_\{k\}\+s\}\-W\_\{\\tau\_\{k\}\}\\bigr\)\.\(58\)
We recall each coefficient below when it enters the argument, together with the term it controls\.

#### A\.7\.1Endpoint and gradient moment estimates

The next lemma states the gradient variation bounds of[Chen et al\. \(2023, Proposition 1, items 1–2\)](https://arxiv.org/html/2610.12865#bib.bib30)and establishes the local regularity properties needed below\. The polynomial and exponential cases must be kept separate because the displacement exponent2/\(1−γ\)2/\(1\-\\gamma\)diverges asγ↑1\\gamma\\uparrow 1\. The local regularity argument additionally shows that the Jacobian bound used in Lemma[A\.16](https://arxiv.org/html/2610.12865#A1.Thmtheorem16)holds almost everywhere without assuming thatVVis twice differentiable\.

###### Lemma A\.15\(Gradient variation bounds and local regularity\)\.

The gradient∇V\\nabla Vis locally Lipschitz\. If0<γ<10<\\gamma<1, then

‖∇V​\(y\)−∇V​\(x\)‖\\displaystyle\\\|\\nabla V\(y\)\-\\nabla V\(x\)\\\|≤L0​\(1\+2γ2/\(1−γ\)\)​‖y−x‖\\displaystyle\\leq L\_\{0\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)\\\|y\-x\\\|\(59\)\+L1​2γ2/\(1−γ\)​3γ​‖∇V​\(x\)‖γ​‖y−x‖\\displaystyle\+L\_\{1\}\\,2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}3^\{\\gamma\}\\\|\\nabla V\(x\)\\\|^\{\\gamma\}\\\|y\-x\\\|\+L11/\(1−γ\)​2γ2/\(1−γ\)​3γ​\(1−γ\)γ/\(1−γ\)​‖y−x‖1/\(1−γ\),\\displaystyle\+L\_\{1\}^\{1/\(1\-\\gamma\)\}2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}3^\{\\gamma\}\(1\-\\gamma\)^\{\\gamma/\(1\-\\gamma\)\}\\\|y\-x\\\|^\{1/\(1\-\\gamma\)\},and

‖∇V​\(y\)−∇V​\(x\)‖2\\displaystyle\\\|\\nabla V\(y\)\-\\nabla V\(x\)\\\|^\{2\}≤3​L02​\(1\+2γ2/\(1−γ\)\)2​‖y−x‖2\\displaystyle\\leq 3L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}\\\|y\-x\\\|^\{2\}\(60\)\+32​γ\+1​22​γ2/\(1−γ\)​L12​‖∇V​\(x\)‖2​γ​‖y−x‖2\\displaystyle\+3^\{2\\gamma\+1\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}L\_\{1\}^\{2\}\\\|\\nabla V\(x\)\\\|^\{2\\gamma\}\\\|y\-x\\\|^\{2\}\+32​γ\+1​22​γ2/\(1−γ\)​L12/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​‖y−x‖2/\(1−γ\)\.\\displaystyle\+3^\{2\\gamma\+1\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}L\_\{1\}^\{2/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}\\\|y\-x\\\|^\{2/\(1\-\\gamma\)\}\.Ifγ=1\\gamma=1, then

‖∇V​\(y\)−∇V​\(x\)‖≤‖y−x‖​\(L0\+L1​‖∇V​\(x\)‖\)​eL1​‖y−x‖\.\\\|\\nabla V\(y\)\-\\nabla V\(x\)\\\|\\leq\\\|y\-x\\\|\\bigl\(L\_\{0\}\+L\_\{1\}\\\|\\nabla V\(x\)\\\|\\bigr\)e^\{L\_\{1\}\\\|y\-x\\\|\}\.\(61\)Moreover,‖D⁡\(∇V\)​\(x\)‖op≤L0\+L1​‖∇V​\(x\)‖γ\\\|D\(\\nabla V\)\(x\)\\\|\_\{\\mathrm\{op\}\}\\leq L\_\{0\}\+L\_\{1\}\\\|\\nabla V\(x\)\\\|^\{\\gamma\}almost everywhere\.

###### Proof\.

The endpoint inequalities \([59](https://arxiv.org/html/2610.12865#A1.E59)\) and \([61](https://arxiv.org/html/2610.12865#A1.E61)\) are given by[Chen et al\. \(2023, Proposition 1, items 1–2\)](https://arxiv.org/html/2610.12865#bib.bib30), applied withf=Vf=Vandα=γ\\alpha=\\gamma\. Their statement takesL0,L1\>0L\_\{0\},L\_\{1\}\>0\. If either constant is zero, Assumption[2\.3](https://arxiv.org/html/2610.12865#S2.Thmtheorem3)remains valid withL0\+δL\_\{0\}\+\\deltaandL1\+δL\_\{1\}\+\\delta\. Applying the cited result and lettingδ↓0\\delta\\downarrow 0gives the displayed bounds\. Squaring \([59](https://arxiv.org/html/2610.12865#A1.E59)\) and using\(a\+b\+c\)2≤3​\(a2\+b2\+c2\)\(a\+b\+c\)^\{2\}\\leq 3\(a^\{2\}\+b^\{2\}\+c^\{2\}\)proves \([60](https://arxiv.org/html/2610.12865#A1.E60)\), since2\+2​γ/\(1−γ\)=2/\(1−γ\)2\+2\\gamma/\(1\-\\gamma\)=2/\(1\-\\gamma\)\.

It remains to establish the local regularity claims\. SinceV∈C1​\(ℝd\)V\\in C^\{1\}\(\\mathbb\{R\}^\{d\}\), its gradient is continuous\. Fix a radiusR\>0R\>0\. Compactness gives

sup‖z‖≤R‖∇V​\(z\)‖<∞\.\\sup\_\{\\\|z\\\|\\leq R\}\\\|\\nabla V\(z\)\\\|<\\infty\.If‖x‖,‖y‖≤R\\\|x\\\|,\\\|y\\\|\\leq R, the line segment\[x,y\]\[x,y\]lies in the same ball\. Assumption[2\.3](https://arxiv.org/html/2610.12865#S2.Thmtheorem3)therefore yields

‖∇V​\(y\)−∇V​\(x\)‖≤\[L0\+L1​\(sup‖z‖≤R‖∇V​\(z\)‖\)γ\]​‖y−x‖\.\\\|\\nabla V\(y\)\-\\nabla V\(x\)\\\|\\leq\\left\[L\_\{0\}\+L\_\{1\}\\left\(\\sup\_\{\\\|z\\\|\\leq R\}\\\|\\nabla V\(z\)\\\|\\right\)^\{\\gamma\}\\right\]\\\|y\-x\\\|\.Thus,∇V\\nabla Vis Lipschitz on every compact ball\. Rademacher’s theorem implies differentiability almost everywhere\. At every differentiability point, apply Assumption[2\.3](https://arxiv.org/html/2610.12865#S2.Thmtheorem3)withy=x\+t​uy=x\+tu, divide by\|t\|\|t\|, and lett→0t\\to 0\. Continuity of∇V\\nabla Valong the shrinking segment gives, for every unit vectoruu,

‖D⁡\(∇V\)​\(x\)​u‖≤L0\+L1​‖∇V​\(x\)‖γ\.\\\|D\(\\nabla V\)\(x\)u\\\|\\leq L\_\{0\}\+L\_\{1\}\\\|\\nabla V\(x\)\\\|^\{\\gamma\}\.Taking the supremum over unit vectors proves the almost everywhere Jacobian bound\. ∎

#### A\.7\.2Gradient second moment bound

The endpoint estimates introduce powers of‖∇V‖\\\|\\nabla V\\\|into the discretization error\. The following lemma bounds the second moment of the gradient using relative Fisher information, extending the corresponding globally smooth bounds of[Chewi et al\. \(2025, Lemma 20\)](https://arxiv.org/html/2610.12865#bib.bib58)and[Balasubramanian et al\. \(2022, Lemma 24\)](https://arxiv.org/html/2610.12865#bib.bib1)\.

###### Lemma A\.16\(Gradient second moment bound\)\.

For every probability measureμ\\muwith finite relative Fisher information,

∫ℝd∥∇V\(x\)∥2μ\(dx\)≤2FI\(μ∥π\)\+4dL0\+\(2−γ\)γγ/\(2−γ\)\(2dL1\)2/\(2−γ\)\.\\int\_\{\\mathbb\{R\}^\{d\}\}\\\|\\nabla V\(x\)\\\|^\{2\}\\,\\mu\(\\mathrm\{d\}x\)\\leq 2\\,\\mathrm\{FI\}\(\\mu\\\|\\pi\)\+4dL\_\{0\}\+\(2\-\\gamma\)\\gamma^\{\\gamma/\(2\-\\gamma\)\}\(2dL\_\{1\}\)^\{2/\(2\-\\gamma\)\}\.\(62\)IfL1=0L\_\{1\}=0, the sharper globally smooth bound

∫ℝd∥∇V\(x\)∥2μ\(dx\)≤FI\(μ∥π\)\+2dL0\\int\_\{\\mathbb\{R\}^\{d\}\}\\\|\\nabla V\(x\)\\\|^\{2\}\\,\\mu\(\\mathrm\{d\}x\)\\leq\\mathrm\{FI\}\(\\mu\\\|\\pi\)\+2dL\_\{0\}\(63\)holds, recovering[Balasubramanian et al\. \(2022, Lemma 24\)](https://arxiv.org/html/2610.12865#bib.bib1)\.

###### Proof\.

First suppose thatd​μ=q2​d​π\\mathrm\{d\}\\mu=q^\{2\}\\,\\mathrm\{d\}\\pi, whereq∈Cc∞​\(ℝd\)q\\in C\_\{c\}^\{\\infty\}\(\\mathbb\{R\}^\{d\}\)and∫q2​𝑑π=1\\int q^\{2\}\\,\\mathrm\{d\}\\pi=1\. Weak integration by parts is valid because∇V\\nabla Vis locally Lipschitz and∇π\(x\)=−π\(x\)∇V\(x\)\\nabla\\pi\(x\)=\-\\pi\(x\)\\nabla V\(x\)\. Let

J:=∫q2​‖∇V‖2​𝑑π\.J:=\\int q^\{2\}\\\|\\nabla V\\\|^\{2\}\\,\\mathrm\{d\}\\pi\.Then

J=∫q2​div⁡\(∇V\)​𝑑π\+2​∫q⁡⟨∇q,∇V⟩​𝑑π\.J=\\int q^\{2\}\\operatorname\{div\}\(\\nabla V\)\\,\\mathrm\{d\}\\pi\+2\\int q\\langle\\nabla q,\\nabla V\\rangle\\,\\mathrm\{d\}\\pi\.The Jacobian bound in Lemma[A\.15](https://arxiv.org/html/2610.12865#A1.Thmtheorem15)impliesdiv⁡\(∇V\)≤d​L0\+d​L1​‖∇V‖γ\\operatorname\{div\}\(\\nabla V\)\\leq dL\_\{0\}\+dL\_\{1\}\\\|\\nabla V\\\|^\{\\gamma\}almost everywhere\. Jensen’s inequality and2​a​b≤a2\+b22ab\\leq a^\{2\}\+b^\{2\}therefore give

J≤dL0\+dL1Jγ/2\+12FI\(μ∥π\)\+12J\.J\\leq dL\_\{0\}\+dL\_\{1\}J^\{\\gamma/2\}\+\\frac\{1\}\{2\}\\,\\mathrm\{FI\}\(\\mu\\\|\\pi\)\+\\frac\{1\}\{2\}J\.WhenL1=0L\_\{1\}=0, rearranging this inequality directly gives \([63](https://arxiv.org/html/2610.12865#A1.E63)\)\. ForL1\>0L\_\{1\}\>0, rearranging the same inequality gives

J≤2dL0\+2dL1Jγ/2\+FI\(μ∥π\)\.J\\leq 2dL\_\{0\}\+2dL\_\{1\}J^\{\\gamma/2\}\+\\mathrm\{FI\}\(\\mu\\\|\\pi\)\.Young’s inequality states thatu​v≤ur/r\+vr′/r′uv\\leq u^\{r\}/r\+v^\{r^\{\\prime\}\}/r^\{\\prime\}foru,v≥0u,v\\geq 0andr,r′\>1r,r^\{\\prime\}\>1, satisfying1/r\+1/r′=11/r\+1/r^\{\\prime\}=1\. Apply it withr=2/γr=2/\\gamma,r′=2/\(2−γ\)r^\{\\prime\}=2/\(2\-\\gamma\),u=\(J/γ\)γ/2u=\(J/\\gamma\)^\{\\gamma/2\}, andv=2​d​L1​γγ/2v=2dL\_\{1\}\\gamma^\{\\gamma/2\}to obtain

2​d​L1​Jγ/2≤J2\+2−γ2​γγ/\(2−γ\)​\(2​d​L1\)2/\(2−γ\)\.2dL\_\{1\}J^\{\\gamma/2\}\\leq\\frac\{J\}\{2\}\+\\frac\{2\-\\gamma\}\{2\}\\gamma^\{\\gamma/\(2\-\\gamma\)\}\(2dL\_\{1\}\)^\{2/\(2\-\\gamma\)\}\.Substituting this bound and movingJ/2J/2to the left gives

J≤2FI\(μ∥π\)\+4dL0\+\(2−γ\)γγ/\(2−γ\)\(2dL1\)2/\(2−γ\),J\\leq 2\\,\\mathrm\{FI\}\(\\mu\\\|\\pi\)\+4dL\_\{0\}\+\(2\-\\gamma\)\\gamma^\{\\gamma/\(2\-\\gamma\)\}\(2dL\_\{1\}\)^\{2/\(2\-\\gamma\)\},which is \([62](https://arxiv.org/html/2610.12865#A1.E62)\)\.

For a generalμ\\muof finite Fisher information,q=d​μ/d​πq=\\sqrt\{\\mathrm\{d\}\\mu/\\mathrm\{d\}\\pi\}belongs to the weighted Sobolev space with energy∫∥∇q∥2dπ=FI\(μ∥π\)/4\\int\\\|\\nabla q\\\|^\{2\}\\,\\mathrm\{d\}\\pi=\\mathrm\{FI\}\(\\mu\\\|\\pi\)/4\. Compact cutoffs followed by local mollification approximateqqin this norm because the positive continuous densityπ⁡\(x\)\\pi\(x\)is bounded above and away from zero on compact sets\. Normalize the approximants inL2​\(π\)L^\{2\}\(\\pi\)\. Their squared densities converge in total variation, their energies converge, and the integral of the nonnegative continuous function‖∇V‖2\\\|\\nabla V\\\|^\{2\}is lower semicontinuous\. Passing to the limit proves the result\. ∎

#### A\.7\.3Exact continuation and KL dissipation

The proof first analyzes a process truncated afternnadaptive steps and then letsn→∞n\\to\\infty\. The next lemma ensures that the truncated process can be continued by the exact Langevin diffusion without imposing a global growth condition on∇V\\nabla V\.

###### Lemma A\.17\(Conservative exact continuation\)\.

For every initial point inℝd\\mathbb\{R\}^\{d\}, the diffusion

d​Yt=−∇V​\(Yt\)​d​t\+2​d​Wt\\mathrm\{d\}Y\_\{t\}=\-\\nabla V\(Y\_\{t\}\)\\,\\mathrm\{d\}t\+\\sqrt\{2\}\\,\\mathrm\{d\}W\_\{t\}has a unique strong solution for allt≥0t\\geq 0, andπ\\piis invariant\. It can continue the interpolation of Definition[A\.1](https://arxiv.org/html/2610.12865#A1.Thmtheorem1)after any fixed finite number of adaptive intervals\.

###### Proof\.

Apply[Röckner et al\. \(2016\)](https://arxiv.org/html/2610.12865#bib.bib61)with densityρ=e−V\\rho=e^\{\-V\}and their additional drift vector field identically zero\. SinceV∈C1V\\in C^\{1\},ρ=e−V/2∈H1,2loc\\sqrt\{\\rho\}=e^\{\-V/2\}\\in H^\{1,2\}\_\{\\mathrm\{loc\}\}and‖∇ρ‖/ρ=‖∇V‖\\\|\\nabla\\rho\\\|/\\rho=\\\|\\nabla V\\\|is locally bounded, so their hypotheses \(H1\)–\(H3\) hold\. Moreover,∫ρ​𝑑x=Z<∞\\int\\rho\\,\\mathrm\{d\}x=Z<\\inftygives \(H4\), conservativity, by their Remark 2\.8\. Their Theorems 3\.6 and 4\.1 therefore give a unique strong solution from every initial point, with infinite lifetime, for

dY~t=−12∇V\(Y~t\)dt\+dW~t\.\\mathrm\{d\}\\widetilde\{Y\}\_\{t\}=\-\\tfrac\{1\}\{2\}\\nabla V\(\\widetilde\{Y\}\_\{t\}\)\\,\\mathrm\{d\}t\+\\mathrm\{d\}\\widetilde\{W\}\_\{t\}\.SettingYt=Y~2​tY\_\{t\}=\\widetilde\{Y\}\_\{2t\}gives the normalization in the statement\. The zero additional drift makes their semigroup symmetric, so their identity \(2\.10\) gives invariance ofπ\\pi\. This invariance is preserved by the constant time rescaling\.

For the continuation, Definition[A\.1](https://arxiv.org/html/2610.12865#A1.Thmtheorem1)selects both the finite forceFkF\_\{k\}and the finite positive lengthHkH\_\{k\}using only information available atτk\\tau\_\{k\}\. Hence eachτn\\tau\_\{n\}is a finite stopping time andXτn∈ℝdX\_\{\\tau\_\{n\}\}\\in\\mathbb\{R\}^\{d\}almost surely for finitenn\. By the strong Markov property of Brownian motion,Wτn\+t−WτnW\_\{\\tau\_\{n\}\+t\}\-W\_\{\\tau\_\{n\}\}is a Brownian motion independent of that history\. Starting the exact strong solution atXτnX\_\{\\tau\_\{n\}\}with these increments therefore gives the required continuation\. ∎

The next lemma is the analytical bridge from drift approximation to Fisher information\. Its one interval mechanism is the frozen drift calculation of[Balasubramanian et al\. \(2022, Lemma 23\)](https://arxiv.org/html/2610.12865#bib.bib1)\. The formulation below permits a progressively selected drift and is therefore suitable for adaptive step sizes\.

###### Lemma A\.18\(Adapted\-drift KL\-dissipation inequality\)\.

Let a conservative continuous process satisfy

d​Xt=Ft​d​t\+2​d​Wt,\\mathrm\{d\}X\_\{t\}=F\_\{t\}\\,\\mathrm\{d\}t\+\\sqrt\{2\}\\,\\mathrm\{d\}W\_\{t\},where the forceFtF\_\{t\}is progressive and locally square integrable\. Writeμt:=Law⁡\(Xt\)\\mu\_\{t\}:=\\operatorname\{Law\}\(X\_\{t\}\)and supposeK0<∞K\_\{0\}<\\inftyand

𝒜T:=𝔼​∫0T‖∇V​\(Xt\)\+Ft‖2​𝑑t<∞\.\\mathcal\{A\}\_\{T\}:=\\mathbb\{E\}\\int\_\{0\}^\{T\}\\\|\\nabla V\(X\_\{t\}\)\+F\_\{t\}\\\|^\{2\}\\,\\mathrm\{d\}t<\\infty\.Then

KL\(μT∥π\)\+34∫0TFI\(μt∥π\)dt≤K0\+𝒜T\.\\mathrm\{KL\}\(\\mu\_\{T\}\\\|\\pi\)\+\\frac\{3\}\{4\}\\int\_\{0\}^\{T\}\\mathrm\{FI\}\(\\mu\_\{t\}\\\|\\pi\)\\,\\mathrm\{d\}t\\leq K\_\{0\}\+\\mathcal\{A\}\_\{T\}\.\(64\)

###### Proof\.

LetPPbe the law of\(Xt\)0≤t≤T\(X\_\{t\}\)\_\{0\\leq t\\leq T\}onC⁡\(\[0,T\],ℝd\)C\(\[0,T\],\\mathbb\{R\}^\{d\}\), and letRRbe the law of the exact Langevin diffusion started fromπ\\pi\. Lemma[A\.17](https://arxiv.org/html/2610.12865#A1.Thmtheorem17)provides this conservative reference process\. The symmetry used in its proof makesRRreversible\. Local Lipschitzness of∇V\\nabla Vgives uniqueness of its martingale problem\.

##### Finite relative entropy of the trajectory\.

Setet:=Ft\+∇V​\(Xt\)e\_\{t\}:=F\_\{t\}\+\\nabla V\(X\_\{t\}\)andσn:=inf\{t∈\[0,T\]:∫0t‖es‖2​𝑑s≥n\}∧T\\sigma\_\{n\}:=\\inf\\\{t\\in\[0,T\]:\\int\_\{0\}^\{t\}\\\|e\_\{s\}\\\|^\{2\}\\,\\mathrm\{d\}s\\geq n\\\}\\wedge T, withinf∅=∞\\inf\\varnothing=\\infty\. FollowXXuntilσn\\sigma\_\{n\}, then continue with the exact Langevin diffusion using the same Brownian motion\. Denote the resulting trajectory law byPnP\_\{n\}\. On the original probability space, the exponential

Zn=exp\(−12∫0σn⟨et,dWt⟩−14∫0σn∥et∥2dt\)Z\_\{n\}=\\exp\\left\(\-\\frac\{1\}\{\\sqrt\{2\}\}\\int\_\{0\}^\{\\sigma\_\{n\}\}\\langle e\_\{t\},\\mathrm\{d\}W\_\{t\}\\rangle\-\\frac\{1\}\{4\}\\int\_\{0\}^\{\\sigma\_\{n\}\}\\\|e\_\{t\}\\\|^\{2\}\\,\\mathrm\{d\}t\\right\)is the density of a new probability measure relative to the original one\. Novikov’s criterion applies because the stochastic integral in its exponent has quadratic variation at mostn/2n/2\. Its conditional expectation given the initial information is one, so the initial law remainsμ0\\mu\_\{0\}\. Girsanov’s theorem turns the continued process into exact Langevin dynamics under this new measure\. WritingRμ0R\_\{\\mu\_\{0\}\}for its law, data processing and the expectation of−log⁡Zn\-\\log Z\_\{n\}give

KL\(Pn∥Rμ0\)≤14𝔼∫0σn∥et∥2dt≤14𝒜T\.\\mathrm\{KL\}\(P\_\{n\}\\\|R\_\{\\mu\_\{0\}\}\)\\leq\\frac\{1\}\{4\}\\mathbb\{E\}\\int\_\{0\}^\{\\sigma\_\{n\}\}\\\|e\_\{t\}\\\|^\{2\}\\,\\mathrm\{d\}t\\leq\\frac\{1\}\{4\}\\mathcal\{A\}\_\{T\}\.SincePnP\_\{n\}has initial lawμ0\\mu\_\{0\}andd​Rμ0/d​R=\(d​μ0/d​π\)​\(X0\)\\mathrm\{d\}R\_\{\\mu\_\{0\}\}/\\mathrm\{d\}R=\(\\mathrm\{d\}\\mu\_\{0\}/\\mathrm\{d\}\\pi\)\(X\_\{0\}\), the relative entropy chain rule addsK0K\_\{0\}when the reference is changed toRR\. The continued and original paths agree on\{∫0T∥et∥2dt<n\}\\\{\\int\_\{0\}^\{T\}\\\|e\_\{t\}\\\|^\{2\}\\,\\mathrm\{d\}t<n\\\}, whose probability tends to one\. ThusPn→PP\_\{n\}\\to Pin total variation, and lower semicontinuity gives

KL\(P∥R\)≤K0\+14𝒜T<∞\.\\mathrm\{KL\}\(P\\\|R\)\\leq K\_\{0\}\+\\frac\{1\}\{4\}\\mathcal\{A\}\_\{T\}<\\infty\.\(65\)

##### A Markov law with the same marginals\.

MinimizeKL\(Q∥R\)\\mathrm\{KL\}\(Q\\\|R\)over trajectory laws satisfyingQt=μtQ\_\{t\}=\\mu\_\{t\}for everyt∈\[0,T\]t\\in\[0,T\]\. Equation \([65](https://arxiv.org/html/2610.12865#A1.E65)\) supplies a feasible law with finite relative entropy\. By[Baradat and Léonard \(2020, Proposition 4\.3 and Theorem 4\.5\)](https://arxiv.org/html/2610.12865#bib.bib62), the minimizer exists and is Markov\. Hence

Qt=μt,KL\(Q∥R\)≤K0\+14𝒜T\.Q\_\{t\}=\\mu\_\{t\},\\qquad\\mathrm\{KL\}\(Q\\\|R\)\\leq K\_\{0\}\+\\frac\{1\}\{4\}\\mathcal\{A\}\_\{T\}\.\(66\)This auxiliary law preserves all quantities in the lemma’s conclusion\. The original process need not be Markov\.

##### Time reversal and Fisher information\.

Write the forward drift ofQQas−∇V​\(x\)\+ut​\(x\)\-\\nabla V\(x\)\+u\_\{t\}\(x\)and the drift of its time reversal at timeT−tT\-tas−∇V​\(x\)\+vt​\(x\)\-\\nabla V\(x\)\+v\_\{t\}\(x\)\. The finite relative entropy theorem of[Léonard \(2012, Theorem 2\.4\)](https://arxiv.org/html/2610.12865#bib.bib63), applied in both directions with covariance2​Id2I\_\{d\}, gives

KL\(Q∥R\)\\displaystyle\\mathrm\{KL\}\(Q\\\|R\)=K0\+14​∫0T∫‖ut‖2​d​μt​𝑑t\\displaystyle=K\_\{0\}\+\\frac\{1\}\{4\}\\int\_\{0\}^\{T\}\\int\\\|u\_\{t\}\\\|^\{2\}\\,\\mathrm\{d\}\\mu\_\{t\}\\,\\mathrm\{d\}t\(67\)=KL\(μT∥π\)\+14∫0T∫∥vt∥2dμtdt\.\\displaystyle=\\mathrm\{KL\}\(\\mu\_\{T\}\\\|\\pi\)\+\\frac\{1\}\{4\}\\int\_\{0\}^\{T\}\\int\\\|v\_\{t\}\\\|^\{2\}\\,\\mathrm\{d\}\\mu\_\{t\}\\,\\mathrm\{d\}t\.The equality condition in that theorem follows from uniqueness of the reference martingale problem\. Reversing both path laws preserves relative entropy, and reversibility makes the reversed reference equal toRR\.

We apply[Cattiaux et al\. \(2023, Theorem 1\.16\)](https://arxiv.org/html/2610.12865#bib.bib64)toQQandRR\. Its reference hypotheses hold because the forward and reversed reference drifts are−∇V\-\\nabla V, which is locally Lipschitz, the covariance is2​Id2I\_\{d\}, and every reference marginal has densityπ\\pi\. The theorem gives, ford​t​μt​\(d​x\)\\mathrm\{d\}t\\,\\mu\_\{t\}\(\\mathrm\{d\}x\)almost every\(t,x\)\(t,x\),

ut\+vt=2∇logd​μtd​π,∫0TFI\(μt∥π\)dt<∞,u\_\{t\}\+v\_\{t\}=2\\nabla\\log\\frac\{\\mathrm\{d\}\\mu\_\{t\}\}\{\\mathrm\{d\}\\pi\},\\qquad\\int\_\{0\}^\{T\}\\mathrm\{FI\}\(\\mu\_\{t\}\\\|\\pi\)\\,\\mathrm\{d\}t<\\infty,\(68\)where the gradient is understood weakly\. Subtracting the two identities in \([67](https://arxiv.org/html/2610.12865#A1.E67)\) and substituting \([68](https://arxiv.org/html/2610.12865#A1.E68)\) yields

KL\(μT∥π\)−K0=\\displaystyle\\mathrm\{KL\}\(\\mu\_\{T\}\\\|\\pi\)\-K\_\{0\}=\{\}−∫0TFI\(μt∥π\)dt\\displaystyle\-\\int\_\{0\}^\{T\}\\mathrm\{FI\}\(\\mu\_\{t\}\\\|\\pi\)\\,\\mathrm\{d\}t\+∫0T∫⟨ut,∇logd​μtd​π⟩dμtdt\.\\displaystyle\+\\int\_\{0\}^\{T\}\\int\\left\\langle u\_\{t\},\\nabla\\log\\frac\{\\mathrm\{d\}\\mu\_\{t\}\}\{\\mathrm\{d\}\\pi\}\\right\\rangle\\,\\mathrm\{d\}\\mu\_\{t\}\\,\\mathrm\{d\}t\.Young’s inequality⟨u,s⟩≤‖u‖2\+‖s‖2/4\\langle u,s\\rangle\\leq\\\|u\\\|^\{2\}\+\\\|s\\\|^\{2\}/4bounds the last term\. Moreover, \([66](https://arxiv.org/html/2610.12865#A1.E66)\) and the first identity in \([67](https://arxiv.org/html/2610.12865#A1.E67)\) give∫0T∫‖ut‖2​d​μt​𝑑t≤𝒜T\\int\_\{0\}^\{T\}\\int\\\|u\_\{t\}\\\|^\{2\}\\,\\mathrm\{d\}\\mu\_\{t\}\\,\\mathrm\{d\}t\\leq\\mathcal\{A\}\_\{T\}\. Together these prove \([64](https://arxiv.org/html/2610.12865#A1.E64)\)\. ∎

#### A\.7\.4Gaussian estimates and a reusable approximate\-drift lemma

Both the exact gradient and robust adaptive\-step analyses require moments of a Gaussian displacement\. We collect the three estimates used below\.

###### Lemma A\.19\(Gaussian moment and exponential bounds\)\.

IfZ∼𝒩⁡\(0,Id\)Z\\sim\\mathcal\{N\}\(0,I\_\{d\}\), then, fora≥2a\\geq 2andλ≥0\\lambda\\geq 0,

𝔼​‖Z‖a≤\(d\+a\)a/2,𝔼​eλ​‖Z‖≤eλ​d\+λ2/2,𝔼⁡\[‖Z‖2​eλ​‖Z‖\]≤\(d\+1\)​eλ​d\+λ2\.\\mathbb\{E\}\\\|Z\\\|^\{a\}\\leq\(d\+a\)^\{a/2\},\\qquad\\mathbb\{E\}e^\{\\lambda\\\|Z\\\|\}\\leq e^\{\\lambda\\sqrt\{d\}\+\\lambda^\{2\}/2\},\\qquad\\mathbb\{E\}\\\!\\left\[\\\|Z\\\|^\{2\}e^\{\\lambda\\\|Z\\\|\}\\right\]\\leq\(d\+1\)e^\{\\lambda\\sqrt\{d\}\+\\lambda^\{2\}\}\.\(69\)

###### Proof\.

LetR:=‖Z‖R:=\\\|Z\\\|, whose density on\[0,∞\)\[0,\\infty\)is proportional tord−1e−r2/2r^\{d\-1\}e^\{\-r^\{2\}/2\}\. Radial integration by parts gives

𝔼​Ra=\(d\+a−2\)​𝔼​Ra−2\.\\mathbb\{E\}R^\{a\}=\(d\+a\-2\)\\mathbb\{E\}R^\{a\-2\}\.Fora\>2a\>2, Hölder’s inequality with conjugate exponentsa/\(a−2\)a/\(a\-2\)anda/2a/2gives

𝔼​Ra−2≤\(𝔼​Ra\)\(a−2\)/a\.\\mathbb\{E\}R^\{a\-2\}\\leq\(\\mathbb\{E\}R^\{a\}\)^\{\(a\-2\)/a\}\.Combining the two displays yields𝔼​Ra≤\(d\+a−2\)a/2≤\(d\+a\)a/2\\mathbb\{E\}R^\{a\}\\leq\(d\+a\-2\)^\{a/2\}\\leq\(d\+a\)^\{a/2\}\. Fora=2a=2, the same claimed bound follows from𝔼​R2=d\\mathbb\{E\}R^\{2\}=d\. Ford\>1d\>1, letM⁡\(λ\):=𝔼​eλ​‖Z‖M\(\\lambda\):=\\mathbb\{E\}e^\{\\lambda\\\|Z\\\|\}\. A second radial integration by parts gives

M′​\(λ\)=λ​M​\(λ\)\+\(d−1\)​𝔼​\[‖Z‖−1​eλ​‖Z‖\]\.M^\{\\prime\}\(\\lambda\)=\\lambda M\(\\lambda\)\+\(d\-1\)\\mathbb\{E\}\[\\\|Z\\\|^\{\-1\}e^\{\\lambda\\\|Z\\\|\}\]\.The covariance of an increasing and a decreasing function of the same real random variable is nonpositive, and therefore

M′​\(λ\)M⁡\(λ\)≤λ\+\(d−1\)​𝔼​‖Z‖−1=λ\+𝔼​‖Z‖≤λ\+d\.\\frac\{M^\{\\prime\}\(\\lambda\)\}\{M\(\\lambda\)\}\\leq\\lambda\+\(d\-1\)\\mathbb\{E\}\\\|Z\\\|^\{\-1\}=\\lambda\+\\mathbb\{E\}\\\|Z\\\|\\leq\\lambda\+\\sqrt\{d\}\.Ford=1d=1, the integration by parts boundary term is2/π\\sqrt\{2/\\pi\}, which gives the same claimed bound withd=1\\sqrt\{d\}=1\. Integration from zero toλ\\lambdaproves the exponential\-moment inequality\. Finally, Cauchy–Schwarz,𝔼​‖Z‖4=d⁡\(d\+2\)≤\(d\+1\)2\\mathbb\{E\}\\\|Z\\\|^\{4\}=d\(d\+2\)\\leq\(d\+1\)^\{2\}, and the exponential bound at2​λ2\\lambdaprove the last inequality\. ∎

The next lemma is not required to complete the exact gradient analysis\. We state it here because it bounds the discretization error under generalized smoothness for any approximate drift\. This result is used directly in the proof of the robust adaptive method\. Its coefficients distinguish the constant terms from those involving the population gradient or the approximate drift\. The frozen gradient bound under global smoothness in the proof of[Balasubramanian et al\. \(2022, Theorem 4\)](https://arxiv.org/html/2610.12865#bib.bib1)provides the basis for this result\. The lemma extends that argument to generalized smoothness and a general approximate drift\.

Recallω:=L1/\(1\+L0\)\\omega:=L\_\{1\}/\(1\+L\_\{0\}\), which is the coefficient of the force term in the adaptive step size\. In the bound below,A~γ\\widetilde\{A\}\_\{\\gamma\}is the constant term,U~γ\\widetilde\{U\}\_\{\\gamma\}multiplies the squared population gradient, andV~γ\\widetilde\{V\}\_\{\\gamma\}multiplies the squared approximate drift\. Their formulas depend on the smoothness regime\. IfL1=0L\_\{1\}=0, they are

A~γ​\(h\)=L02​d​h,U~γ​\(h\)=0,V~γ​\(h\)=13​L02​h2\.\\widetilde\{A\}\_\{\\gamma\}\(h\)=L\_\{0\}^\{2\}dh,\\qquad\\widetilde\{U\}\_\{\\gamma\}\(h\)=0,\\qquad\\widetilde\{V\}\_\{\\gamma\}\(h\)=\\tfrac\{1\}\{3\}L\_\{0\}^\{2\}h^\{2\}\.IfL1\>0L\_\{1\}\>0and0<γ<10<\\gamma<1, the coefficients are

A~γ​\(h\)\\displaystyle\\widetilde\{A\}\_\{\\gamma\}\(h\)=3​\[L02​\(1\+2γ2/\(1−γ\)\)2\+32​γ​22​γ2/\(1−γ\)​L12\]​d​h\\displaystyle=3\\left\[L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}\+3^\{2\\gamma\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}L\_\{1\}^\{2\}\\right\]dh\+32​γ\+1​2\(2​γ2\+γ\+2\)/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​L12/\(1−γ\)​\[\(d\+21−γ\)​h\]1/\(1−γ\),\\displaystyle\\quad\+3^\{2\\gamma\+1\}2^\{\(2\\gamma^\{2\}\+\\gamma\+2\)/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}L\_\{1\}^\{2/\(1\-\\gamma\)\}\\left\[\\left\(d\+\\frac\{2\}\{1\-\\gamma\}\\right\)h\\right\]^\{1/\(1\-\\gamma\)\},U~γ​\(h\)\\displaystyle\\widetilde\{U\}\_\{\\gamma\}\(h\)=32​γ​22​γ2/\(1−γ\)​\[γ​\(1\+L0\)2​h2\+3​L12​d​h\],\\displaystyle=3^\{2\\gamma\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}\\left\[\\gamma\(1\+L\_\{0\}\)^\{2\}h^\{2\}\+3L\_\{1\}^\{2\}dh\\right\],V~γ​\(h\)\\displaystyle\\widetilde\{V\}\_\{\\gamma\}\(h\)=\[L02​\(1\+2γ2/\(1−γ\)\)2\+\(1−γ\)​32​γ​22​γ2/\(1−γ\)​\(1\+L0\)2\]​h2\\displaystyle=\\left\[L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}\+\(1\-\\gamma\)3^\{2\\gamma\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}\(1\+L\_\{0\}\)^\{2\}\\right\]h^\{2\}\+32​γ\+1​2\(2​γ2\+γ\+1\)/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​\[\(1\+L0\)​h\]2/\(1−γ\)\.\\displaystyle\\quad\+3^\{2\\gamma\+1\}2^\{\(2\\gamma^\{2\}\+\\gamma\+1\)/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}\[\(1\+L\_\{0\}\)h\]^\{2/\(1\-\\gamma\)\}\.ForL1\>0L\_\{1\}\>0andγ=1\\gamma=1, the Gaussian exponential bound introduces

E1ex​\(h\):=exp⁡\{2​\(1\+L0\)​h\+2​2​L1​d​h\+8​L12​h\},E\_\{1\}^\{\\mathrm\{ex\}\}\(h\):=\\exp\\\{2\(1\+L\_\{0\}\)h\+2\\sqrt\{2\}L\_\{1\}\\sqrt\{dh\}\+8L\_\{1\}^\{2\}h\\\},and the corresponding coefficients are

A~1​\(h\)\\displaystyle\\widetilde\{A\}\_\{1\}\(h\)=4​E1ex​\(h\)​L02​\(d\+1\)​h,\\displaystyle=4E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)L\_\{0\}^\{2\}\(d\+1\)h,U~1​\(h\)\\displaystyle\\widetilde\{U\}\_\{1\}\(h\)=E1ex​\(h\)​\{43​\(1\+L0\)2​h2\+4​L12​\(d\+1\)​h\},\\displaystyle=E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)\\\{\\tfrac\{4\}\{3\}\(1\+L\_\{0\}\)^\{2\}h^\{2\}\+4L\_\{1\}^\{2\}\(d\+1\)h\\\},V~1​\(h\)\\displaystyle\\widetilde\{V\}\_\{1\}\(h\)=43​E1ex​\(h\)​L02​h2\.\\displaystyle=\\tfrac\{4\}\{3\}E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)L\_\{0\}^\{2\}h^\{2\}\.
###### Lemma A\.20\(Adaptive\-step discretization bound for approximate drifts\)\.

Letℱ\\mathcal\{F\}be a sigma\-field and letX,FX,Fbeℱ\\mathcal\{F\}\-measurableℝd\\mathbb\{R\}^\{d\}\-valued random vectors\. LetH,ℓH,\\ellbe nonnegativeℱ\\mathcal\{F\}\-measurable random variables satisfying

0≤ℓ≤H≤h1\+ω​‖F‖γ\.0\\leq\\ell\\leq H\\leq\\frac\{h\}\{1\+\\omega\\\|F\\\|^\{\\gamma\}\}\.Let\(Ws\)s≥0\(W\_\{s\}\)\_\{s\\geq 0\}be a standarddd\-dimensional Brownian motion independent ofℱ\\mathcal\{F\}, and define

𝒟s:=s​F\+2​Ws\.\\mathcal\{D\}\_\{s\}:=sF\+\\sqrt\{2\}W\_\{s\}\.Use the coefficientsA~γ,U~γ,V~γ\\widetilde\{A\}\_\{\\gamma\},\\widetilde\{U\}\_\{\\gamma\},\\widetilde\{V\}\_\{\\gamma\}defined above for the applicable regime\. Then, almost surely,

𝔼⁡\[∫0ℓ‖∇V​\(X\+𝒟s\)−∇V​\(X\)‖2​𝑑s\|ℱ\]≤ℓ⁡\{A~γ​\(h\)\+U~γ​\(h\)​‖∇V​\(X\)‖2\+V~γ​\(h\)​‖F‖2\}\.\\mathbb\{E\}\\\!\\left\[\\int\_\{0\}^\{\\ell\}\\\|\\nabla V\(X\+\\mathcal\{D\}\_\{s\}\)\-\\nabla V\(X\)\\\|^\{2\}\\,\\mathrm\{d\}s\\;\\middle\|\\;\\mathcal\{F\}\\right\]\\leq\\ell\\left\\\{\\widetilde\{A\}\_\{\\gamma\}\(h\)\+\\widetilde\{U\}\_\{\\gamma\}\(h\)\\\|\\nabla V\(X\)\\\|^\{2\}\+\\widetilde\{V\}\_\{\\gamma\}\(h\)\\\|F\\\|^\{2\}\\right\\\}\.Each coefficient is nondecreasing inhhand, for0<h≤10<h\\leq 1, is at mosthhtimes its value at one\.

###### Proof\.

Condition onℱ\\mathcal\{F\}and writeg:=‖∇V​\(X\)‖g:=\\\|\\nabla V\(X\)\\\|andw:=‖F‖w:=\\\|F\\\|\. All expectations in this proof are conditional onℱ\\mathcal\{F\}\. Direct Gaussian calculation gives

𝔼⁡\[‖𝒟s‖2∣ℱ\]=s2​w2\+2​d​s\.\\mathbb\{E\}\[\\\|\\mathcal\{D\}\_\{s\}\\\|^\{2\}\\mid\\mathcal\{F\}\]=s^\{2\}w^\{2\}\+2ds\.\(70\)
Suppose first thatL1\>0L\_\{1\}\>0and0<γ<10<\\gamma<1\. Substituting \([70](https://arxiv.org/html/2610.12865#A1.E70)\) into the first two terms of \([60](https://arxiv.org/html/2610.12865#A1.E60)\), integrating, and using the step\-size bound gives

3​L02​\(1\+2γ2/\(1−γ\)\)2​∫0ℓ𝔼​‖𝒟s‖2​𝑑s≤ℓ​L02​\(1\+2γ2/\(1−γ\)\)2​\(h2​w2\+3​𝑑h\),\\displaystyle 3L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}\\int\_\{0\}^\{\\ell\}\\mathbb\{E\}\\\|\\mathcal\{D\}\_\{s\}\\\|^\{2\}\\,\\mathrm\{d\}s\\leq\\ell L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}\(h^\{2\}w^\{2\}\+3dh\),32​γ\+1​22​γ2/\(1−γ\)​L12​g2​γ​∫0ℓ𝔼​‖𝒟s‖2​𝑑s≤ℓ​22​γ2/\(1−γ\)​32​γ​\(1\+L0\)2​h2​\{γ​g2\+\(1−γ\)​w2\}\\displaystyle 3^\{2\\gamma\+1\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}L\_\{1\}^\{2\}g^\{2\\gamma\}\\int\_\{0\}^\{\\ell\}\\mathbb\{E\}\\\|\\mathcal\{D\}\_\{s\}\\\|^\{2\}\\,\\mathrm\{d\}s\\leq\\ell\\,2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}3^\{2\\gamma\}\(1\+L\_\{0\}\)^\{2\}h^\{2\}\\\{\\gamma g^\{2\}\+\(1\-\\gamma\)w^\{2\}\\\}\+32​γ\+1​22​γ2/\(1−γ\)​ℓ​L12​d​h​\(1\+g2\)\.\\displaystyle\\qquad\\qquad\\qquad\\qquad\\quad\\qquad\\qquad\\qquad\\qquad\+3^\{2\\gamma\+1\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}\\ell L\_\{1\}^\{2\}dh\(1\+g^\{2\}\)\.For the first term on the right of the second inequality, we used

ℓ2​g2​γ​w2≤h2​ω−2​g2​γ​w2−2​γ≤h2​ω−2​\{γ​g2\+\(1−γ\)​w2\},\\ell^\{2\}g^\{2\\gamma\}w^\{2\}\\leq h^\{2\}\\omega^\{\-2\}g^\{2\\gamma\}w^\{2\-2\\gamma\}\\leq h^\{2\}\\omega^\{\-2\}\\\{\\gamma g^\{2\}\+\(1\-\\gamma\)w^\{2\}\\\},where the last inequality is weighted Young’s inequality\. For the Brownian term, we usedg2​γ≤1\+g2g^\{2\\gamma\}\\leq 1\+g^\{2\}\.

For everya≥2a\\geq 2, Lemma[A\.19](https://arxiv.org/html/2610.12865#A1.Thmtheorem19)gives

𝔼​‖𝒟s‖a≤2a−1​sa​wa\+23​a/2−1​sa/2​\(d\+a\)a/2\.\\mathbb\{E\}\\\|\\mathcal\{D\}\_\{s\}\\\|^\{a\}\\leq 2^\{a\-1\}s^\{a\}w^\{a\}\+2^\{3a/2\-1\}s^\{a/2\}\(d\+a\)^\{a/2\}\.Seta=2/\(1−γ\)a=2/\(1\-\\gamma\), integrate over\[0,ℓ\]\[0,\\ell\], and bound the integration denominators below by one\. The adaptive step bound gives

ℓ2/\(1−γ\)​w2/\(1−γ\)≤\(h/ω\)2/\(1−γ\)​w2,ℓ1/\(1−γ\)≤h1/\(1−γ\)\.\\ell^\{2/\(1\-\\gamma\)\}w^\{2/\(1\-\\gamma\)\}\\leq\(h/\\omega\)^\{2/\(1\-\\gamma\)\}w^\{2\},\\qquad\\ell^\{1/\(1\-\\gamma\)\}\\leq h^\{1/\(1\-\\gamma\)\}\.These are exactly the final terms in the coefficientsV~γ\\widetilde\{V\}\_\{\\gamma\}andA~γ\\widetilde\{A\}\_\{\\gamma\}defined above, proving the claim when0<γ<10<\\gamma<1\.

Now suppose thatL1\>0L\_\{1\}\>0andγ=1\\gamma=1\. Sinces​w≤h/ω=\(1\+L0\)​h/L1sw\\leq h/\\omega=\(1\+L\_\{0\}\)h/L\_\{1\}, Lemma[A\.19](https://arxiv.org/html/2610.12865#A1.Thmtheorem19)implies

𝔼⁡\[‖𝒟s‖2​e2​L1​‖𝒟s‖\|ℱ\]≤E1ex​\(h\)​\{2​s2​w2\+4​s​\(d\+1\)\}\.\\mathbb\{E\}\\\!\\left\[\\\|\\mathcal\{D\}\_\{s\}\\\|^\{2\}e^\{2L\_\{1\}\\\|\\mathcal\{D\}\_\{s\}\\\|\}\\;\\middle\|\\;\\mathcal\{F\}\\right\]\\leq E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)\\\{2s^\{2\}w^\{2\}\+4s\(d\+1\)\\\}\.Square \([61](https://arxiv.org/html/2610.12865#A1.E61)\), use\(L0\+L1​g\)2≤2​\(L02\+L12​g2\)\(L\_\{0\}\+L\_\{1\}g\)^\{2\}\\leq 2\(L\_\{0\}^\{2\}\+L\_\{1\}^\{2\}g^\{2\}\), and integrate to obtain

𝔼⁡\[∫0ℓ‖∇V​\(X\+𝒟s\)−∇V​\(X\)‖2​𝑑s\|ℱ\]≤ℓ​E1ex​\(h\)​\(L02\+L12​g2\)​\{43​ℓ2​w2\+4​\(d\+1\)​ℓ\}\.\\mathbb\{E\}\\\!\\left\[\\int\_\{0\}^\{\\ell\}\\\|\\nabla V\(X\+\\mathcal\{D\}\_\{s\}\)\-\\nabla V\(X\)\\\|^\{2\}\\,\\mathrm\{d\}s\\;\\middle\|\\;\\mathcal\{F\}\\right\]\\leq\\ell E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)\(L\_\{0\}^\{2\}\+L\_\{1\}^\{2\}g^\{2\}\)\\left\\\{\\frac\{4\}\{3\}\\ell^\{2\}w^\{2\}\+4\(d\+1\)\\ell\\right\\\}\.For terms multiplied byL02L\_\{0\}^\{2\}, useℓ≤h\\ell\\leq h\. For the mixed term, useℓ2​w2≤h2​ω−2\\ell^\{2\}w^\{2\}\\leq h^\{2\}\\omega^\{\-2\}\. This gives the coefficientsA~1,U~1,V~1\\widetilde\{A\}\_\{1\},\\widetilde\{U\}\_\{1\},\\widetilde\{V\}\_\{1\}defined above\.

Finally, ifL1=0L\_\{1\}=0, globalL0L\_\{0\}\-smoothness and \([70](https://arxiv.org/html/2610.12865#A1.E70)\) give directly

𝔼⁡\[∫0ℓ‖∇V​\(X\+𝒟s\)−∇V​\(X\)‖2​𝑑s\|ℱ\]≤ℓ⁡\{L02​𝑑h\+13​L02​h2​w2\}\.\\mathbb\{E\}\\\!\\left\[\\int\_\{0\}^\{\\ell\}\\\|\\nabla V\(X\+\\mathcal\{D\}\_\{s\}\)\-\\nabla V\(X\)\\\|^\{2\}\\,\\mathrm\{d\}s\\;\\middle\|\\;\\mathcal\{F\}\\right\]\\leq\\ell\\left\\\{L\_\{0\}^\{2\}dh\+\\frac\{1\}\{3\}L\_\{0\}^\{2\}h^\{2\}w^\{2\}\\right\\\}\.This gives the coefficients defined above forL1=0L\_\{1\}=0\. The stated monotonicity follows directly from the formulas\. Atγ=1\\gamma=1,E1ex​\(h\)E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)is increasing\. ∎

#### A\.7\.5Discretization error for adaptive LMC with exact gradients

We now exploit that the drift and the step\-size rule use the same exact gradient\. This produces sharper coefficients than the reusable approximate drift lemma and is essential for recovering the globally smooth rate whenL1=0L\_\{1\}=0\. We use the coefficientsAγ,Bγ,QγA\_\{\\gamma\},B\_\{\\gamma\},Q\_\{\\gamma\}defined in Subsection[A\.2](https://arxiv.org/html/2610.12865#A1.SS2)\. We restate their formulas where they are used in the proof\.

###### Lemma A\.21\(Discretization error for adaptive LMC with exact gradients\)\.

FixT\>0T\>0and consider a path on\[0,T\]\[0,T\]formed by finitely many intervals of the adaptive step method \([7](https://arxiv.org/html/2610.12865#S3.E7)\), where the decision to take an interval is made using its starting history\. If these intervals end at a timeσ<T\\sigma<T, continue the path by the exact Langevin diffusion on\[σ,T\]\[\\sigma,T\]\. Define the piecewise force, with the sign convention of \([4](https://arxiv.org/html/2610.12865#S3.E4)\), by

Ftfr:=\{Fk=−∇V​\(Xτk\),t∈\[τk,τk\+ℓk\),−∇V​\(Xt\),t∈\[σ,T\]​on the exact continuation\.F\_\{t\}^\{\\mathrm\{fr\}\}:=\\begin\{cases\}F\_\{k\}=\-\\nabla V\(X\_\{\\tau\_\{k\}\}\),&t\\in\[\\tau\_\{k\},\\tau\_\{k\}\+\\ell\_\{k\}\),\\\\ \-\\nabla V\(X\_\{t\}\),&t\\in\[\\sigma,T\]\\text\{ on the exact continuation\}\.\\end\{cases\}A final adaptive interval cut off atTTuses the first line untilTT\. Define

eT:=1T​𝔼​∫0T‖∇V​\(Xt\)\+Ftfr‖2​𝑑tandmT:=1T​𝔼​∫0T‖Ftfr‖2​𝑑t\.e\_\{T\}:=\\frac\{1\}\{T\}\\mathbb\{E\}\\int\_\{0\}^\{T\}\\\|\\nabla V\(X\_\{t\}\)\+F\_\{t\}^\{\\mathrm\{fr\}\}\\\|^\{2\}\\,\\mathrm\{d\}t\\quad\\text\{and\}\\quad m\_\{T\}:=\\frac\{1\}\{T\}\\mathbb\{E\}\\int\_\{0\}^\{T\}\\\|F\_\{t\}^\{\\mathrm\{fr\}\}\\\|^\{2\}\\,\\mathrm\{d\}t\.\(71\)Then

eT≤Aγ​\(h\)\+Bγ​\(h\)​mT\+Qγ​\(h\)​mTγ/2\.e\_\{T\}\\leq A\_\{\\gamma\}\(h\)\+B\_\{\\gamma\}\(h\)m\_\{T\}\+Q\_\{\\gamma\}\(h\)m\_\{T\}^\{\\gamma/2\}\.\(72\)

###### Proof\.

On an adaptive interval, condition on the complete information at timeτk\\tau\_\{k\}, and letxxdenote the resulting value ofXτkX\_\{\\tau\_\{k\}\}\. Putℓ:=ℓk\\ell:=\\ell\_\{k\}andg:=‖∇V​\(x\)‖g:=\\\|\\nabla V\(x\)\\\|\. All expectations until the summation over intervals below are conditional on this history\. For every fixed0≤s≤ℓ0\\leq s\\leq\\ell, equation \([58](https://arxiv.org/html/2610.12865#A1.E58)\) gives

Xτk\+s−Xτk=d−s∇V\(x\)\+2​sZ,Z∼𝒩\(0,Id\)\.X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\stackrel\{\{\\scriptstyle\\mathrm\{d\}\}\}\{\{=\}\}\-s\\nabla V\(x\)\+\\sqrt\{2s\}Z,\\qquad Z\\sim\\mathcal\{N\}\(0,I\_\{d\}\)\.\(73\)The step\-size rule and final\-interval truncation imply

0≤ℓ≤Hk=h1\+ω​gγ≤h,ω​ℓ​gγ≤h\.0\\leq\\ell\\leq H\_\{k\}=\\frac\{h\}\{1\+\\omega g^\{\\gamma\}\}\\leq h,\\qquad\\omega\\ell g^\{\\gamma\}\\leq h\.\(74\)Consequently,

𝔼​‖Xτk\+s−Xτk‖2=s2​g2\+2​d​s\.\\mathbb\{E\}\\\|X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\\|^\{2\}=s^\{2\}g^\{2\}\+2ds\.\(75\)
IfL1=0L\_\{1\}=0, globalL0L\_\{0\}\-smoothness and \([75](https://arxiv.org/html/2610.12865#A1.E75)\) give

𝔼​∫0ℓ‖∇V​\(Xτk\+s\)−∇V​\(Xτk\)‖2​𝑑s≤L02​\(ℓ3​g23\+d​ℓ2\)≤ℓ⁡\{d​L02​h\+13​L02​h2​g2\}\.\\mathbb\{E\}\\int\_\{0\}^\{\\ell\}\\\|\\nabla V\(X\_\{\\tau\_\{k\}\+s\}\)\-\\nabla V\(X\_\{\\tau\_\{k\}\}\)\\\|^\{2\}\\,\\mathrm\{d\}s\\leq L\_\{0\}^\{2\}\\left\(\\frac\{\\ell^\{3\}g^\{2\}\}\{3\}\+d\\ell^\{2\}\\right\)\\leq\\ell\\left\\\{dL\_\{0\}^\{2\}h\+\\frac\{1\}\{3\}L\_\{0\}^\{2\}h^\{2\}g^\{2\}\\right\\\}\.\(76\)Now supposeL1\>0L\_\{1\}\>0and0<γ<10<\\gamma<1\. For everya≥2a\\geq 2, the triangle inequality, scalar convexity, and Lemma[A\.19](https://arxiv.org/html/2610.12865#A1.Thmtheorem19)give

𝔼​‖Xτk\+s−Xτk‖a\\displaystyle\\mathbb\{E\}\\\|X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\\|^\{a\}≤2a−1​sa​ga\+23​a/2−1​sa/2​\(d\+a\)a/2\.\\displaystyle\\leq 2^\{a\-1\}s^\{a\}g^\{a\}\+2^\{3a/2\-1\}s^\{a/2\}\(d\+a\)^\{a/2\}\.\(77\)For eachs∈\[0,ℓ\]s\\in\[0,\\ell\], apply \([60](https://arxiv.org/html/2610.12865#A1.E60)\) from Lemma[A\.15](https://arxiv.org/html/2610.12865#A1.Thmtheorem15)withx=Xτkx=X\_\{\\tau\_\{k\}\}andy=Xτk\+sy=X\_\{\\tau\_\{k\}\+s\}\. The bound holds for every pair of points, so it applies to these states on each realization of the path\. Using‖∇V​\(Xτk\)‖=g\\\|\\nabla V\(X\_\{\\tau\_\{k\}\}\)\\\|=g, we obtain

‖∇V​\(Xτk\+s\)−∇V​\(Xτk\)‖2\\displaystyle\\\|\\nabla V\(X\_\{\\tau\_\{k\}\+s\}\)\-\\nabla V\(X\_\{\\tau\_\{k\}\}\)\\\|^\{2\}≤3​L02​\(1\+2γ2/\(1−γ\)\)2​‖Xτk\+s−Xτk‖2\\displaystyle\\leq 3L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}\\\|X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\\|^\{2\}\+32​γ\+1​22​γ2/\(1−γ\)​L12​g2​γ​‖Xτk\+s−Xτk‖2\\displaystyle\+3^\{2\\gamma\+1\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}L\_\{1\}^\{2\}g^\{2\\gamma\}\\\|X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\\|^\{2\}\+32​γ\+1​22​γ2/\(1−γ\)​L12/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​‖Xτk\+s−Xτk‖2/\(1−γ\)\.\\displaystyle\+3^\{2\\gamma\+1\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}L\_\{1\}^\{2/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}\\\|X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\\|^\{2/\(1\-\\gamma\)\}\.Conditioning on the history atτk\\tau\_\{k\}fixesggandℓ\\ell, so the factors outside the displacement norms can be taken outside the conditional expectations\. Taking these expectations and integrating overs∈\[0,ℓ\]s\\in\[0,\\ell\]gives the following bound\. Tonelli’s theorem permits interchanging the expectation and integral because the integrands are nonnegative\.

𝔼​∫0ℓ‖∇V​\(Xτk\+s\)−∇V​\(Xτk\)‖2​𝑑s\\displaystyle\\mathbb\{E\}\\int\_\{0\}^\{\\ell\}\\\|\\nabla V\(X\_\{\\tau\_\{k\}\+s\}\)\-\\nabla V\(X\_\{\\tau\_\{k\}\}\)\\\|^\{2\}\\,\\mathrm\{d\}s≤3​L02​\(1\+2γ2/\(1−γ\)\)2​∫0ℓ𝔼​‖Xτk\+s−Xτk‖2​𝑑s\\displaystyle\\quad\\leq 3L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}\\int\_\{0\}^\{\\ell\}\\mathbb\{E\}\\\|X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\\|^\{2\}\\,\\mathrm\{d\}s\+32​γ\+122​γ2/\(1−γ\)L12g2​γ∫0ℓ𝔼∥Xτk\+s−Xτk∥2ds\\displaystyle\\quad\+3^\{2\\gamma\+1\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}L\_\{1\}^\{2\}g^\{2\\gamma\}\\int\_\{0\}^\{\\ell\}\\mathbb\{E\}\\\|X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\\|^\{2\}\\,\\mathrm\{d\}s\+32​γ\+122​γ2/\(1−γ\)L12/\(1−γ\)\(1−γ\)2​γ/\(1−γ\)∫0ℓ𝔼∥Xτk\+s−Xτk∥2/\(1−γ\)ds\.\\displaystyle\\quad\+3^\{2\\gamma\+1\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}L\_\{1\}^\{2/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}\\int\_\{0\}^\{\\ell\}\\mathbb\{E\}\\\|X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\\|^\{2/\(1\-\\gamma\)\}\\,\\mathrm\{d\}s\.\(78\)By \([75](https://arxiv.org/html/2610.12865#A1.E75)\) and \([74](https://arxiv.org/html/2610.12865#A1.E74)\), the first term on the right of \([78](https://arxiv.org/html/2610.12865#A1.E78)\) is at most

ℓ​L02​\(1\+2γ2/\(1−γ\)\)2​\(h2​g2\+3​d​h\)\.\\ell L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}\(h^\{2\}g^\{2\}\+3dh\)\.\(79\)For the second term,

ℓ2​g2\+2​γ≤h2ω2​g2andℓ​g2​γ≤hω​gγ,\\ell^\{2\}g^\{2\+2\\gamma\}\\leq\\frac\{h^\{2\}\}\{\\omega^\{2\}\}g^\{2\}\\quad\\text\{and\}\\quad\\ell g^\{2\\gamma\}\\leq\\frac\{h\}\{\\omega\}g^\{\\gamma\},and hence the second term is at most

ℓ​22​γ2/\(1−γ\)​32​γ​\{\(1\+L0\)2​h2​g2\+3​L1​\(1\+L0\)​d​h​gγ\}\.\\ell\\,2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}3^\{2\\gamma\}\\left\\\{\(1\+L\_\{0\}\)^\{2\}h^\{2\}g^\{2\}\+3L\_\{1\}\(1\+L\_\{0\}\)dhg^\{\\gamma\}\\right\\\}\.\(80\)For the third term, seta=2/\(1−γ\)a=2/\(1\-\\gamma\)in \([77](https://arxiv.org/html/2610.12865#A1.E77)\) and integrate\. Bounding the two integration denominators below by one gives

32​γ\+1​22​γ2/\(1−γ\)​L12/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​∫0ℓ𝔼​‖Xτk\+s−Xτk‖2/\(1−γ\)​𝑑s\\displaystyle 3^\{2\\gamma\+1\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}L\_\{1\}^\{2/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}\\int\_\{0\}^\{\\ell\}\\mathbb\{E\}\\\|X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\\|^\{2/\(1\-\\gamma\)\}\\,\\mathrm\{d\}s≤ℓ​32​γ\+1​2\(2​γ2\+γ\+1\)/\(1−γ\)​L12/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​\(ℓ​g\)2/\(1−γ\)\\displaystyle\\quad\\leq\\ell\\,3^\{2\\gamma\+1\}2^\{\(2\\gamma^\{2\}\+\\gamma\+1\)/\(1\-\\gamma\)\}L\_\{1\}^\{2/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}\(\\ell g\)^\{2/\(1\-\\gamma\)\}\+ℓ​32​γ\+1​2\(2​γ2\+γ\+2\)/\(1−γ\)​L12/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​\[\(d\+21−γ\)​ℓ\]1/\(1−γ\)\.\\displaystyle\\qquad\+\\ell\\,3^\{2\\gamma\+1\}2^\{\(2\\gamma^\{2\}\+\\gamma\+2\)/\(1\-\\gamma\)\}L\_\{1\}^\{2/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}\\left\[\\left\(d\+\\frac\{2\}\{1\-\\gamma\}\\right\)\\ell\\right\]^\{1/\(1\-\\gamma\)\}\.\(81\)Since2​γ/\(1−γ\)=2/\(1−γ\)−22\\gamma/\(1\-\\gamma\)=2/\(1\-\\gamma\)\-2, \([74](https://arxiv.org/html/2610.12865#A1.E74)\) implies

ℓ2/\(1−γ\)​g2/\(1−γ\)≤\(h/ω\)2/\(1−γ\)​g2,ℓ1/\(1−γ\)≤h1/\(1−γ\)\.\\ell^\{2/\(1\-\\gamma\)\}g^\{2/\(1\-\\gamma\)\}\\leq\(h/\\omega\)^\{2/\(1\-\\gamma\)\}g^\{2\},\\qquad\\ell^\{1/\(1\-\\gamma\)\}\\leq h^\{1/\(1\-\\gamma\)\}\.\(82\)Substituting \([82](https://arxiv.org/html/2610.12865#A1.E82)\) into \([81](https://arxiv.org/html/2610.12865#A1.E81)\) shows that the third term is at most

ℓ​32​γ\+1​2\(2​γ2\+γ\+1\)/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​\[\(1\+L0\)​h\]2/\(1−γ\)​g2\\displaystyle\\ell\\,3^\{2\\gamma\+1\}2^\{\(2\\gamma^\{2\}\+\\gamma\+1\)/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}\[\(1\+L\_\{0\}\)h\]^\{2/\(1\-\\gamma\)\}g^\{2\}\(83\)\+ℓ​32​γ\+1​2\(2​γ2\+γ\+2\)/\(1−γ\)​L12/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​\[\(d\+21−γ\)​h\]1/\(1−γ\)\.\\displaystyle\+\\ell\\,3^\{2\\gamma\+1\}2^\{\(2\\gamma^\{2\}\+\\gamma\+2\)/\(1\-\\gamma\)\}L\_\{1\}^\{2/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}\\left\[\\left\(d\+\\frac\{2\}\{1\-\\gamma\}\\right\)h\\right\]^\{1/\(1\-\\gamma\)\}\.Combining \([79](https://arxiv.org/html/2610.12865#A1.E79)\), \([80](https://arxiv.org/html/2610.12865#A1.E80)\), and \([83](https://arxiv.org/html/2610.12865#A1.E83)\) gives the coefficients

Aγ​\(h\)\\displaystyle A\_\{\\gamma\}\(h\)=3​L02​\(1\+2γ2/\(1−γ\)\)2​d​h\\displaystyle=3L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}dh\+32​γ\+1​2\(2​γ2\+γ\+2\)/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​L12/\(1−γ\)​\[\(d\+21−γ\)​h\]1/\(1−γ\),\\displaystyle\\quad\+3^\{2\\gamma\+1\}2^\{\(2\\gamma^\{2\}\+\\gamma\+2\)/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}L\_\{1\}^\{2/\(1\-\\gamma\)\}\\left\[\\left\(d\+\\frac\{2\}\{1\-\\gamma\}\\right\)h\\right\]^\{1/\(1\-\\gamma\)\},Bγ​\(h\)\\displaystyle B\_\{\\gamma\}\(h\)=\[L02​\(1\+2γ2/\(1−γ\)\)2\+32​γ​22​γ2/\(1−γ\)​\(1\+L0\)2\]​h2\\displaystyle=\\left\[L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}\+3^\{2\\gamma\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}\(1\+L\_\{0\}\)^\{2\}\\right\]h^\{2\}\+32​γ\+1​2\(2​γ2\+γ\+1\)/\(1−γ\)​\(1−γ\)2​γ/\(1−γ\)​\[\(1\+L0\)​h\]2/\(1−γ\),\\displaystyle\\quad\+3^\{2\\gamma\+1\}2^\{\(2\\gamma^\{2\}\+\\gamma\+1\)/\(1\-\\gamma\)\}\(1\-\\gamma\)^\{2\\gamma/\(1\-\\gamma\)\}\[\(1\+L\_\{0\}\)h\]^\{2/\(1\-\\gamma\)\},Qγ​\(h\)\\displaystyle Q\_\{\\gamma\}\(h\)=32​γ\+1​22​γ2/\(1−γ\)​L1​\(1\+L0\)​d​h,\\displaystyle=3^\{2\\gamma\+1\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}L\_\{1\}\(1\+L\_\{0\}\)dh,and therefore

𝔼​∫0ℓ‖∇V​\(Xτk\+s\)−∇V​\(Xτk\)‖2​𝑑s≤ℓ⁡\{Aγ​\(h\)\+Bγ​\(h\)​g2\+Qγ​\(h\)​gγ\}\.\\mathbb\{E\}\\int\_\{0\}^\{\\ell\}\\\|\\nabla V\(X\_\{\\tau\_\{k\}\+s\}\)\-\\nabla V\(X\_\{\\tau\_\{k\}\}\)\\\|^\{2\}\\,\\mathrm\{d\}s\\leq\\ell\\left\\\{A\_\{\\gamma\}\(h\)\+B\_\{\\gamma\}\(h\)g^\{2\}\+Q\_\{\\gamma\}\(h\)g^\{\\gamma\}\\right\\\}\.\(84\)
It remains to treatL1\>0L\_\{1\}\>0andγ=1\\gamma=1\. Equations \([73](https://arxiv.org/html/2610.12865#A1.E73)\) and \([74](https://arxiv.org/html/2610.12865#A1.E74)\) imply

L1​‖Xτk\+s−Xτk‖≤\(1\+L0\)​h\+L1​2​h​‖Z‖,‖Xτk\+s−Xτk‖2≤2​s2​g2\+4​s​‖Z‖2\.L\_\{1\}\\\|X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\\|\\leq\(1\+L\_\{0\}\)h\+L\_\{1\}\\sqrt\{2h\}\\\|Z\\\|,\\qquad\\\|X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\\|^\{2\}\\leq 2s^\{2\}g^\{2\}\+4s\\\|Z\\\|^\{2\}\.Recall the exponential factor

E1ex​\(h\)=exp⁡\{2​\(1\+L0\)​h\+2​2​L1​d​h\+8​L12​h\}\.E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)=\\exp\\\{2\(1\+L\_\{0\}\)h\+2\\sqrt\{2\}L\_\{1\}\\sqrt\{dh\}\+8L\_\{1\}^\{2\}h\\\}\.Lemma[A\.19](https://arxiv.org/html/2610.12865#A1.Thmtheorem19)then gives

𝔼⁡\[‖Xτk\+s−Xτk‖2​e2​L1​‖Xτk\+s−Xτk‖\]≤E1ex​\(h\)​\{2​s2​g2\+4​s​\(d\+1\)\}\.\\mathbb\{E\}\\\!\\left\[\\\|X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\\|^\{2\}e^\{2L\_\{1\}\\\|X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\\|\}\\right\]\\leq E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)\\\{2s^\{2\}g^\{2\}\+4s\(d\+1\)\\\}\.\(85\)Square \([61](https://arxiv.org/html/2610.12865#A1.E61)\), use\(L0\+L1​g\)2≤2​\(L02\+L12​g2\)\(L\_\{0\}\+L\_\{1\}g\)^\{2\}\\leq 2\(L\_\{0\}^\{2\}\+L\_\{1\}^\{2\}g^\{2\}\), and integrate \([85](https://arxiv.org/html/2610.12865#A1.E85)\) to obtain

𝔼​∫0ℓ‖∇V​\(Xτk\+s\)−∇V​\(Xτk\)‖2​𝑑s≤E1ex​\(h\)​\(L02\+L12​g2\)​\{43​ℓ3​g2\+4​\(d\+1\)​ℓ2\}\.\\mathbb\{E\}\\int\_\{0\}^\{\\ell\}\\\|\\nabla V\(X\_\{\\tau\_\{k\}\+s\}\)\-\\nabla V\(X\_\{\\tau\_\{k\}\}\)\\\|^\{2\}\\,\\mathrm\{d\}s\\leq E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)\(L\_\{0\}^\{2\}\+L\_\{1\}^\{2\}g^\{2\}\)\\left\\\{\\frac\{4\}\{3\}\\ell^\{3\}g^\{2\}\+4\(d\+1\)\\ell^\{2\}\\right\\\}\.\(86\)The step\-size bound gives

ℓ≤h,ℓ2​g2≤h2​g2,L12​ℓ2​g4≤\(1\+L0\)2​h2​g2,L12​ℓ​g2≤L1​\(1\+L0\)​h​g\.\\ell\\leq h,\\qquad\\ell^\{2\}g^\{2\}\\leq h^\{2\}g^\{2\},\\qquad L\_\{1\}^\{2\}\\ell^\{2\}g^\{4\}\\leq\(1\+L\_\{0\}\)^\{2\}h^\{2\}g^\{2\},\\qquad L\_\{1\}^\{2\}\\ell g^\{2\}\\leq L\_\{1\}\(1\+L\_\{0\}\)hg\.Substituting these bounds into \([86](https://arxiv.org/html/2610.12865#A1.E86)\) gives

𝔼​∫0ℓ‖∇V​\(Xτk\+s\)−∇V​\(Xτk\)‖2​𝑑s≤ℓ⁡\{A1​\(h\)\+B1​\(h\)​g2\+Q1​\(h\)​g\},\\mathbb\{E\}\\int\_\{0\}^\{\\ell\}\\\|\\nabla V\(X\_\{\\tau\_\{k\}\+s\}\)\-\\nabla V\(X\_\{\\tau\_\{k\}\}\)\\\|^\{2\}\\,\\mathrm\{d\}s\\leq\\ell\\\{A\_\{1\}\(h\)\+B\_\{1\}\(h\)g^\{2\}\+Q\_\{1\}\(h\)g\\\},\(87\)whereA1​\(h\)=4​E1ex​\(h\)​L02​\(d\+1\)​hA\_\{1\}\(h\)=4E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)L\_\{0\}^\{2\}\(d\+1\)h,B1​\(h\)=43​E1ex​\(h\)​\{L02\+\(1\+L0\)2\}​h2B\_\{1\}\(h\)=\\tfrac\{4\}\{3\}E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)\\\{L\_\{0\}^\{2\}\+\(1\+L\_\{0\}\)^\{2\}\\\}h^\{2\}, andQ1​\(h\)=4​E1ex​\(h\)​L1​\(1\+L0\)​\(d\+1\)​hQ\_\{1\}\(h\)=4E\_\{1\}^\{\\mathrm\{ex\}\}\(h\)L\_\{1\}\(1\+L\_\{0\}\)\(d\+1\)h\.

Take expectations over the histories at interval starts and sum \([76](https://arxiv.org/html/2610.12865#A1.E76)\), \([84](https://arxiv.org/html/2610.12865#A1.E84)\), or \([87](https://arxiv.org/html/2610.12865#A1.E87)\) over the adaptive intervals\. The discretization error is zero on the exact continuation\. This gives

eT≤Aγ​\(h\)\+Bγ​\(h\)​mT\+Qγ​\(h\)​1T​𝔼​∫0T‖Ftfr‖γ​𝑑t\.e\_\{T\}\\leq A\_\{\\gamma\}\(h\)\+B\_\{\\gamma\}\(h\)m\_\{T\}\+Q\_\{\\gamma\}\(h\)\\frac\{1\}\{T\}\\mathbb\{E\}\\int\_\{0\}^\{T\}\\\|F\_\{t\}^\{\\mathrm\{fr\}\}\\\|^\{\\gamma\}\\,\\mathrm\{d\}t\.Jensen’s inequality for the probability measureT−1​d​t​d​ℙT^\{\-1\}\\mathrm\{d\}t\\,\\mathrm\{d\}\\mathbb\{P\}yields

1T​𝔼​∫0T‖Ftfr‖γ​𝑑t≤mTγ/2\.\\frac\{1\}\{T\}\\mathbb\{E\}\\int\_\{0\}^\{T\}\\\|F\_\{t\}^\{\\mathrm\{fr\}\}\\\|^\{\\gamma\}\\,\\mathrm\{d\}t\\leq m\_\{T\}^\{\\gamma/2\}\.\(88\)This proves \([72](https://arxiv.org/html/2610.12865#A1.E72)\)\. ∎

#### A\.7\.6Closure, nonaccumulation, and finite horizon bounds

We now use the preceding bounds and the constraint on the base step size to control the average discretization erroreTe\_\{T\}\.

###### Proof of the finite horizon claims in Theorem[A\.4](https://arxiv.org/html/2610.12865#A1.Thmtheorem4)\.

We first perform the calculation when all terms are finite, and then justify it using a finite step approximation\. This order avoids assuming the gradient integrability that the argument is meant to establish\. Recall the constant term in the gradient second moment bound in Lemma[A\.16](https://arxiv.org/html/2610.12865#A1.Thmtheorem16),

Cγ:=4​d​L0\+\(2−γ\)​γγ/\(2−γ\)​\(2​d​L1\)2/\(2−γ\)\.C\_\{\\gamma\}:=4dL\_\{0\}\+\(2\-\\gamma\)\\gamma^\{\\gamma/\(2\-\\gamma\)\}\(2dL\_\{1\}\)^\{2/\(2\-\\gamma\)\}\.WriteMT:=1\+Cγ\+K0/TM\_\{T\}:=1\+C\_\{\\gamma\}\+K\_\{0\}/TandFI¯T:=T−1∫0TFI\(μt∥π\)dt\\overline\{\\mathrm\{FI\}\}\_\{T\}:=T^\{\-1\}\\int\_\{0\}^\{T\}\\mathrm\{FI\}\(\\mu\_\{t\}\\\|\\pi\)\\,\\mathrm\{d\}t\. For the interpolated process, the drift isFtfrF\_\{t\}^\{\\mathrm\{fr\}\}and the error action in Lemma[A\.18](https://arxiv.org/html/2610.12865#A1.Thmtheorem18)is𝒜T=T​eT\\mathcal\{A\}\_\{T\}=Te\_\{T\}\. Dividing \([64](https://arxiv.org/html/2610.12865#A1.E64)\) byTTtherefore gives

KL\(μT∥π\)T\+34​FI¯T≤K0T\+eT\.\\frac\{\\mathrm\{KL\}\(\\mu\_\{T\}\\\|\\pi\)\}\{T\}\+\\frac\{3\}\{4\}\\overline\{\\mathrm\{FI\}\}\_\{T\}\\leq\\frac\{K\_\{0\}\}\{T\}\+e\_\{T\}\.SinceKL\(μT∥π\)≥0\\mathrm\{KL\}\(\\mu\_\{T\}\\\|\\pi\)\\geq 0, we obtain

FI¯T≤43​\(K0T\+eT\)\.\\overline\{\\mathrm\{FI\}\}\_\{T\}\\leq\\frac\{4\}\{3\}\\left\(\\frac\{K\_\{0\}\}\{T\}\+e\_\{T\}\\right\)\.\(89\)
We next boundmTm\_\{T\}, the average squared norm of the applied drift\. WriteFtfr=−∇V​\(Xt\)\+\(Ftfr\+∇V​\(Xt\)\)F\_\{t\}^\{\\mathrm\{fr\}\}=\-\\nabla V\(X\_\{t\}\)\+\(F\_\{t\}^\{\\mathrm\{fr\}\}\+\\nabla V\(X\_\{t\}\)\)and use‖u\+v‖2≤2​‖u‖2\+2​‖v‖2\\\|u\+v\\\|^\{2\}\\leq 2\\\|u\\\|^\{2\}\+2\\\|v\\\|^\{2\}to obtain

‖Ftfr‖2≤2​‖∇V​\(Xt\)‖2\+2​‖Ftfr\+∇V​\(Xt\)‖2\.\\\|F\_\{t\}^\{\\mathrm\{fr\}\}\\\|^\{2\}\\leq 2\\\|\\nabla V\(X\_\{t\}\)\\\|^\{2\}\+2\\\|F\_\{t\}^\{\\mathrm\{fr\}\}\+\\nabla V\(X\_\{t\}\)\\\|^\{2\}\.Taking expectations, integrating over\[0,T\]\[0,T\], and dividing byTTgives

mT≤2T​𝔼​∫0T‖∇V​\(Xt\)‖2​𝑑t\+2​eT\.m\_\{T\}\\leq\\frac\{2\}\{T\}\\mathbb\{E\}\\int\_\{0\}^\{T\}\\\|\\nabla V\(X\_\{t\}\)\\\|^\{2\}\\,\\mathrm\{d\}t\+2e\_\{T\}\.Lemma[A\.16](https://arxiv.org/html/2610.12865#A1.Thmtheorem16), applied to the lawμt\\mu\_\{t\}ofXtX\_\{t\}, bounds the gradient second moment by𝔼∥∇V\(Xt\)∥2≤2FI\(μt∥π\)\+Cγ\\mathbb\{E\}\\\|\\nabla V\(X\_\{t\}\)\\\|^\{2\}\\leq 2\\mathrm\{FI\}\(\\mu\_\{t\}\\\|\\pi\)\+C\_\{\\gamma\}\. Substituting this bound into the preceding inequality yields

mT≤4​FI¯T\+2​Cγ\+2​eT\.m\_\{T\}\\leq 4\\overline\{\\mathrm\{FI\}\}\_\{T\}\+2C\_\{\\gamma\}\+2e\_\{T\}\.\(90\)Finally, substituting \([89](https://arxiv.org/html/2610.12865#A1.E89)\) into \([90](https://arxiv.org/html/2610.12865#A1.E90)\) gives

mT\\displaystyle m\_\{T\}≤163​K0T\+2​Cγ\+223​eT\\displaystyle\\leq\\frac\{16\}\{3\}\\frac\{K\_\{0\}\}\{T\}\+2C\_\{\\gamma\}\+\\frac\{22\}\{3\}e\_\{T\}\(91\)≤8​\(MT\+eT\)\.\\displaystyle\\leq 8\(M\_\{T\}\+e\_\{T\}\)\.The last inequality uses16/3≤816/3\\leq 8,2≤82\\leq 8, and22/3≤822/3\\leq 8, together with nonnegativity andMT=1\+Cγ\+K0/TM\_\{T\}=1\+C\_\{\\gamma\}\+K\_\{0\}/T\. Since0<γ/2≤1/20<\\gamma/2\\leq 1/2andMT≥1M\_\{T\}\\geq 1, concavity gives

\(MT\+eT\)γ/2≤MTγ/2\+γ2​MTγ/2−1​eT≤MTγ/2\+eT\.\(M\_\{T\}\+e\_\{T\}\)^\{\\gamma/2\}\\leq M\_\{T\}^\{\\gamma/2\}\+\\frac\{\\gamma\}\{2\}M\_\{T\}^\{\\gamma/2\-1\}e\_\{T\}\\leq M\_\{T\}^\{\\gamma/2\}\+e\_\{T\}\.\(92\)
After applying the moment bound, we define

Rγ​\(h,M\):=Aγ​\(h\)\+8​Bγ​\(h\)​M\+8γ/2​Qγ​\(h\)​Mγ/2R\_\{\\gamma\}\(h,M\):=A\_\{\\gamma\}\(h\)\+8B\_\{\\gamma\}\(h\)M\+8^\{\\gamma/2\}Q\_\{\\gamma\}\(h\)M^\{\\gamma/2\}to collect the terms independent of the discretization erroreTe\_\{T\}\. The coefficient multiplyingeTe\_\{T\}is

Δγ​\(h\):=8​Bγ​\(h\)\+8γ/2​Qγ​\(h\)\.\\Delta\_\{\\gamma\}\(h\):=8B\_\{\\gamma\}\(h\)\+8^\{\\gamma/2\}Q\_\{\\gamma\}\(h\)\.Substitute the bound formTm\_\{T\}in \([91](https://arxiv.org/html/2610.12865#A1.E91)\) into the discretization bound for exact gradients \([72](https://arxiv.org/html/2610.12865#A1.E72)\), and then apply \([92](https://arxiv.org/html/2610.12865#A1.E92)\) to obtain

eT\\displaystyle e\_\{T\}≤Aγ​\(h\)\+8​Bγ​\(h\)​\(MT\+eT\)\+8γ/2​Qγ​\(h\)​\(MT\+eT\)γ/2\\displaystyle\\leq A\_\{\\gamma\}\(h\)\+8B\_\{\\gamma\}\(h\)\(M\_\{T\}\+e\_\{T\}\)\+8^\{\\gamma/2\}Q\_\{\\gamma\}\(h\)\(M\_\{T\}\+e\_\{T\}\)^\{\\gamma/2\}≤Rγ​\(h,MT\)\+Δγ​\(h\)​eT\.\\displaystyle\\leq R\_\{\\gamma\}\(h,M\_\{T\}\)\+\\Delta\_\{\\gamma\}\(h\)e\_\{T\}\.\(93\)IfΔγ​\(h\)≤1/2\\Delta\_\{\\gamma\}\(h\)\\leq 1/2, then

eT≤2​Rγ​\(h,MT\)\.e\_\{T\}\\leq 2R\_\{\\gamma\}\(h,M\_\{T\}\)\.\(94\)Substituting \([94](https://arxiv.org/html/2610.12865#A1.E94)\) into the Fisher information bound \([89](https://arxiv.org/html/2610.12865#A1.E89)\) proves

FI¯T≤4​K03​T\+83​Rγ​\(h,MT\)\.\\overline\{\\mathrm\{FI\}\}\_\{T\}\\leq\\frac\{4K\_\{0\}\}\{3T\}\+\\frac\{8\}\{3\}R\_\{\\gamma\}\(h,M\_\{T\}\)\.Convexity of relative Fisher information givesFI\(μ¯T∥π\)≤FI¯T\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\overline\{\\mathrm\{FI\}\}\_\{T\}, proving \([19](https://arxiv.org/html/2610.12865#A1.E19)\)\. The coefficientsAγ,Bγ,QγA\_\{\\gamma\},B\_\{\\gamma\},Q\_\{\\gamma\}defined above are continuous and nondecreasing inhh, vanish at zero, and for0<h≤10<h\\leq 1are at mosthhtimes their values at one\. Consequently,Δγ​\(h\)≤h​Δγ​\(1\)\\Delta\_\{\\gamma\}\(h\)\\leq h\\Delta\_\{\\gamma\}\(1\)\. The base step constraint in Theorem[A\.4](https://arxiv.org/html/2610.12865#A1.Thmtheorem4), namely0<h≤\[1\+2​Δγ​\(1\)\]−10<h\\leq\[1\+2\\Delta\_\{\\gamma\}\(1\)\]^\{\-1\}, therefore implies

Δγ​\(h\)≤Δγ​\(1\)1\+2​Δγ​\(1\)<12,\\Delta\_\{\\gamma\}\(h\)\\leq\\frac\{\\Delta\_\{\\gamma\}\(1\)\}\{1\+2\\Delta\_\{\\gamma\}\(1\)\}<\\frac\{1\}\{2\},which verifies the stated explicit sufficient condition\.

##### Finite step approximation\.

The rearrangement leading to \([94](https://arxiv.org/html/2610.12865#A1.E94)\) requireseT<∞e\_\{T\}<\\infty, which has not yet been established for the full adaptive process\. We therefore first truncate the adaptive scheme afternnsteps and prove finiteness for each fixednn\. The preceding inequalities then give bounds independent ofnn, allowing us to letn→∞n\\to\\inftyin the next paragraph\.

First suppose thatμ0\\mu\_\{0\}has compact support\. Run at mostnnadaptive intervals and then append the exact diffusion from Lemma[A\.17](https://arxiv.org/html/2610.12865#A1.Thmtheorem17)\. This produces a conservative process on every finite horizon\. Equations \([60](https://arxiv.org/html/2610.12865#A1.E60)\) and \([61](https://arxiv.org/html/2610.12865#A1.E61)\), together with Lemma[A\.19](https://arxiv.org/html/2610.12865#A1.Thmtheorem19), imply at the end of each complete adaptive interval that

𝔼⁡\[‖∇V​\(Xτk\+1\)‖2\|ℱk\]≤C⁡\(h,d,L0,L1,γ\)​\(1\+‖∇V​\(Xτk\)‖2\),\\mathbb\{E\}\\\!\\left\[\\\|\\nabla V\(X\_\{\\tau\_\{k\+1\}\}\)\\\|^\{2\}\\;\\middle\|\\;\\mathcal\{F\}\_\{k\}\\right\]\\leq C\(h,d,L\_\{0\},L\_\{1\},\\gamma\)\\left\(1\+\\\|\\nabla V\(X\_\{\\tau\_\{k\}\}\)\\\|^\{2\}\\right\),where the finite constant does not depend onkk\. For0<γ<10<\\gamma<1, this uses2​γ/\(1−γ\)=2/\(1−γ\)−22\\gamma/\(1\-\\gamma\)=2/\(1\-\\gamma\)\-2\. Forγ=1\\gamma=1, it uses the exponential moments in \([69](https://arxiv.org/html/2610.12865#A1.E69)\)\. Continuity bounds the initial gradient on the compact support, so induction gives finite gradient second moments through every one of the firstnnintervals\.

The single\-interval estimates in \([76](https://arxiv.org/html/2610.12865#A1.E76)\), \([84](https://arxiv.org/html/2610.12865#A1.E84)\), and \([87](https://arxiv.org/html/2610.12865#A1.E87)\) give finite total error action over the adaptive intervals\. This error is zero on the exact continuation\. Lemma[A\.18](https://arxiv.org/html/2610.12865#A1.Thmtheorem18)therefore first establishes a finite Fisher integral, and Lemma[A\.16](https://arxiv.org/html/2610.12865#A1.Thmtheorem16)then gives a finite current\-gradient integral\. All terms in \([89](https://arxiv.org/html/2610.12865#A1.E89)\)–\([91](https://arxiv.org/html/2610.12865#A1.E91)\) are consequently finite, justifying the preceding absorption\. The preliminary moment bounds are used only to establish finiteness\. Applying \([93](https://arxiv.org/html/2610.12865#A1.E93)\) and \([89](https://arxiv.org/html/2610.12865#A1.E89)\) now gives \([94](https://arxiv.org/html/2610.12865#A1.E94)\) and the resulting Fisher information bound with constants independent ofnn\.

##### Passing to the limit and excluding finite\-time accumulation\.

Let

NT,n:=\#⁡\{0≤k<n:τk<T\}\.N\_\{T,n\}:=\\\#\\\{0\\leq k<n:\\tau\_\{k\}<T\\\}\.Every complete interval satisfies

1=h−1​Hk​\(1\+ω​‖∇V​\(Xτk\)‖γ\),1=h^\{\-1\}H\_\{k\}\\bigl\(1\+\\omega\\\|\\nabla V\(X\_\{\\tau\_\{k\}\}\)\\\|^\{\\gamma\}\\bigr\),and at most one occupied interval is truncated byTT\. Thus, pathwise,

NT,n≤1\+1h​∫0T∧τn\(1\+ω​‖Ftfr‖γ\)​𝑑t\.N\_\{T,n\}\\leq 1\+\\frac\{1\}\{h\}\\int\_\{0\}^\{T\\wedge\\tau\_\{n\}\}\\left\(1\+\\omega\\\|F\_\{t\}^\{\\mathrm\{fr\}\}\\\|^\{\\gamma\}\\right\)\\,\\mathrm\{d\}t\.\(95\)HereT∧τn=min⁡\{T,τn\}T\\wedge\\tau\_\{n\}=\\min\\\{T,\\tau\_\{n\}\\\}\. Bounding the number of steps by integrating the reciprocal step duration follows the argument of[Fang and Giles \(2020, Lemma 2\)](https://arxiv.org/html/2610.12865#bib.bib65)for adaptive Euler–Maruyama schemes\. Take expectations in \([95](https://arxiv.org/html/2610.12865#A1.E95)\), apply \([88](https://arxiv.org/html/2610.12865#A1.E88)\), \([91](https://arxiv.org/html/2610.12865#A1.E91)\), and \([94](https://arxiv.org/html/2610.12865#A1.E94)\) to obtain, uniformly innn,

𝔼​NT,n≤1\+Th​\[1\+ω​\{8​\(MT\+2​Rγ​\(h,MT\)\)\}γ/2\]\.\\mathbb\{E\}N\_\{T,n\}\\leq 1\+\\frac\{T\}\{h\}\\left\[1\+\\omega\\\{8\(M\_\{T\}\+2R\_\{\\gamma\}\(h,M\_\{T\}\)\)\\\}^\{\\gamma/2\}\\right\]\.\(96\)
Construct all finite prefixes using the same Brownian motion and initial state\. On\{τn≤T\}\\\{\\tau\_\{n\}\\leq T\\\}, every one of the firstnnintervals starts beforeTT, soNT,n=nN\_\{T,n\}=n\. Markov’s inequality and \([96](https://arxiv.org/html/2610.12865#A1.E96)\) give

ℙ⁡\(τn≤T\)≤𝔼​NT,nn⟶0\.\\mathbb\{P\}\(\\tau\_\{n\}\\leq T\)\\leq\\frac\{\\mathbb\{E\}N\_\{T,n\}\}\{n\}\\longrightarrow 0\.Applying this argument to every positive integer horizon provesτn↑∞\\tau\_\{n\}\\uparrow\\inftyalmost surely\. For each fixedTT, the truncated and original path laws agree on\{τn\>T\}\\\{\\tau\_\{n\}\>T\\\}and hence converge in total variation on\[0,T\]\[0,T\]\. Lower semicontinuity of Fisher information and Fatou’s lemma pass \([19](https://arxiv.org/html/2610.12865#A1.E19)\)\. Monotone convergence passes \([96](https://arxiv.org/html/2610.12865#A1.E96)\) and proves \([20](https://arxiv.org/html/2610.12865#A1.E20)\)\. Because the step durations are strictly positive, the definition ofNTN\_\{T\}in Definition[A\.1](https://arxiv.org/html/2610.12865#A1.Thmtheorem1)now agrees with the number of occupied intervals\#​\{k:τk<T\}\\\#\\\{k:\\tau\_\{k\}<T\\\}\.

##### General finite KL initialization\.

We now remove the temporary compact support assumption onμ0\\mu\_\{0\}and extend the preceding bounds to any initial law withK0=KL\(μ0∥π\)<∞K\_\{0\}=\\mathrm\{KL\}\(\\mu\_\{0\}\\\|\\pi\)<\\infty, as required by the theorem\. We approximateμ0\\mu\_\{0\}by conditioning it on expanding balls and then pass to the limit\. Keeph,Th,Tfixed and use the same update rule and Brownian driving noise throughout\. The base step constraint in Theorem[A\.4](https://arxiv.org/html/2610.12865#A1.Thmtheorem4)does not depend on the initial law, so it remains valid for every approximating initialization\.

Choose increasing ballsARA\_\{R\}withpR:=μ0​\(AR\)\>0p\_\{R\}:=\\mu\_\{0\}\(A\_\{R\}\)\>0andpR↑1p\_\{R\}\\uparrow 1, and setμ0R:=μ0\(⋅∣AR\)\\mu\_\{0\}^\{R\}:=\\mu\_\{0\}\(\\cdot\\mid A\_\{R\}\)\. Iff:=d​μ0/d​πf:=\\mathrm\{d\}\\mu\_\{0\}/\\mathrm\{d\}\\pi, then

KL\(μ0R∥π\)=pR−1∫ARflogfdπ−logpR⟶K0\.\\mathrm\{KL\}\(\\mu\_\{0\}^\{R\}\\\|\\pi\)=p\_\{R\}^\{\-1\}\\int\_\{A\_\{R\}\}f\\log f\\,\\mathrm\{d\}\\pi\-\\log p\_\{R\}\\longrightarrow K\_\{0\}\.The negative part is integrable becausez​log⁡z≥−e−1z\\log z\\geq\-e^\{\-1\}, andK0<∞K\_\{0\}<\\inftyensures that the positive part is integrable as well\. This justifies the convergence of the restricted integrals\. Apply the compact support argument to eachμ0R\\mu\_\{0\}^\{R\}, usingKL\(μ0R∥π\)\\mathrm\{KL\}\(\\mu\_\{0\}^\{R\}\\\|\\pi\)in place ofK0K\_\{0\}\. Conditioning on the initial state gives

ℙμ0​\(τ∞<∞,X0∈AR\)=0\.\\mathbb\{P\}\_\{\\mu\_\{0\}\}\(\\tau\_\{\\infty\}<\\infty,\\ X\_\{0\}\\in A\_\{R\}\)=0\.Since the balls exhaust the initial law,τ∞=∞\\tau\_\{\\infty\}=\\inftyalmost surely also underμ0\\mu\_\{0\}\. The conditional law of the interpolated path given its initial state defines a probability kernel, defined arbitrarily on any exceptionalμ0\\mu\_\{0\}\-null set of initial states\. Applying this same kernel toμ0R\\mu\_\{0\}^\{R\}andμ0\\mu\_\{0\}gives convergence of their path laws in total variation on every finite time horizon\. Lower semicontinuity of Fisher information and Fatou’s lemma then pass the integrated Fisher bound \([19](https://arxiv.org/html/2610.12865#A1.E19)\)\. Its right side converges becauseKL\(μ0R∥π\)→K0\\mathrm\{KL\}\(\\mu\_\{0\}^\{R\}\\\|\\pi\)\\to K\_\{0\}andRγ​\(h,M\)R\_\{\\gamma\}\(h,M\)is continuous inMM\. Furthermore,

pR𝔼μ0RNT=𝔼μ0\[NT𝟏\{X0∈AR\}\]↑𝔼μ0NT\.p\_\{R\}\\mathbb\{E\}\_\{\\mu\_\{0\}^\{R\}\}N\_\{T\}=\\mathbb\{E\}\_\{\\mu\_\{0\}\}\[N\_\{T\}\\mathbf\{1\}\_\{\\\{X\_\{0\}\\in A\_\{R\}\\\}\}\]\\uparrow\\mathbb\{E\}\_\{\\mu\_\{0\}\}N\_\{T\}\.Here monotone convergence applies toNT𝟏\{X0∈AR\}N\_\{T\}\\mathbf\{1\}\_\{\\\{X\_\{0\}\\in A\_\{R\}\\\}\}\. SincepR→1p\_\{R\}\\to 1and the right side of \([20](https://arxiv.org/html/2610.12865#A1.E20)\) is continuous in the initial KL divergence, the expected iteration bound passes to the limit as well\. Convexity and lower semicontinuity of Fisher information follow, for example, from

FI\(μ∥π\)=supφ∈Cc∞​\(ℝd,ℝd\)∫\{2⟨∇V,φ⟩−2divφ−∥φ∥2\}dμ\.\\mathrm\{FI\}\(\\mu\\\|\\pi\)=\\sup\_\{\\varphi\\in C\_\{c\}^\{\\infty\}\(\\mathbb\{R\}^\{d\};\\mathbb\{R\}^\{d\}\)\}\\int\\left\\\{2\\langle\\nabla V,\\varphi\\rangle\-2\\operatorname\{div\}\\varphi\-\\\|\\varphi\\\|^\{2\}\\right\\\}\\,\\mathrm\{d\}\\mu\.This also passes the time\-averaged\-law bound\. Finally, each occupied interval uses exactly one exact gradient query at its start and no unused terminal query\. HenceQT=NTQ\_\{T\}=N\_\{T\}pathwise\. This completes the proof of all finite horizon claims\. ∎

#### A\.7\.7Accuracy schedule and complexity

We finish by verifying the explicit schedule in the full theorem and deriving the stated complexity bounds\.

###### Proof of the accuracy and complexity claims in Theorem[A\.4](https://arxiv.org/html/2610.12865#A1.Thmtheorem4)\.

AssumeK0\>0K\_\{0\}\>0and0<ε≤10<\\varepsilon\\leq 1\. Recall the accuracy schedule

T:=4​K0ε,h:=max⁡\{u∈\[0,\(1\+2​Δγ​\(1\)\)−1\]:Rγ​\(u,MT\)≤3​ε16\},T:=\\frac\{4K\_\{0\}\}\{\\varepsilon\},\\qquad h:=\\max\\left\\\{u\\in\[0,\(1\+2\\Delta\_\{\\gamma\}\(1\)\)^\{\-1\}\]:R\_\{\\gamma\}\(u,M\_\{T\}\)\\leq\\frac\{3\\varepsilon\}\{16\}\\right\\\},whereMT=1\+Cγ\+K0/TM\_\{T\}=1\+C\_\{\\gamma\}\+K\_\{0\}/T\. Hence

K0T=ε4,MT=1\+Cγ\+ε4,Rγ​\(h,MT\)≤3​ε16\.\\frac\{K\_\{0\}\}\{T\}=\\frac\{\\varepsilon\}\{4\},\\qquad M\_\{T\}=1\+C\_\{\\gamma\}\+\\frac\{\\varepsilon\}\{4\},\\qquad R\_\{\\gamma\}\(h,M\_\{T\}\)\\leq\\frac\{3\\varepsilon\}\{16\}\.The set defininghhhas a positive maximum becauseRγ​\(⋅,MT\)R\_\{\\gamma\}\(\\cdot,M\_\{T\}\)is continuous, nondecreasing, vanishes at zero, and the threshold is positive\. SinceMT≥1M\_\{T\}\\geq 1and all coefficients are nonnegative,

Rγ​\(h,MT\)−Δγ​\(h\)=Aγ​\(h\)\+8​Bγ​\(h\)​\(MT−1\)\+8γ/2​Qγ​\(h\)​\(MTγ/2−1\)≥0\.R\_\{\\gamma\}\(h,M\_\{T\}\)\-\\Delta\_\{\\gamma\}\(h\)=A\_\{\\gamma\}\(h\)\+8B\_\{\\gamma\}\(h\)\(M\_\{T\}\-1\)\+8^\{\\gamma/2\}Q\_\{\\gamma\}\(h\)\(M\_\{T\}^\{\\gamma/2\}\-1\)\\geq 0\.\(97\)Therefore

Δγ​\(h\)≤3​ε16<12\.\\Delta\_\{\\gamma\}\(h\)\\leq\\frac\{3\\varepsilon\}\{16\}<\\frac\{1\}\{2\}\.The finite horizon bound then gives the slightly sharper intermediate result

FI\(μ¯T∥π\)≤ε3\+833​ε16=5​ε6≤ε\.\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\frac\{\\varepsilon\}\{3\}\+\\frac\{8\}\{3\}\\frac\{3\\varepsilon\}\{16\}=\\frac\{5\\varepsilon\}\{6\}\\leq\\varepsilon\.
Equations \([93](https://arxiv.org/html/2610.12865#A1.E93)\), \([97](https://arxiv.org/html/2610.12865#A1.E97)\), and \([91](https://arxiv.org/html/2610.12865#A1.E91)\) giveeT≤3​ε/8e\_\{T\}\\leq 3\\varepsilon/8andmT≤16​MTm\_\{T\}\\leq 16M\_\{T\}\. Since16γ/2≤416^\{\\gamma/2\}\\leq 4, the expected count bound becomes

𝔼​NT≤1\+4​K0ε​h​\(1\+4​ω​MTγ/2\),\\mathbb\{E\}N\_\{T\}\\leq 1\+\\frac\{4K\_\{0\}\}\{\\varepsilon h\}\\left\(1\+4\\omega M\_\{T\}^\{\\gamma/2\}\\right\),\(98\)which is \([22](https://arxiv.org/html/2610.12865#A1.E22)\)\.

##### The regimeL1\>0L\_\{1\}\>0and0<γ<10<\\gamma<1\.

The formulas forAγ,Bγ,QγA\_\{\\gamma\},B\_\{\\gamma\},Q\_\{\\gamma\}in the proof of Lemma[A\.21](https://arxiv.org/html/2610.12865#A1.Thmtheorem21)andRγ​\(u,MT\)=Aγ​\(u\)\+8​Bγ​\(u\)​MT\+8γ/2​Qγ​\(u\)​MTγ/2R\_\{\\gamma\}\(u,M\_\{T\}\)=A\_\{\\gamma\}\(u\)\+8B\_\{\\gamma\}\(u\)M\_\{T\}\+8^\{\\gamma/2\}Q\_\{\\gamma\}\(u\)M\_\{T\}^\{\\gamma/2\}give, asu↓0u\\downarrow 0,

Rγ​\(u,MT\)\\displaystyle R\_\{\\gamma\}\(u,M\_\{T\}\)=3​d​\(L02​\(1\+2γ2/\(1−γ\)\)2\+8γ/2​22​γ2/\(1−γ\)​32​γ​L1​\(1\+L0\)​MTγ/2\)​u\+o⁡\(u\),\\displaystyle=3d\\left\(L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}\+8^\{\\gamma/2\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}3^\{2\\gamma\}L\_\{1\}\(1\+L\_\{0\}\)M\_\{T\}^\{\\gamma/2\}\\right\)u\+o\(u\),uniformly for0<ε≤10<\\varepsilon\\leq 1\. Maximality ofhhtherefore implies, for sufficiently smallε\\varepsilonat fixed model parameters,

h−1=𝒪⁡\(1\+d⁡\[L02\+L1​\(1\+L0\)​MTγ/2\]ε\)\.h^\{\-1\}=\\mathcal\{O\}\\\!\\left\(1\+\\frac\{d\[L\_\{0\}^\{2\}\+L\_\{1\}\(1\+L\_\{0\}\)M\_\{T\}^\{\\gamma/2\}\]\}\{\\varepsilon\}\\right\)\.\(99\)In this regime, the numerical constants hidden by𝒪\\mathcal\{O\}depend only onγ\\gamma\. To justify this inversion explicitly, the positive linear coefficient above boundsRγ​\(u,MT\)/uR\_\{\\gamma\}\(u,M\_\{T\}\)/ufrom above by twice that coefficient onceuuis sufficiently small\. A step proportional toε\\varepsilondivided by that coefficient is consequently admissible and, for sufficiently smallε\\varepsilon, lies below the fixed upper bound in \([21](https://arxiv.org/html/2610.12865#A1.E21)\)\. Hence the maximal admissible step is at least as large\. The threshold onε\\varepsilonmay depend on the fixed smoothness parameters and dimension\. Combining \([98](https://arxiv.org/html/2610.12865#A1.E98)\) and \([99](https://arxiv.org/html/2610.12865#A1.E99)\), and recallingω=L1/\(1\+L0\)\\omega=L\_\{1\}/\(1\+L\_\{0\}\), bounds the leadingε−2\\varepsilon^\{\-2\}term by

𝒪⁡\(K0​d\)ε2\\displaystyle\\frac\{\\mathcal\{O\}\(K\_\{0\}d\)\}\{\\varepsilon^\{2\}\}\(1\+L11\+L0​MTγ/2\)​\(L02\+L1​\(1\+L0\)​MTγ/2\)\.\\displaystyle\\left\(1\+\\frac\{L\_\{1\}\}\{1\+L\_\{0\}\}M\_\{T\}^\{\\gamma/2\}\\right\)\\left\(L\_\{0\}^\{2\}\+L\_\{1\}\(1\+L\_\{0\}\)M\_\{T\}^\{\\gamma/2\}\\right\)\.\(100\)BecauseL02/\(1\+L0\)≤1\+L0L\_\{0\}^\{2\}/\(1\+L\_\{0\}\)\\leq 1\+L\_\{0\}, the product is bounded, up to a constant, by

L02\+L1​\(1\+L0\)​MTγ/2\+L12​MTγ\.L\_\{0\}^\{2\}\+L\_\{1\}\(1\+L\_\{0\}\)M\_\{T\}^\{\\gamma/2\}\+L\_\{1\}^\{2\}M\_\{T\}^\{\\gamma\}\.WithSγ:=1\+d​L0\+\(d​L1\)2/\(2−γ\)S\_\{\\gamma\}:=1\+dL\_\{0\}\+\(dL\_\{1\}\)^\{2/\(2\-\\gamma\)\}, the definition ofCγC\_\{\\gamma\}givesMT=𝒪⁡\(Sγ\)M\_\{T\}=\\mathcal\{O\}\(S\_\{\\gamma\}\)\. Consequently,

𝔼​NT=𝔼​QT=𝒪⁡\(K0​dε2​\[L02\+L1​\(1\+L0\)​Sγγ/2\+L12​Sγγ\]\)\.\\mathbb\{E\}N\_\{T\}=\\mathbb\{E\}Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{K\_\{0\}d\}\{\\varepsilon^\{2\}\}\\left\[L\_\{0\}^\{2\}\+L\_\{1\}\(1\+L\_\{0\}\)S\_\{\\gamma\}^\{\\gamma/2\}\+L\_\{1\}^\{2\}S\_\{\\gamma\}^\{\\gamma\}\\right\]\\right\)\.Using\(x\+y\)a≤xa\+ya\(x\+y\)^\{a\}\\leq x^\{a\}\+y^\{a\}forx,y≥0x,y\\geq 0and0<a≤10<a\\leq 1, we can further bound the expression in brackets by

L02\+L1​\(1\+L0\)​\{1\+\(d​L0\)γ/2\+\(d​L1\)γ/\(2−γ\)\}\+L12​\{1\+\(d​L0\)γ\+\(d​L1\)2​γ/\(2−γ\)\}\.\\displaystyle L\_\{0\}^\{2\}\+L\_\{1\}\(1\+L\_\{0\}\)\\\{1\+\(dL\_\{0\}\)^\{\\gamma/2\}\+\(dL\_\{1\}\)^\{\\gamma/\(2\-\\gamma\)\}\\\}\+L\_\{1\}^\{2\}\\\{1\+\(dL\_\{0\}\)^\{\\gamma\}\+\(dL\_\{1\}\)^\{2\\gamma/\(2\-\\gamma\)\}\\\}\.

##### The endpointγ=1\\gamma=1\.

This endpoint is proved directly\. SinceE1ex​\(u\)=1\+𝒪⁡\(u\)E\_\{1\}^\{\\mathrm\{ex\}\}\(u\)=1\+\\mathcal\{O\}\(\\sqrt\{u\}\), the formulas forA1,B1,Q1A\_\{1\},B\_\{1\},Q\_\{1\}in the proof of the discretization error bound in Lemma[A\.21](https://arxiv.org/html/2610.12865#A1.Thmtheorem21)give

R1​\(u,MT\)\\displaystyle R\_\{1\}\(u,M\_\{T\}\)=\(d\+1\)​\(4​L02\+8​2​L1​\(1\+L0\)​MT1/2\)​u\+o⁡\(u\)\.\\displaystyle=\(d\+1\)\\left\(4L\_\{0\}^\{2\}\+8\\sqrt\{2\}L\_\{1\}\(1\+L\_\{0\}\)M\_\{T\}^\{1/2\}\\right\)u\+o\(u\)\.Consequently,

h−1=𝒪⁡\(1\+\(d\+1\)​\[L02\+L1​\(1\+L0\)​MT1/2\]ε\)\.h^\{\-1\}=\\mathcal\{O\}\\\!\\left\(1\+\\frac\{\(d\+1\)\[L\_\{0\}^\{2\}\+L\_\{1\}\(1\+L\_\{0\}\)M\_\{T\}^\{1/2\}\]\}\{\\varepsilon\}\\right\)\.Applying the same product calculation as in \([100](https://arxiv.org/html/2610.12865#A1.E100)\) and usingS1=1\+d​L0\+d2​L12S\_\{1\}=1\+dL\_\{0\}\+d^\{2\}L\_\{1\}^\{2\}gives

𝔼​NT=𝔼​QT=𝒪⁡\(K0​\(d\+1\)ε2​\[L02\+L1​\(1\+L0\)​S11/2\+L12​S1\]\)\.\\mathbb\{E\}N\_\{T\}=\\mathbb\{E\}Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{K\_\{0\}\(d\+1\)\}\{\\varepsilon^\{2\}\}\\left\[L\_\{0\}^\{2\}\+L\_\{1\}\(1\+L\_\{0\}\)S\_\{1\}^\{1/2\}\+L\_\{1\}^\{2\}S\_\{1\}\\right\]\\right\)\.This establishes the endpoint directly, without taking a limit in theγ<1\\gamma<1constants\.

RecallΛ:=max⁡\{1,L0,L1\}\\Lambda:=\\max\\\{1,L\_\{0\},L\_\{1\}\\\}\. For0<γ<10<\\gamma<1,Sγ=𝒪⁡\(\(d​Λ\)2/\(2−γ\)\)S\_\{\\gamma\}=\\mathcal\{O\}\(\(d\\Lambda\)^\{2/\(2\-\\gamma\)\}\), and atγ=1\\gamma=1,d\+1≤2​dd\+1\\leq 2d\. In the expanded bounds, the largest powers are supplied byL12​SγγL\_\{1\}^\{2\}S\_\{\\gamma\}^\{\\gamma\}, giving

𝔼​NT=𝔼​QT=𝒪⁡\(K0​Λ4/\(2−γ\)​d\(2\+γ\)/\(2−γ\)ε2\)\.\\mathbb\{E\}N\_\{T\}=\\mathbb\{E\}Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{K\_\{0\}\\Lambda^\{4/\(2\-\\gamma\)\}d^\{\(2\+\\gamma\)/\(2\-\\gamma\)\}\}\{\\varepsilon^\{2\}\}\\right\)\.The hidden numerical constant depends only onγ\\gamma\. This proves the accuracy and complexity claims in Theorem[A\.4](https://arxiv.org/html/2610.12865#A1.Thmtheorem4)\. ∎

### A\.8Proof of Corollary[A\.5](https://arxiv.org/html/2610.12865#A1.Thmtheorem5)

###### Proof\.

The first item follows from \([23](https://arxiv.org/html/2610.12865#A1.E23)\) after imposingK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)\.

##### The globallyL0L\_\{0\}\-smooth caseL1=0L\_\{1\}=0\.

Hereω=0\\omega=0, soHk=hH\_\{k\}=hdeterministically\. Recall thatAγ​\(h\)=d​L02​hA\_\{\\gamma\}\(h\)=dL\_\{0\}^\{2\}h,Bγ​\(h\)=L02​h2/3B\_\{\\gamma\}\(h\)=L\_\{0\}^\{2\}h^\{2\}/3, andQγ​\(h\)=0Q\_\{\\gamma\}\(h\)=0in this regime\. Substituting these intoRγ​\(h,MT\)=Aγ​\(h\)\+8​Bγ​\(h\)​MT\+8γ/2​Qγ​\(h\)​MTγ/2R\_\{\\gamma\}\(h,M\_\{T\}\)=A\_\{\\gamma\}\(h\)\+8B\_\{\\gamma\}\(h\)M\_\{T\}\+8^\{\\gamma/2\}Q\_\{\\gamma\}\(h\)M\_\{T\}^\{\\gamma/2\}gives, independently ofγ\\gamma,

Rγ​\(h,MT\)=d​L02​h\+83​L02​MT​h2\.R\_\{\\gamma\}\(h,M\_\{T\}\)=dL\_\{0\}^\{2\}h\+\\frac\{8\}\{3\}L\_\{0\}^\{2\}M\_\{T\}h^\{2\}\.\(101\)LetM¯:=1\+4​d​L0\+1/4\\overline\{M\}:=1\+4dL\_\{0\}\+1/4\. Whenever

ε≤min⁡\{1,32​d​L023,4​d2​L02M¯\},\\varepsilon\\leq\\min\\left\\\{1,\\frac\{32dL\_\{0\}^\{2\}\}\{3\},\\frac\{4d^\{2\}L\_\{0\}^\{2\}\}\{\\overline\{M\}\}\\right\\\},the candidateu0=3​ε/\(32​d​L02\)u\_\{0\}=3\\varepsilon/\(32dL\_\{0\}^\{2\}\)belongs to\[0,1\]\[0,1\]and makes each term on the right of \([101](https://arxiv.org/html/2610.12865#A1.E101)\) at most3​ε/323\\varepsilon/32\. Maximality in \([21](https://arxiv.org/html/2610.12865#A1.E21)\) therefore gives

h≥u0=3​ε32​d​L02\.h\\geq u\_\{0\}=\\frac\{3\\varepsilon\}\{32dL\_\{0\}^\{2\}\}\.The count is deterministic and equalsNT=⌈T/h⌉N\_\{T\}=\\lceil T/h\\rceil\. Hence

QT=NT≤1\+128​K0​d​L023​ε2,Q\_\{T\}=N\_\{T\}\\leq 1\+\\frac\{128K\_\{0\}dL\_\{0\}^\{2\}\}\{3\\varepsilon^\{2\}\},which proves \([24](https://arxiv.org/html/2610.12865#A1.E24)\)\. UnderK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\), this is𝒪⁡\(d2​L02/ε2\)\\mathcal\{O\}\(d^\{2\}L\_\{0\}^\{2\}/\\varepsilon^\{2\}\), recovering the globally smooth exact gradient LMC order\. Neither the algorithm nor this specialization of its accuracy schedule depends onγ\\gamma\. ∎

### A\.9Proof of Theorem[A\.6](https://arxiv.org/html/2610.12865#A1.Thmtheorem6)

The robust estimator and the adaptive step size control different errors\. The former turns a finite\-ppstochastic oracle into a gradient estimate with a finite conditional second moment, while the latter controls the variation of∇V\\nabla Vduring an iteration\. The adaptive interval length depends on the same gradient estimator used in the update\. Consequently, controlling the estimator error at each iteration does not directly control its accumulation over time\. The termκ​sp​\(Xτk\)2\\kappa s\_\{p\}\(X\_\{\\tau\_\{k\}\}\)^\{2\}in the step\-size denominator allows us to bound this accumulated error through \([112](https://arxiv.org/html/2610.12865#A1.E112)\)\. We first bound the discretization and estimation errors, then derive the iteration and oracle complexities\.

#### A\.9\.1Finite iteration localization and proof quantities

We first suppose that the initial law is supported in a ball, while retaining its actual finite KL divergenceK0:=KL\(μ0∥π\)K\_\{0\}:=\\mathrm\{KL\}\(\\mu\_\{0\}\\\|\\pi\), and truncate the RA\-LMC construction aftermmiterations\. If the truncation timeτm\\tau\_\{m\}occurs beforeTT, we continue with the conservative exact Langevin diffusion from Lemma[A\.17](https://arxiv.org/html/2610.12865#A1.Thmtheorem17)\. This continuation is only a proof device and is not part of RA\-LMC\. The state at the truncation time has an absolutely continuous law because its last Brownian increment is nondegenerate\.

On an RA\-LMC interval beginning atτk\\tau\_\{k\}, use the force from \([8](https://arxiv.org/html/2610.12865#S3.E8)\) and write

Ftfr:=Fk=−1b∑i=1b𝒢^k,i,Xtfr:=Xτk,t∈\[τk,τk\+1\),F\_\{t\}^\{\\mathrm\{fr\}\}:=F\_\{k\}=\-\\frac\{1\}\{b\}\\sum\_\{i=1\}^\{b\}\\widehat\{\\mathcal\{G\}\}\_\{k,i\},\\qquad X\_\{t\}^\{\\mathrm\{fr\}\}:=X\_\{\\tau\_\{k\}\},\\qquad t\\in\[\\tau\_\{k\},\\tau\_\{k\+1\}\),HereFtfrF\_\{t\}^\{\\mathrm\{fr\}\}andXtfrX\_\{t\}^\{\\mathrm\{fr\}\}denote the drift and state selected at the beginning of the interval and held constant throughout it\. After the truncation time, setFtfr=−∇V​\(Xt\)F\_\{t\}^\{\\mathrm\{fr\}\}=\-\\nabla V\(X\_\{t\}\)andXtfr=XtX\_\{t\}^\{\\mathrm\{fr\}\}=X\_\{t\}\. Ifτk<T\\tau\_\{k\}<T, letℓk:=min⁡\{Hk,T−τk\}\\ell\_\{k\}:=\\min\\\{H\_\{k\},T\-\\tau\_\{k\}\\\}be the duration of that update beforeTT\. Recall the adaptive rule and the definition ofω\\omega,

Hk=h1\+ω​‖Fk‖γ\+κ​sp​\(Xτk\)2,ω:=L11\+L0\.H\_\{k\}=\\frac\{h\}\{1\+\\omega\\\|F\_\{k\}\\\|^\{\\gamma\}\+\\kappa s\_\{p\}\(X\_\{\\tau\_\{k\}\}\)^\{2\}\},\\qquad\\omega:=\\frac\{L\_\{1\}\}\{1\+L\_\{0\}\}\.They imply

Hk≤h,ω​Hk​‖Fk‖γ≤h,κ​Hk​sp​\(Xτk\)2≤h\.H\_\{k\}\\leq h,\\qquad\\omega H\_\{k\}\\\|F\_\{k\}\\\|^\{\\gamma\}\\leq h,\\qquad\\kappa H\_\{k\}s\_\{p\}\(X\_\{\\tau\_\{k\}\}\)^\{2\}\\leq h\.\(102\)Letℱk\\mathcal\{F\}\_\{k\}denote the history available immediately before the fresh oracle batch at iterationkkis drawn\. In particular,XτkX\_\{\\tau\_\{k\}\}andτk\\tau\_\{k\}areℱk\\mathcal\{F\}\_\{k\}\-measurable, whereasFkF\_\{k\}andHkH\_\{k\}also depend on the fresh batch\. The clipping radius isλk=sp​\(Xτk\)​b1/p\\lambda\_\{k\}=s\_\{p\}\(X\_\{\\tau\_\{k\}\}\)b^\{1/p\}\. Lemma[A\.9](https://arxiv.org/html/2610.12865#A1.Thmtheorem9)gives

𝔼\[∥Fk\+∇V\(Xτk\)∥2\|ℱk\]≤ρbsp\(Xτk\)2,ρb=Cpb−2\(p−1\)/p\.\\mathbb\{E\}\\\!\\left\[\\\|F\_\{k\}\+\\nabla V\(X\_\{\\tau\_\{k\}\}\)\\\|^\{2\}\\,\\middle\|\\,\\mathcal\{F\}\_\{k\}\\right\]\\leq\\rho\_\{b\}s\_\{p\}\(X\_\{\\tau\_\{k\}\}\)^\{2\},\\qquad\\rho\_\{b\}=C\_\{p\}b^\{\-2\(p\-1\)/p\}\.\(103\)HereCpC\_\{p\}is the estimator error coefficient established in Lemma[A\.9](https://arxiv.org/html/2610.12865#A1.Thmtheorem9)\. The moment justification proceeds in a fixed order\. Bounded initial support gives finite initial state and gradient second moments\. The conditional estimation bound above andsp​\(x\)2≤C⁡\(1\+‖x−xref‖2\)s\_\{p\}\(x\)^\{2\}\\leq C\(1\+\\\|x\-x\_\{\\mathrm\{ref\}\}\\\|^\{2\}\)give a finite second moment for each selected force\. The gradient variation bounds in Lemma[A\.15](https://arxiv.org/html/2610.12865#A1.Thmtheorem15), together with the adaptive step bound, then propagate finite state and gradient second moments through any finite number of updates\. ForL1\>0L\_\{1\}\>0and0<γ<10<\\gamma<1, the adaptive step bound gives

\(Hk​‖Fk‖\)2/\(1−γ\)≤\(h/ω\)2/\(1−γ\)​‖Fk‖2\.\(H\_\{k\}\\\|F\_\{k\}\\\|\)^\{2/\(1\-\\gamma\)\}\\leq\(h/\\omega\)^\{2/\(1\-\\gamma\)\}\\\|F\_\{k\}\\\|^\{2\}\.Thus the required displacement moment is controlled by the force second moment\. ForL1\>0L\_\{1\}\>0andγ=1\\gamma=1, the drift displacement is bounded byh/ωh/\\omega, and Gaussian exponential moments give the same conclusion\. WhenL1=0L\_\{1\}=0, global smoothness suffices\.

Thus the expected integral of the squared drift error is finite before truncation, and the drift error is zero on the exact continuation\. Lemma[A\.18](https://arxiv.org/html/2610.12865#A1.Thmtheorem18)gives finite integrated FI\. Lemma[A\.16](https://arxiv.org/html/2610.12865#A1.Thmtheorem16)then gives a finite integral of the gradient second moment over time, including on the continuation\. The path identity yields the bounds on expected squared distances fromxrefx\_\{\\mathrm\{ref\}\}used below\. This order avoids assuming the moment bounds that the subsequent closure will prove\.

#### A\.9\.2Discretization, FI, and gradient and drift moment bounds

For the process constructed above, which follows RA\-LMC for at mostmmiterations and then continues with the exact Langevin diffusion if needed, define the following quantities\.

FI¯T\\displaystyle\\overline\{\\mathrm\{FI\}\}\_\{T\}:=1T∫0TFI\(μt∥π\)dt,\\displaystyle:=\\frac\{1\}\{T\}\\int\_\{0\}^\{T\}\\mathrm\{FI\}\(\\mu\_\{t\}\\\|\\pi\)\\,\\mathrm\{d\}t,e¯\\displaystyle\\overline\{e\}:=1T​𝔼​∫0T‖Ftfr\+∇V​\(Xtfr\)‖2​𝑑t,\\displaystyle:=\\frac\{1\}\{T\}\\mathbb\{E\}\\int\_\{0\}^\{T\}\\\|F\_\{t\}^\{\\mathrm\{fr\}\}\+\\nabla V\(X\_\{t\}^\{\\mathrm\{fr\}\}\)\\\|^\{2\}\\,\\mathrm\{d\}t,g¯\\displaystyle\\overline\{g\}:=1T​𝔼​∫0T‖∇V​\(Xtfr\)‖2​𝑑t,\\displaystyle:=\\frac\{1\}\{T\}\\mathbb\{E\}\\int\_\{0\}^\{T\}\\\|\\nabla V\(X\_\{t\}^\{\\mathrm\{fr\}\}\)\\\|^\{2\}\\,\\mathrm\{d\}t,a¯\\displaystyle\\overline\{a\}:=1T​𝔼​∫0T‖Ftfr‖2​𝑑t,\\displaystyle:=\\frac\{1\}\{T\}\\mathbb\{E\}\\int\_\{0\}^\{T\}\\\|F\_\{t\}^\{\\mathrm\{fr\}\}\\\|^\{2\}\\,\\mathrm\{d\}t,Ddisc\\displaystyle D\_\{\\mathrm\{disc\}\}:=𝔼​∫0T‖∇V​\(Xt\)−∇V​\(Xtfr\)‖2​𝑑t\.\\displaystyle:=\\mathbb\{E\}\\int\_\{0\}^\{T\}\\\|\\nabla V\(X\_\{t\}\)\-\\nabla V\(X\_\{t\}^\{\\mathrm\{fr\}\}\)\\\|^\{2\}\\,\\mathrm\{d\}t\.These quantities represent, respectively, FI, the average squared estimation error, the average squared norm of∇V​\(Xtfr\)\\nabla V\(X\_\{t\}^\{\\mathrm\{fr\}\}\), the average squared drift norm, and the accumulated discretization error\.

Condition after the gradient estimate and step size have been selected but before the Brownian increment is generated\. By \([102](https://arxiv.org/html/2610.12865#A1.E102)\),ℓk≤h/\(1\+ω​‖Fk‖γ\)\\ell\_\{k\}\\leq h/\(1\+\\omega\\\|F\_\{k\}\\\|^\{\\gamma\}\), so Lemma[A\.20](https://arxiv.org/html/2610.12865#A1.Thmtheorem20)applies\. Summing its local bound over the occupied intervals gives

DdiscT≤A~γ​\(h\)\+U~γ​\(h\)​g¯\+V~γ​\(h\)​a¯\.\\frac\{D\_\{\\mathrm\{disc\}\}\}\{T\}\\leq\\widetilde\{A\}\_\{\\gamma\}\(h\)\+\\widetilde\{U\}\_\{\\gamma\}\(h\)\\overline\{g\}\+\\widetilde\{V\}\_\{\\gamma\}\(h\)\\overline\{a\}\.HereA~γ\\widetilde\{A\}\_\{\\gamma\},U~γ\\widetilde\{U\}\_\{\\gamma\}, andV~γ\\widetilde\{V\}\_\{\\gamma\}are the coefficients defined above for Lemma[A\.20](https://arxiv.org/html/2610.12865#A1.Thmtheorem20)\. They multiply the constant term, the average squared gradient normg¯\\overline\{g\}, and the average squared drift norma¯\\overline\{a\}, respectively\. Sincea¯≤2​e¯\+2​g¯\\overline\{a\}\\leq 2\\overline\{e\}\+2\\overline\{g\},

DdiscT≤A~γ​\(h\)\+\{U~γ​\(h\)\+2​V~γ​\(h\)\}​g¯\+2​V~γ​\(h\)​e¯\.\\frac\{D\_\{\\mathrm\{disc\}\}\}\{T\}\\leq\\widetilde\{A\}\_\{\\gamma\}\(h\)\+\\\{\\widetilde\{U\}\_\{\\gamma\}\(h\)\+2\\widetilde\{V\}\_\{\\gamma\}\(h\)\\\}\\overline\{g\}\+2\\widetilde\{V\}\_\{\\gamma\}\(h\)\\overline\{e\}\.\(104\)
To boundg¯\\overline\{g\}, compare∇V​\(Xtfr\)\\nabla V\(X\_\{t\}^\{\\mathrm\{fr\}\}\)with the current gradient∇V​\(Xt\)\\nabla V\(X\_\{t\}\)\.

g¯≤2​DdiscT\+2T​∫0T𝔼​‖∇V​\(Xt\)‖2​𝑑t\.\\overline\{g\}\\leq\\frac\{2D\_\{\\mathrm\{disc\}\}\}\{T\}\+\\frac\{2\}\{T\}\\int\_\{0\}^\{T\}\\mathbb\{E\}\\\|\\nabla V\(X\_\{t\}\)\\\|^\{2\}\\,\\mathrm\{d\}t\.The gradient moment bound in Lemma[A\.16](https://arxiv.org/html/2610.12865#A1.Thmtheorem16)uses

Cγ:=4​d​L0\+\(2−γ\)​γγ/\(2−γ\)​\(2​d​L1\)2/\(2−γ\)\.C\_\{\\gamma\}:=4dL\_\{0\}\+\(2\-\\gamma\)\\gamma^\{\\gamma/\(2\-\\gamma\)\}\(2dL\_\{1\}\)^\{2/\(2\-\\gamma\)\}\.That lemma and \([104](https://arxiv.org/html/2610.12865#A1.E104)\) therefore yield

\[1−2​\{U~γ​\(h\)\+2​V~γ​\(h\)\}\]​g¯≤4​FI¯T\+2​Cγ\+2​A~γ​\(h\)\+4​V~γ​\(h\)​e¯\.\\displaystyle\[1\-2\\\{\\widetilde\{U\}\_\{\\gamma\}\(h\)\+2\\widetilde\{V\}\_\{\\gamma\}\(h\)\\\}\]\\overline\{g\}\\leq 4\\overline\{\\mathrm\{FI\}\}\_\{T\}\+2C\_\{\\gamma\}\+2\\widetilde\{A\}\_\{\\gamma\}\(h\)\+4\\widetilde\{V\}\_\{\\gamma\}\(h\)\\overline\{e\}\.\(105\)Next apply the KL\-dissipation inequality in Lemma[A\.18](https://arxiv.org/html/2610.12865#A1.Thmtheorem18)and use

‖∇V​\(Xt\)\+Ftfr‖2≤2​‖∇V​\(Xt\)−∇V​\(Xtfr\)‖2\+2​‖∇V​\(Xtfr\)\+Ftfr‖2\.\\displaystyle\\\|\\nabla V\(X\_\{t\}\)\+F\_\{t\}^\{\\mathrm\{fr\}\}\\\|^\{2\}\\leq 2\\\|\\nabla V\(X\_\{t\}\)\-\\nabla V\(X\_\{t\}^\{\\mathrm\{fr\}\}\)\\\|^\{2\}\+2\\\|\\nabla V\(X\_\{t\}^\{\\mathrm\{fr\}\}\)\+F\_\{t\}^\{\\mathrm\{fr\}\}\\\|^\{2\}\.Together with \([104](https://arxiv.org/html/2610.12865#A1.E104)\), this gives

34​FI¯T≤K0T\+2​A~γ​\(h\)\+2​\{U~γ​\(h\)\+2​V~γ​\(h\)\}​g¯\+\{4​V~γ​\(h\)\+2\}​e¯\.\\displaystyle\\frac\{3\}\{4\}\\overline\{\\mathrm\{FI\}\}\_\{T\}\\leq\\frac\{K\_\{0\}\}\{T\}\+2\\widetilde\{A\}\_\{\\gamma\}\(h\)\+2\\\{\\widetilde\{U\}\_\{\\gamma\}\(h\)\+2\\widetilde\{V\}\_\{\\gamma\}\(h\)\\\}\\overline\{g\}\+\\\{4\\widetilde\{V\}\_\{\\gamma\}\(h\)\+2\\\}\\overline\{e\}\.\(106\)Suppose for now thatU~γ​\(h\)\+2​V~γ​\(h\)≤1/64\\widetilde\{U\}\_\{\\gamma\}\(h\)\+2\\widetilde\{V\}\_\{\\gamma\}\(h\)\\leq 1/64\. The theorem’s step\-size choice will imply this sufficient condition below\. Substitute \([105](https://arxiv.org/html/2610.12865#A1.E105)\) into \([106](https://arxiv.org/html/2610.12865#A1.E106)\)\. For clarity, putch:=U~γ​\(h\)\+2​V~γ​\(h\)c\_\{h\}:=\\widetilde\{U\}\_\{\\gamma\}\(h\)\+2\\widetilde\{V\}\_\{\\gamma\}\(h\)in this calculation\. Collecting the terms involvingFI¯T\\overline\{\\mathrm\{FI\}\}\_\{T\}on the left gives the coefficient3/4−8​ch/\(1−2​ch\)≥1/23/4\-8c\_\{h\}/\(1\-2c\_\{h\}\)\\geq 1/2\. AlsoV~γ​\(h\)≤ch/2\\widetilde\{V\}\_\{\\gamma\}\(h\)\\leq c\_\{h\}/2and1−2​ch≥31/321\-2c\_\{h\}\\geq 31/32\. Recall the discretization coefficient

𝒟γ​\(h\):=A~γ​\(h\)\+\{U~γ​\(h\)\+2​V~γ​\(h\)\}​\(1\+Cγ\)\.\\mathcal\{D\}\_\{\\gamma\}\(h\):=\\widetilde\{A\}\_\{\\gamma\}\(h\)\+\\\{\\widetilde\{U\}\_\{\\gamma\}\(h\)\+2\\widetilde\{V\}\_\{\\gamma\}\(h\)\\\}\(1\+C\_\{\\gamma\}\)\.Bounding the remaining coefficients gives

FI¯T\\displaystyle\\overline\{\\mathrm\{FI\}\}\_\{T\}≤2​K0T\+5​A~γ​\(h\)\+9​\{U~γ​\(h\)\+2​V~γ​\(h\)\}​Cγ\+5​e¯\\displaystyle\\leq\\frac\{2K\_\{0\}\}\{T\}\+5\\widetilde\{A\}\_\{\\gamma\}\(h\)\+9\\\{\\widetilde\{U\}\_\{\\gamma\}\(h\)\+2\\widetilde\{V\}\_\{\\gamma\}\(h\)\\\}C\_\{\\gamma\}\+5\\overline\{e\}≤2​K0T\+9​𝒟γ​\(h\)\+5​e¯\.\\displaystyle\\leq\\frac\{2K\_\{0\}\}\{T\}\+9\\mathcal\{D\}\_\{\\gamma\}\(h\)\+5\\overline\{e\}\.\(107\)The last inequality uses this definition of𝒟γ\\mathcal\{D\}\_\{\\gamma\}\. Recall the quantity defined in Theorem[A\.6](https://arxiv.org/html/2610.12865#A1.Thmtheorem6),

MT:=1\+Cγ\+K0T\.M\_\{T\}:=1\+C\_\{\\gamma\}\+\\frac\{K\_\{0\}\}\{T\}\.Substituting \([107](https://arxiv.org/html/2610.12865#A1.E107)\) back into \([105](https://arxiv.org/html/2610.12865#A1.E105)\) also gives the following bounds on the average squared gradient and drift norms,

g¯≤9​MT\+40​𝒟γ​\(h\)\+21​e¯,a¯≤18​MT\+80​𝒟γ​\(h\)\+44​e¯\.\\overline\{g\}\\leq 9M\_\{T\}\+40\\mathcal\{D\}\_\{\\gamma\}\(h\)\+21\\overline\{e\},\\qquad\\overline\{a\}\\leq 18M\_\{T\}\+80\\mathcal\{D\}\_\{\\gamma\}\(h\)\+44\\overline\{e\}\.\(108\)The numerical constants in these displays are universal upper bounds and they hide no dependence on the problem parameters\.

#### A\.9\.3Bounds on squared distance and local noise

The average squared local noise scale is

s¯:=1T​𝔼​∫0Tsp​\(Xtfr\)2​𝑑t\.\\overline\{s\}:=\\frac\{1\}\{T\}\\mathbb\{E\}\\int\_\{0\}^\{T\}s\_\{p\}\(X\_\{t\}^\{\\mathrm\{fr\}\}\)^\{2\}\\,\\mathrm\{d\}t\.To bound it, define the term independent ofe¯\\overline\{e\}in the squared distance bound \([110](https://arxiv.org/html/2610.12865#A1.E110)\),

RT,h:=6​M2,0\+6​d​T\+2​d​h\+\(3​T2\+2​h2/3\)​\{18​MT\+80​𝒟γ​\(h\)\}\.R\_\{T,h\}:=6M\_\{2,0\}\+6dT\+2dh\+\(3T^\{2\}\+2h^\{2\}/3\)\\\{18M\_\{T\}\+80\\mathcal\{D\}\_\{\\gamma\}\(h\)\\\}\.For the theorem’s accuracy schedule,T≥8T\\geq 8,h≤1h\\leq 1, and𝒟γ​\(h\)≤ε/64≤1/64\\mathcal\{D\}\_\{\\gamma\}\(h\)\\leq\\varepsilon/64\\leq 1/64\. SinceMT≥1M\_\{T\}\\geq 1,

18​MT\+80​𝒟γ​\(h\)≤20​MT,RT,h≤ST:=6​M2,0\+8​d​T\+80​T2​MT\.18M\_\{T\}\+80\\mathcal\{D\}\_\{\\gamma\}\(h\)\\leq 20M\_\{T\},\\qquad R\_\{T,h\}\\leq S\_\{T\}:=6M\_\{2,0\}\+8dT\+80T^\{2\}M\_\{T\}\.Indeed,h≤Th\\leq Tgives6​M2,0\+6​d​T\+2​d​h≤6​M2,0\+8​d​T6M\_\{2,0\}\+6dT\+2dh\\leq 6M\_\{2,0\}\+8dTand bounds the remaining term by\(11/3\)​T2​\(20​MT\)≤80​T2​MT\(11/3\)T^\{2\}\(20M\_\{T\}\)\\leq 80T^\{2\}M\_\{T\}\. Thus the theorem’s noise scale and step\-size coefficient are

s¯T2:=\[B​STp/2\+G\]2/p,κ:=\(1\+s¯T2\)−1\.\\bar\{s\}\_\{T\}^\{2\}:=\[BS\_\{T\}^\{p/2\}\+G\]^\{2/p\},\\qquad\\kappa:=\(1\+\\bar\{s\}\_\{T\}^\{2\}\)^\{\-1\}\.ForB\>0B\>0,M2,0M\_\{2,0\}satisfies𝔼​‖X0−xref‖2≤M2,0\\mathbb\{E\}\\\|X\_\{0\}\-x\_\{\\mathrm\{ref\}\}\\\|^\{2\}\\leq M\_\{2,0\}\. ForB=0B=0, the convention isM2,0=0M\_\{2,0\}=0, ands¯T2=G2/p\\bar\{s\}\_\{T\}^\{\\,2\}=G^\{2/p\}does not depend onSTS\_\{T\}\. The local noise scale is then constant, sos¯=G2/p=s¯T2\\overline\{s\}=G^\{2/p\}=\\bar\{s\}\_\{T\}^\{\\,2\}without a bound on the expected squared distance fromxrefx\_\{\\mathrm\{ref\}\}\. We now treatB\>0B\>0\. The path identity and Cauchy–Schwarz imply

Xt−xref\\displaystyle X\_\{t\}\-x\_\{\\mathrm\{ref\}\}=X0−xref\+∫0tFufr​𝑑u\+2​Wt,\\displaystyle=X\_\{0\}\-x\_\{\\mathrm\{ref\}\}\+\\int\_\{0\}^\{t\}F\_\{u\}^\{\\mathrm\{fr\}\}\\,\\mathrm\{d\}u\+\\sqrt\{2\}W\_\{t\},1T​∫0T𝔼​‖Xt−xref‖2​𝑑t\\displaystyle\\frac\{1\}\{T\}\\int\_\{0\}^\{T\}\\mathbb\{E\}\\\|X\_\{t\}\-x\_\{\\mathrm\{ref\}\}\\\|^\{2\}\\,\\mathrm\{d\}t≤3​M2,0\+3​d​T\+32​T2​a¯\.\\displaystyle\\leq 3M\_\{2,0\}\+3dT\+\\frac\{3\}\{2\}T^\{2\}\\overline\{a\}\.\(109\)On an occupied interval, conditional Gaussian calculation gives

∫0ℓk𝔼\[∥Xτk\+s−Xτk∥2∣Xτk,Fk,τk\]ds=ℓk33∥Fk∥2\+dℓk2≤ℓk\{h23∥Fk∥2\+dh\}\.\\int\_\{0\}^\{\\ell\_\{k\}\}\\mathbb\{E\}\\\!\\left\[\\\|X\_\{\\tau\_\{k\}\+s\}\-X\_\{\\tau\_\{k\}\}\\\|^\{2\}\\mid X\_\{\\tau\_\{k\}\},F\_\{k\},\\tau\_\{k\}\\right\]\\mathrm\{d\}s=\\frac\{\\ell\_\{k\}^\{3\}\}\{3\}\\\|F\_\{k\}\\\|^\{2\}\+d\\ell\_\{k\}^\{2\}\\leq\\ell\_\{k\}\\left\\\{\\frac\{h^\{2\}\}\{3\}\\\|F\_\{k\}\\\|^\{2\}\+dh\\right\\\}\.Combining this calculation with \([109](https://arxiv.org/html/2610.12865#A1.E109)\) and \([108](https://arxiv.org/html/2610.12865#A1.E108)\) yields

1T​𝔼​∫0T‖Xtfr−xref‖2​𝑑t≤RT,h\+44​\(3​T2\+2​h23\)​e¯\.\\displaystyle\\frac\{1\}\{T\}\\mathbb\{E\}\\int\_\{0\}^\{T\}\\\|X\_\{t\}^\{\\mathrm\{fr\}\}\-x\_\{\\mathrm\{ref\}\}\\\|^\{2\}\\,\\mathrm\{d\}t\\leq R\_\{T,h\}\+44\\left\(3T^\{2\}\+\\frac\{2h^\{2\}\}\{3\}\\right\)\\overline\{e\}\.\(110\)
Defineϕ⁡\(u\):=\(B​up/2\+G\)2/p\\phi\(u\):=\(Bu^\{p/2\}\+G\)^\{2/p\}foru≥0u\\geq 0, so thatϕ⁡\(‖x−xref‖2\)=sp​\(x\)2\\phi\(\\\|x\-x\_\{\\mathrm\{ref\}\}\\\|^\{2\}\)=s\_\{p\}\(x\)^\{2\}\. Lemma[A\.10](https://arxiv.org/html/2610.12865#A1.Thmtheorem10)shows thatϕ\\phiis nonnegative, nondecreasing, and concave\. Apply Jensen’s inequality toϕ\\phiunder the probability measureT−1​d​t​d​ℙT^\{\-1\}\\mathrm\{d\}t\\,\\mathrm\{d\}\\mathbb\{P\}\. Moreover, these properties implyϕ⁡\(u\+v\)≤\(1\+v/u\)​ϕ​\(u\)\\phi\(u\+v\)\\leq\(1\+v/u\)\\phi\(u\)foru\>0u\>0\. SinceST≥RT,h≥18​\(3​T2\+2​h2/3\)S\_\{T\}\\geq R\_\{T,h\}\\geq 18\(3T^\{2\}\+2h^\{2\}/3\), \([110](https://arxiv.org/html/2610.12865#A1.E110)\) gives

s¯≤s¯T2​\(1\+3​e¯\)\.\\overline\{s\}\\leq\\bar\{s\}\_\{T\}^\{\\,2\}\(1\+3\\overline\{e\}\)\.\(111\)ForB=0B=0, \([111](https://arxiv.org/html/2610.12865#A1.E111)\) holds directly from the constant scale identity above\.

#### A\.9\.4From per\-update estimation error to error over elapsed time

LetNT,mN\_\{T,m\}be the number of RA\-LMC intervals among the firstmmwhose starting times are strictly belowTT\. The indicator that iterationkkis occupied is measurable before its fresh oracle batch is drawn\. Althoughℓk\\ell\_\{k\}depends on that batch through the estimated gradient, the predictable noise term in the adaptive denominator and \([103](https://arxiv.org/html/2610.12865#A1.E103)\) imply

𝔼⁡\[ℓk​‖Fk\+∇V​\(Xτk\)‖2\|ℱk\]\\displaystyle\\mathbb\{E\}\\\!\\left\[\\ell\_\{k\}\\\|F\_\{k\}\+\\nabla V\(X\_\{\\tau\_\{k\}\}\)\\\|^\{2\}\\,\\middle\|\\,\\mathcal\{F\}\_\{k\}\\right\]≤h1\+κ​sp​\(Xτk\)2​𝔼​\[‖Fk\+∇V​\(Xτk\)‖2\|ℱk\]\\displaystyle\\leq\\frac\{h\}\{1\+\\kappa s\_\{p\}\(X\_\{\\tau\_\{k\}\}\)^\{2\}\}\\mathbb\{E\}\\\!\\left\[\\\|F\_\{k\}\+\\nabla V\(X\_\{\\tau\_\{k\}\}\)\\\|^\{2\}\\,\\middle\|\\,\\mathcal\{F\}\_\{k\}\\right\]≤h​ρb​sp​\(Xτk\)21\+κ​sp​\(Xτk\)2\\displaystyle\\leq h\\rho\_\{b\}\\frac\{s\_\{p\}\(X\_\{\\tau\_\{k\}\}\)^\{2\}\}\{1\+\\kappa s\_\{p\}\(X\_\{\\tau\_\{k\}\}\)^\{2\}\}≤h​ρbκ\.\\displaystyle\\leq\\frac\{h\\rho\_\{b\}\}\{\\kappa\}\.Consequently,

T​e¯≤h​ρbκ​𝔼​NT,m\.T\\overline\{e\}\\leq\\frac\{h\\rho\_\{b\}\}\{\\kappa\}\\mathbb\{E\}N\_\{T,m\}\.\(112\)Every complete interval satisfies

Hk​\{1\+ω​‖Fk‖γ\+κ​sp​\(Xτk\)2\}=h,H\_\{k\}\\\{1\+\\omega\\\|F\_\{k\}\\\|^\{\\gamma\}\+\\kappa s\_\{p\}\(X\_\{\\tau\_\{k\}\}\)^\{2\}\\\}=h,and at most the last occupied interval is truncated atTT\. It follows pathwise that

NT,m≤1\+1h​∫0T∧τm\{1\+ω​‖Ftfr‖γ\+κ​sp​\(Xtfr\)2\}​𝑑t\.\\displaystyle N\_\{T,m\}\\leq 1\+\\frac\{1\}\{h\}\\int\_\{0\}^\{T\\wedge\\tau\_\{m\}\}\\\{1\+\\omega\\\|F\_\{t\}^\{\\mathrm\{fr\}\}\\\|^\{\\gamma\}\+\\kappa s\_\{p\}\(X\_\{t\}^\{\\mathrm\{fr\}\}\)^\{2\}\\\}\\,\\mathrm\{d\}t\.Enlarge the integral to\[0,T\]\[0,T\]\. Jensen’s inequality and \([108](https://arxiv.org/html/2610.12865#A1.E108)\) give

a¯γ/2\\displaystyle\\overline\{a\}^\{\\gamma/2\}≤\[18​MT\+80​𝒟γ​\(h\)\]γ/2\+44γ/2​e¯γ/2\\displaystyle\\leq\[18M\_\{T\}\+80\\mathcal\{D\}\_\{\\gamma\}\(h\)\]^\{\\gamma/2\}\+44^\{\\gamma/2\}\\overline\{e\}^\{\\gamma/2\}≤\[18​MT\+80​𝒟γ​\(h\)\]γ/2\+7​\(1\+e¯\)\.\\displaystyle\\leq\[18M\_\{T\}\+80\\mathcal\{D\}\_\{\\gamma\}\(h\)\]^\{\\gamma/2\}\+7\(1\+\\overline\{e\}\)\.The resulting count bound uses the deterministic coefficients

JT,h\\displaystyle J\_\{T,h\}:=1\+ω⁡\{\[18​MT\+80​𝒟γ​\(h\)\]γ/2\+7\}\+κ​s¯T2,\\displaystyle:=1\+\\omega\\\{\[18M\_\{T\}\+80\\mathcal\{D\}\_\{\\gamma\}\(h\)\]^\{\\gamma/2\}\+7\\\}\+\\kappa\\bar\{s\}\_\{T\}^\{2\},χT,h\\displaystyle\\chi\_\{T,h\}:=7​ω\+3​κ​s¯T2\.\\displaystyle:=7\\omega\+3\\kappa\\bar\{s\}\_\{T\}^\{2\}\.Combining the preceding bound with \([111](https://arxiv.org/html/2610.12865#A1.E111)\) proves

𝔼​NT,m≤1\+Th​\{JT,h\+χT,h​e¯\}\.\\mathbb\{E\}N\_\{T,m\}\\leq 1\+\\frac\{T\}\{h\}\\\{J\_\{T,h\}\+\\chi\_\{T,h\}\\overline\{e\}\\\}\.\(113\)Substituting \([113](https://arxiv.org/html/2610.12865#A1.E113)\) into \([112](https://arxiv.org/html/2610.12865#A1.E112)\) yields the scalar feedback inequality

e¯≤ρb​κ−1​\{h/T\+JT,h\+χT,h​e¯\}\.\\overline\{e\}\\leq\\rho\_\{b\}\\kappa^\{\-1\}\\\{h/T\+J\_\{T,h\}\+\\chi\_\{T,h\}\\overline\{e\}\\\}\.To state the bound one¯\\overline\{e\}, define

ET,henv:=κ−1​\(h/T\+JT,h\)\.E\_\{T,h\}^\{\\mathrm\{env\}\}:=\\kappa^\{\-1\}\(h/T\+J\_\{T,h\}\)\.If2​ρb​κ−1​χT,h≤12\\rho\_\{b\}\\kappa^\{\-1\}\\chi\_\{T,h\}\\leq 1, the final term can be absorbed\. We verify this sufficient restriction for the theorem’s batch size below\. It gives

e¯≤2​ρb​ET,henv\.\\overline\{e\}\\leq 2\\rho\_\{b\}E\_\{T,h\}^\{\\mathrm\{env\}\}\.\(114\)The same condition givesχT,h​e¯≤h/T\+JT,h\\chi\_\{T,h\}\\overline\{e\}\\leq h/T\+J\_\{T,h\}\. Inserting this sharper consequence into \([113](https://arxiv.org/html/2610.12865#A1.E113)\) gives

𝔼​NT,m≤2\+2​T​JT,hh\.\\mathbb\{E\}N\_\{T,m\}\\leq 2\+\\frac\{2TJ\_\{T,h\}\}\{h\}\.\(115\)Under these sufficient restrictions, equations \([107](https://arxiv.org/html/2610.12865#A1.E107)\), \([114](https://arxiv.org/html/2610.12865#A1.E114)\), and \([115](https://arxiv.org/html/2610.12865#A1.E115)\) prove the finite horizon bounds uniformly in the truncation levelmm\.

#### A\.9\.5Removing the iteration truncation and the bounded support restriction

For each integerm≥1m\\geq 1, consider the process constructed above that follows RA\-LMC formmiterations and then continues with the exact Langevin diffusion ifτm<T\\tau\_\{m\}<T\. Keep the initial law and algorithm parameters fixed, and construct these processes using the same initial state, Brownian motion, and sequence of oracle samples\. Every finite sequence of updates is well defined because each robust estimate is finite and each step size is positive\. On\{τm≤T\}\\\{\\tau\_\{m\}\\leq T\\\}, all firstmmiterations start strictly beforeTT\. Markov’s inequality and \([115](https://arxiv.org/html/2610.12865#A1.E115)\) therefore give

ℙ⁡\(τm≤T\)≤𝔼​NT,mm⟶0\.\\mathbb\{P\}\(\\tau\_\{m\}\\leq T\)\\leq\\frac\{\\mathbb\{E\}N\_\{T,m\}\}\{m\}\\longrightarrow 0\.Becauseτm\\tau\_\{m\}is increasing, the iteration times cannot accumulate beforeTT\. The truncated and original paths agree on\[0,T\]\[0,T\]wheneverτm\>T\\tau\_\{m\}\>T, so their path laws converge in total variation\. The nonnegative estimation error accumulated over the firstmmupdates increases to the full error, andNT,m↑NTN\_\{T,m\}\\uparrow N\_\{T\}\. Monotone convergence therefore passes the error and count bounds to the original process\. Lower semicontinuity of relative FI at each time and Fatou’s lemma pass the FI bound as well\. For the original process, letk⁡\(t\):=max⁡\{k:τk≤t\}k\(t\):=\\max\\\{k:\\tau\_\{k\}\\leq t\\\}for0≤t<T0\\leq t<Tand define its duration\-weighted estimation error by

ℰT:=1T​𝔼​∫0T‖Fk⁡\(t\)\+∇V​\(Xτk⁡\(t\)\)‖2​𝑑t\.\\mathcal\{E\}\_\{T\}:=\\frac\{1\}\{T\}\\mathbb\{E\}\\int\_\{0\}^\{T\}\\\|F\_\{k\(t\)\}\+\\nabla V\(X\_\{\\tau\_\{k\(t\)\}\}\)\\\|^\{2\}\\,\\mathrm\{d\}t\.Thene¯=ℰT\\overline\{e\}=\\mathcal\{E\}\_\{T\}in the limiting bounds\. Convexity of relative Fisher information yields

FI\(μ¯T∥π\)≤1T∫0TFI\(μt∥π\)dt\.\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\frac\{1\}\{T\}\\int\_\{0\}^\{T\}\\mathrm\{FI\}\(\\mu\_\{t\}\\\|\\pi\)\\,\\mathrm\{d\}t\.
It remains to remove the temporary bounded\-support condition onμ0\\mu\_\{0\}\. Letμ0\(R0\)\\mu\_\{0\}^\{\(R\_\{0\}\)\}beμ0\\mu\_\{0\}conditioned on\{∥X0−xref∥≤R0\}\\\{\\\|X\_\{0\}\-x\_\{\\mathrm\{ref\}\}\\\|\\leq R\_\{0\}\\\}\. Thenμ0\(R0\)→μ0\\mu\_\{0\}^\{\(R\_\{0\}\)\}\\to\\mu\_\{0\}in total variation and

KL\(μ0\(R0\)∥π\)⟶K0\.\\mathrm\{KL\}\(\\mu\_\{0\}^\{\(R\_\{0\}\)\}\\\|\\pi\)\\longrightarrow K\_\{0\}\.\(116\)ForB\>0B\>0, conditioning on this ball also gives𝔼μ0\(R0\)​‖X0−xref‖2≤M2,0\\mathbb\{E\}\_\{\\mu\_\{0\}^\{\(R\_\{0\}\)\}\}\\\|X\_\{0\}\-x\_\{\\mathrm\{ref\}\}\\\|^\{2\}\\leq M\_\{2,0\}\. ForB=0B=0, no bound on the expected squared distance fromxrefx\_\{\\mathrm\{ref\}\}that is uniform inR0R\_\{0\}is needed\. The noise scale is constant, so the squared distance bound is not used\. KeepT,h,b,κ,MT,STT,h,b,\\kappa,M\_\{T\},S\_\{T\}and all proof coefficients computed from the original declared parameters fixed\. For all sufficiently largeR0R\_\{0\}, \([116](https://arxiv.org/html/2610.12865#A1.E116)\) implies

KL\(μ0\(R0\)∥π\)/T≤K0/T\+1\.\\mathrm\{KL\}\(\\mu\_\{0\}^\{\(R\_\{0\}\)\}\\\|\\pi\)/T\\leq K\_\{0\}/T\+1\.To see why this suffices without enlarging the schedules, retain the actual initial KL in the derivation of the bounds \([108](https://arxiv.org/html/2610.12865#A1.E108)\) ong¯\\overline\{g\}anda¯\\overline\{a\}\. WritingK0,R0:=KL\(μ0\(R0\)∥π\)K\_\{0,R\_\{0\}\}:=\\mathrm\{KL\}\(\\mu\_\{0\}^\{\(R\_\{0\}\)\}\\\|\\pi\)gives

g¯\\displaystyle\\overline\{g\}≤9​\(Cγ\+K0,R0/T\)\+40​𝒟γ​\(h\)\+21​e¯,\\displaystyle\\leq 9\(C\_\{\\gamma\}\+K\_\{0,R\_\{0\}\}/T\)\+40\\mathcal\{D\}\_\{\\gamma\}\(h\)\+21\\overline\{e\},a¯\\displaystyle\\overline\{a\}≤18​\(Cγ\+K0,R0/T\)\+80​𝒟γ​\(h\)\+44​e¯\.\\displaystyle\\leq 18\(C\_\{\\gamma\}\+K\_\{0,R\_\{0\}\}/T\)\+80\\mathcal\{D\}\_\{\\gamma\}\(h\)\+44\\overline\{e\}\.The added one inMT=1\+Cγ\+K0/TM\_\{T\}=1\+C\_\{\\gamma\}\+K\_\{0\}/Tabsorbs the cutoff’s KL difference\. Thus the originalRT,h≤STR\_\{T,h\}\\leq S\_\{T\}and the same noise, step\-size, and count bounds apply to every sufficiently large cutoff, without recomputing any algorithm parameter\. The finite\-update laws depend on the initial law through a probability kernel, so total\-variation convergence of the initial laws passes to every finite sequence of updates\. In particular, the uniform bound onℙ⁡\(τm≤T\)\\mathbb\{P\}\(\\tau\_\{m\}\\leq T\)passes to the original initialization and rules out accumulation there\. The error integrals and update counts are nonnegative path functionals under the same algorithm\. Their conditional expectations converge to the original expectations by monotone convergence on the increasing initial balls, followed by division by the ball probability\. Thus the estimation error itself converges, which allows the FI inequality to retain the actualℰT\\mathcal\{E\}\_\{T\}on its right\-hand side\. Lower semicontinuity of FI andKL\(μ0\(R0\)∥π\)→K0\\mathrm\{KL\}\(\\mu\_\{0\}^\{\(R\_\{0\}\)\}\\\|\\pi\)\\to K\_\{0\}now prove the finite horizon statements forμ0\\mu\_\{0\}\. Finally, every occupied stochastic interval uses exactly4​b4bindividual oracle evaluations, soQT=4​b​NTQ\_\{T\}=4bN\_\{T\}pathwise and𝔼​QT=4​b​𝔼​NT\\mathbb\{E\}Q\_\{T\}=4b\\mathbb\{E\}N\_\{T\}\.

#### A\.9\.6Accuracy schedule and general complexity

We now verify the parameter choices in the theorem\. For0<ε≤10<\\varepsilon\\leq 1, the prescribed time horizon and base step size are

T:=8​\(K0\+1\)εandh:=max⁡\{u∈\[0,1\]:𝒟γ​\(u\)≤ε/64\}\.T:=\\frac\{8\(K\_\{0\}\+1\)\}\{\\varepsilon\}\\quad\\text\{and\}\\quad h:=\\max\\\{u\\in\[0,1\]:\\mathcal\{D\}\_\{\\gamma\}\(u\)\\leq\\varepsilon/64\\\}\.WithMTM\_\{T\}andκ=\(1\+s¯T2\)−1\\kappa=\(1\+\\bar\{s\}\_\{T\}^\{2\}\)^\{\-1\}as above, the batch size is

b:=⌈\[260​Cp​κ−1​\(1\+ω​MTγ/2\)ε\]p/\[2​\(p−1\)\]⌉\.b:=\\left\\lceil\\left\[\\frac\{260C\_\{p\}\\kappa^\{\-1\}\(1\+\\omega M\_\{T\}^\{\\gamma/2\}\)\}\{\\varepsilon\}\\right\]^\{p/\[2\(p\-1\)\]\}\\right\\rceil\.By continuity and monotonicity of the coefficient functions, the maximalhhis positive\. Since

𝒟γ​\(h\)≥U~γ​\(h\)\+2​V~γ​\(h\),\\mathcal\{D\}\_\{\\gamma\}\(h\)\\geq\\widetilde\{U\}\_\{\\gamma\}\(h\)\+2\\widetilde\{V\}\_\{\\gamma\}\(h\),the choice𝒟γ​\(h\)≤ε/64\\mathcal\{D\}\_\{\\gamma\}\(h\)\\leq\\varepsilon/64with0<ε≤10<\\varepsilon\\leq 1enforces the discretization stability condition\. Moreover,18​MT\+80​𝒟γ​\(h\)≤20​MT18M\_\{T\}\+80\\mathcal\{D\}\_\{\\gamma\}\(h\)\\leq 20M\_\{T\},MT≥1M\_\{T\}\\geq 1, andκ​s¯T2≤1\\kappa\\bar\{s\}\_\{T\}^\{2\}\\leq 1imply

JT,h\\displaystyle J\_\{T,h\}≤2\+\(20\+7\)​ω​MTγ/2≤12​\(1\+ω​MTγ/2\),\\displaystyle\\leq 2\+\(\\sqrt\{20\}\+7\)\\omega M\_\{T\}^\{\\gamma/2\}\\leq 12\(1\+\\omega M\_\{T\}^\{\\gamma/2\}\),χT,h\\displaystyle\\chi\_\{T,h\}≤3\+7​ω≤7​\(1\+ω​MTγ/2\)\.\\displaystyle\\leq 3\+7\\omega\\leq 7\(1\+\\omega M\_\{T\}^\{\\gamma/2\}\)\.Sinceh/T≤1/8h/T\\leq 1/8, it follows that

ET,henv≤13​κ−1​\(1\+ω​MTγ/2\)\.E\_\{T,h\}^\{\\mathrm\{env\}\}\\leq 13\\kappa^\{\-1\}\(1\+\\omega M\_\{T\}^\{\\gamma/2\}\)\.The specified batch size gives

ρb≤ε260​κ−1​\(1\+ω​MTγ/2\)\.\\rho\_\{b\}\\leq\\frac\{\\varepsilon\}\{260\\kappa^\{\-1\}\(1\+\\omega M\_\{T\}^\{\\gamma/2\}\)\}\.Consequently,

2​ρb​κ−1​χT,h≤14​ε260<1,2​ρb​ET,henv≤ε10\.2\\rho\_\{b\}\\kappa^\{\-1\}\\chi\_\{T,h\}\\leq\\frac\{14\\varepsilon\}\{260\}<1,\\qquad 2\\rho\_\{b\}E\_\{T,h\}^\{\\mathrm\{env\}\}\\leq\\frac\{\\varepsilon\}\{10\}\.All sufficient restrictions used above therefore hold, andℰT≤ε/10\\mathcal\{E\}\_\{T\}\\leq\\varepsilon/10\. The FI bound becomes

FI\(μ¯T∥π\)\\displaystyle\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)≤FI¯T≤2​K0T\+9​𝒟γ​\(h\)\+ε2\\displaystyle\\leq\\overline\{\\mathrm\{FI\}\}\_\{T\}\\leq\\frac\{2K\_\{0\}\}\{T\}\+9\\mathcal\{D\}\_\{\\gamma\}\(h\)\+\\frac\{\\varepsilon\}\{2\}≤ε4\+9​ε64\+ε2=57​ε64<ε\.\\displaystyle\\leq\\frac\{\\varepsilon\}\{4\}\+\\frac\{9\\varepsilon\}\{64\}\+\\frac\{\\varepsilon\}\{2\}=\\frac\{57\\varepsilon\}\{64\}<\\varepsilon\.Likewise, the count bound gives

𝔼​NT≤2\+24​Th​\(1\+ω​MTγ/2\),𝔼​QT=4​b​𝔼​NT\.\\mathbb\{E\}N\_\{T\}\\leq 2\+\\frac\{24T\}\{h\}\(1\+\\omega M\_\{T\}^\{\\gamma/2\}\),\\qquad\\mathbb\{E\}Q\_\{T\}=4b\\mathbb\{E\}N\_\{T\}\.This proves the theorem\.

For the complexity corollary, we retain parameter\-sensitive estimates\. Define the coefficient of the term linear inhhby

Dγ\(1\):=\{d​L02,L1=0,3​d​\[L02​\(1\+2γ2/\(1−γ\)\)2\+32​γ​22​γ2/\(1−γ\)​L12​\(2\+Cγ\)\],L1\>0,0<γ<1,4​\(d\+1\)​\{L02\+L12​\(1\+C1\)\},L1\>0,γ=1\.D\_\{\\gamma\}^\{\(1\)\}:=\\begin\{cases\}dL\_\{0\}^\{2\},&L\_\{1\}=0,\\\\ \\begin\{aligned\} 3d\\bigl\[L\_\{0\}^\{2\}\(1\+2^\{\\gamma^\{2\}/\(1\-\\gamma\)\}\)^\{2\}\+3^\{2\\gamma\}2^\{2\\gamma^\{2\}/\(1\-\\gamma\)\}L\_\{1\}^\{2\}\(2\+C\_\{\\gamma\}\)\\bigr\],\\end\{aligned\}&L\_\{1\}\>0,\\ 0<\\gamma<1,\\\\ 4\(d\+1\)\\\{L\_\{0\}^\{2\}\+L\_\{1\}^\{2\}\(1\+C\_\{1\}\)\\\},&L\_\{1\}\>0,\\ \\gamma=1\.\\end\{cases\}The explicit coefficient formulas imply

limu↓0𝒟γ​\(u\)u=Dγ\(1\)\.\\lim\_\{u\\downarrow 0\}\\frac\{\\mathcal\{D\}\_\{\\gamma\}\(u\)\}\{u\}=D\_\{\\gamma\}^\{\(1\)\}\.The limit is positive whenever a normalizable target satisfying the stated assumptions exists\. Thus, for fixed problem parameters,h−1=𝒪⁡\(Dγ\(1\)/ε\)h^\{\-1\}=\\mathcal\{O\}\(D\_\{\\gamma\}^\{\(1\)\}/\\varepsilon\)\. The hidden constant in this step\-size estimate and the iteration estimate below depends only onγ\\gamma\. Constants in the batch\-size and oracle estimates may additionally depend onpp\. The small accuracy thresholds may depend on the other fixed problem parameters\. To boundJT,hJ\_\{T,h\}andSTS\_\{T\}, define

Jγ∗\\displaystyle J\_\{\\gamma\}^\{\*\}:=2\+ω⁡\{\[22​\(1\+Cγ\)\]γ/2\+7\},\\displaystyle:=2\+\\omega\\\{\[22\(1\+C\_\{\\gamma\}\)\]^\{\\gamma/2\}\+7\\\},𝖱∗2\\displaystyle\\mathsf\{R\}\_\{\*\}^\{2\}:=6​M2,0\+64​\(K0\+1\)​d\+5760​\(K0\+1\)2​\(1\+Cγ\)\.\\displaystyle:=6M\_\{2,0\}\+64\(K\_\{0\}\+1\)d\+5760\(K\_\{0\}\+1\)^\{2\}\(1\+C\_\{\\gamma\}\)\.The schedule gives

MT≤98\(1\+Cγ\),18MT\+80𝒟γ\(h\)≤22\(1\+Cγ\),ST≤𝖱∗2/ε2,JT,h≤Jγ∗\.\\displaystyle M\_\{T\}\\leq\\frac\{9\}\{8\}\(1\+C\_\{\\gamma\}\),\\quad 18M\_\{T\}\+80\\mathcal\{D\}\_\{\\gamma\}\(h\)\\leq 22\(1\+C\_\{\\gamma\}\),\\quad S\_\{T\}\\leq\\mathsf\{R\}\_\{\*\}^\{2\}/\\varepsilon^\{2\}\\quad,J\_\{T,h\}\\leq J\_\{\\gamma\}^\{\*\}\.Consequently, we have

1\+ω​MTγ/2≤Jγ∗,ands¯T2≤\[B​\(𝖱∗/ε\)p\+G\]2/p\.1\+\\omega M\_\{T\}^\{\\gamma/2\}\\leq J\_\{\\gamma\}^\{\*\},\\quad\\text\{and\}\\quad\\bar\{s\}\_\{T\}^\{\\,2\}\\leq\[B\(\\mathsf\{R\}\_\{\*\}/\\varepsilon\)^\{p\}\+G\]^\{2/p\}\.Substitution into the prescribed batch size therefore gives

b=𝒪\(\(Jγ∗\)p/\[2​\(p−1\)\]ε−p/\[2\(p−1\)\]\[1\+B\(𝖱∗/ε\)p\+G\]1/\(p−1\)\)\.\\displaystyle b=\\mathcal\{O\}\\\!\\left\(\(J\_\{\\gamma\}^\{\*\}\)^\{p/\[2\(p\-1\)\]\}\\varepsilon^\{\-p/\[2\(p\-1\)\]\}\[1\+B\(\\mathsf\{R\}\_\{\*\}/\\varepsilon\)^\{p\}\+G\]^\{1/\(p\-1\)\}\\right\)\.Multiplying by the expected iteration bound proves

𝔼​NT\\displaystyle\\mathbb\{E\}N\_\{T\}=𝒪⁡\(\(K0\+1\)​Dγ\(1\)​Jγ∗​ε−2\),\\displaystyle=\\mathcal\{O\}\\\!\\left\(\(K\_\{0\}\+1\)D\_\{\\gamma\}^\{\(1\)\}J\_\{\\gamma\}^\{\*\}\\varepsilon^\{\-2\}\\right\),\(117\)𝔼​QT\\displaystyle\\mathbb\{E\}Q\_\{T\}=𝒪⁡\(\(K0\+1\)​Dγ\(1\)​\(Jγ∗\)1\+p/\[2​\(p−1\)\]ε2\+p/\[2​\(p−1\)\]​\[1\+B​\(𝖱∗/ε\)p\+G\]1/\(p−1\)\)\.\\displaystyle=\\mathcal\{O\}\\\!\\left\(\\frac\{\(K\_\{0\}\+1\)D\_\{\\gamma\}^\{\(1\)\}\(J\_\{\\gamma\}^\{\*\}\)^\{1\+p/\[2\(p\-1\)\]\}\}\{\\varepsilon^\{2\+p/\[2\(p\-1\)\]\}\}\[1\+B\(\\mathsf\{R\}\_\{\*\}/\\varepsilon\)^\{p\}\+G\]^\{1/\(p\-1\)\}\\right\)\.\(118\)These estimates will give the stated complexity specializations\.

### A\.10Proof of Corollary[A\.7](https://arxiv.org/html/2610.12865#A1.Thmtheorem7)

We derive the stated complexity specializations\. WhenB\>0B\>0is fixed, the final factor in \([118](https://arxiv.org/html/2610.12865#A1.E118)\) is, for sufficiently smallε\\varepsilon,

𝒪\(B1/\(p−1\)𝖱∗p/\(p−1\)ε−p/\(p−1\)\)\.\\mathcal\{O\}\\\!\\left\(B^\{1/\(p\-1\)\}\\mathsf\{R\}\_\{\*\}^\{p/\(p\-1\)\}\\varepsilon^\{\-p/\(p\-1\)\}\\right\)\.Therefore

𝔼​QT=𝒪⁡\(B1/\(p−1\)​\(K0\+1\)​Dγ\(1\)​\(Jγ∗\)1\+p/\[2​\(p−1\)\]​𝖱∗p/\(p−1\)ε2\+3​p/\[2​\(p−1\)\]\)\.\\displaystyle\\mathbb\{E\}Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{B^\{1/\(p\-1\)\}\(K\_\{0\}\+1\)D\_\{\\gamma\}^\{\(1\)\}\(J\_\{\\gamma\}^\{\*\}\)^\{1\+p/\[2\(p\-1\)\]\}\\mathsf\{R\}\_\{\*\}^\{p/\(p\-1\)\}\}\{\\varepsilon^\{2\+3p/\[2\(p\-1\)\]\}\}\\right\)\.\(119\)IfB=0B=0, instead,

𝔼​QT=𝒪⁡\(\(1\+G\)1/\(p−1\)​\(K0\+1\)​Dγ\(1\)​\(Jγ∗\)1\+p/\[2​\(p−1\)\]ε2\+p/\[2​\(p−1\)\]\)\.\\displaystyle\\mathbb\{E\}Q\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{\(1\+G\)^\{1/\(p\-1\)\}\(K\_\{0\}\+1\)D\_\{\\gamma\}^\{\(1\)\}\(J\_\{\\gamma\}^\{\*\}\)^\{1\+p/\[2\(p\-1\)\]\}\}\{\\varepsilon^\{2\+p/\[2\(p\-1\)\]\}\}\\right\)\.\(120\)In particular, forp=2p=2and fixed remaining parameters, these bounds give𝔼​QT=𝒪⁡\(ε−5\)\\mathbb\{E\}Q\_\{T\}=\\mathcal\{O\}\(\\varepsilon^\{\-5\}\)under BG\-0 noise \(B\>0B\>0\) and𝔼​QT=𝒪⁡\(ε−3\)\\mathbb\{E\}Q\_\{T\}=\\mathcal\{O\}\(\\varepsilon^\{\-3\}\)under uniformly bounded variance \(B=0B=0\)\.

We next make the dimension and smoothness dependence explicit\. LetΛ=max⁡\{1,L0,L1\}\\Lambda=\\max\\\{1,L\_\{0\},L\_\{1\}\\\}\. For fixed0<γ<10<\\gamma<1, the definitions of the generalized\-smoothness coefficients imply the following bounds, with hidden constants depending only onγ\\gamma:

1\+Cγ\\displaystyle 1\+C\_\{\\gamma\}=𝒪⁡\(\(d​Λ\)2/\(2−γ\)\),\\displaystyle=\\mathcal\{O\}\(\(d\\Lambda\)^\{2/\(2\-\\gamma\)\}\),Dγ\(1\)\\displaystyle D\_\{\\gamma\}^\{\(1\)\}=𝒪⁡\(d​Λ2​\(1\+Cγ\)\),\\displaystyle=\\mathcal\{O\}\(d\\Lambda^\{2\}\(1\+C\_\{\\gamma\}\)\),Jγ∗\\displaystyle J\_\{\\gamma\}^\{\*\}=𝒪⁡\(dγ/\(2−γ\)​Λ2/\(2−γ\)\),\\displaystyle=\\mathcal\{O\}\(d^\{\\gamma/\(2\-\\gamma\)\}\\Lambda^\{2/\(2\-\\gamma\)\}\),\(121\)𝖱∗\\displaystyle\\mathsf\{R\}\_\{\*\}=𝒪⁡\(M2,0\+\(K0\+1\)​d\+\(K0\+1\)​\(d​Λ\)1/\(2−γ\)\)\.\\displaystyle=\\mathcal\{O\}\\\!\\left\(\\sqrt\{M\_\{2,0\}\}\+\\sqrt\{\(K\_\{0\}\+1\)d\}\+\(K\_\{0\}\+1\)\(d\\Lambda\)^\{1/\(2\-\\gamma\)\}\\right\)\.Under the conventionK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)stated in Section[2](https://arxiv.org/html/2610.12865#S2), substituting \([121](https://arxiv.org/html/2610.12865#A1.E121)\) into \([117](https://arxiv.org/html/2610.12865#A1.E117)\) gives

𝔼​NT=𝒪⁡\(Λ\(8−2​γ\)/\(2−γ\)​d\(6−γ\)/\(2−γ\)ε2\)\.\\mathbb\{E\}N\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{\\Lambda^\{\(8\-2\\gamma\)/\(2\-\\gamma\)\}d^\{\(6\-\\gamma\)/\(2\-\\gamma\)\}\}\{\\varepsilon^\{2\}\}\\right\)\.
For the compactB\>0B\>0oracle rate, we can sharpen the radius estimate in the small\-ε\\varepsilonregime\. For each fixed finiteM2,0M\_\{2,0\},MT→1\+CγM\_\{T\}\\to 1\+C\_\{\\gamma\}, so

ε2​ST⟶5120​\(K0\+1\)2​\(1\+Cγ\)\.\\varepsilon^\{2\}S\_\{T\}\\longrightarrow 5120\(K\_\{0\}\+1\)^\{2\}\(1\+C\_\{\\gamma\}\)\.\(122\)In particular, for all sufficiently smallε\\varepsilon,

ST≤R~∗ε,R~∗:=80​\(K0\+1\)​1\+Cγ=𝒪⁡\(d​\(d​Λ\)1/\(2−γ\)\)\.\\sqrt\{S\_\{T\}\}\\leq\\frac\{\\widetilde\{R\}\_\{\*\}\}\{\\varepsilon\},\\qquad\\widetilde\{R\}\_\{\*\}:=80\(K\_\{0\}\+1\)\\sqrt\{1\+C\_\{\\gamma\}\}=\\mathcal\{O\}\\\!\\left\(d\(d\\Lambda\)^\{1/\(2\-\\gamma\)\}\\right\)\.Repeating the noise\-envelope estimate withR~∗\\widetilde\{R\}\_\{\*\}in place of𝖱∗\\mathsf\{R\}\_\{\*\}and using \([121](https://arxiv.org/html/2610.12865#A1.E121)\) gives

𝔼​QT=𝒪⁡\(B1/\(p−1\)​Λ\(8−2​γ\)​\(p−1\)\+2​p\(p−1\)​\(2−γ\)​d\(3​p−2\)​\(6−γ\)2​\(p−1\)​\(2−γ\)ε2\+3​p/\[2​\(p−1\)\]\)\.\\displaystyle\\mathbb\{E\}Q\_\{T\}=\\mathcal\{O\}\\\!\\bigg\(\\frac\{B^\{1/\(p\-1\)\}\\Lambda^\{\\frac\{\(8\-2\\gamma\)\(p\-1\)\+2p\}\{\(p\-1\)\(2\-\\gamma\)\}\}d^\{\\frac\{\(3p\-2\)\(6\-\\gamma\)\}\{2\(p\-1\)\(2\-\\gamma\)\}\}\}\{\\varepsilon^\{2\+3p/\[2\(p\-1\)\]\}\}\\bigg\)\.This is the compactB\>0B\>0rate in Corollary[A\.7](https://arxiv.org/html/2610.12865#A1.Thmtheorem7)\. Its small\-accuracy threshold may depend onM2,0M\_\{2,0\},BB,GG, and the other fixed problem parameters\. Thus it does not replace the explicit initialization dependence in the finite\-accuracy schedule\. In particular, finiteK0K\_\{0\}alone does not imply the finite initial second moment used whenB\>0B\>0\. ForB=0B=0, the corresponding substitution in \([120](https://arxiv.org/html/2610.12865#A1.E120)\) gives

𝔼​QT=𝒪⁡\(\(1\+G\)1/\(p−1\)​Λ\(8−2​γ\)​\(p−1\)\+p\(p−1\)​\(2−γ\)​d1\+8​\(p−1\)\+γ​p2​\(p−1\)​\(2−γ\)ε2\+p/\[2​\(p−1\)\]\)\.\\displaystyle\\mathbb\{E\}Q\_\{T\}=\\mathcal\{O\}\\\!\\bigg\(\\frac\{\(1\+G\)^\{1/\(p\-1\)\}\\Lambda^\{\\frac\{\(8\-2\\gamma\)\(p\-1\)\+p\}\{\(p\-1\)\(2\-\\gamma\)\}\}d^\{1\+\\frac\{8\(p\-1\)\+\\gamma p\}\{2\(p\-1\)\(2\-\\gamma\)\}\}\}\{\\varepsilon^\{2\+p/\[2\(p\-1\)\]\}\}\\bigg\)\.With the constant inK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)fixed, the hidden constants in these compact oracle formulas depend only onppandγ\\gamma\. They are pointwise inγ\\gammaand are not uniform asγ↑1\\gamma\\uparrow 1\. Atγ=1\\gamma=1, direct use of the corresponding coefficients gives the same displayed powers evaluated atγ=1\\gamma=1\.

WhenL1=0L\_\{1\}=0,ω=0\\omega=0,Dγ\(1\)=d​L02D\_\{\\gamma\}^\{\(1\)\}=dL\_\{0\}^\{2\}, andJγ∗=2J\_\{\\gamma\}^\{\*\}=2\. Thus

𝔼​NT=𝒪⁡\(\(K0\+1\)​d​L02ε2\)=𝒪⁡\(d2​L02ε2\)\\mathbb\{E\}N\_\{T\}=\\mathcal\{O\}\\\!\\left\(\\frac\{\(K\_\{0\}\+1\)dL\_\{0\}^\{2\}\}\{\\varepsilon^\{2\}\}\\right\)=\\mathcal\{O\}\\\!\\left\(\\frac\{d^\{2\}L\_\{0\}^\{2\}\}\{\\varepsilon^\{2\}\}\\right\)underK0=𝒪⁡\(d\)K\_\{0\}=\\mathcal\{O\}\(d\)\. The first equality concerns the noise\-aware RA\-LMC rule and its iteration count can remain random\. This proves Corollary[A\.7](https://arxiv.org/html/2610.12865#A1.Thmtheorem7)\.

### A\.11Verification of the algorithmic reductions

WhenL1=0L\_\{1\}=0,ω=0\\omega=0\. If one additionally setsκ=0\\kappa=0, the adaptive duration in \([8](https://arxiv.org/html/2610.12865#S3.E8)\) becomes the constanthh, and the algorithm is the fixed\-step robust method of Theorem[A\.2](https://arxiv.org/html/2610.12865#A1.Thmtheorem2)\. This changes the positive value ofκ\\kappaprescribed by Theorem[A\.6](https://arxiv.org/html/2610.12865#A1.Thmtheorem6), so the sharper fixed\-step bounds follow from Theorem[A\.2](https://arxiv.org/html/2610.12865#A1.Thmtheorem2)and Corollary[A\.3](https://arxiv.org/html/2610.12865#A1.Thmtheorem3), rather than by substituting into the combined theorem’s accuracy schedule\.

IfB=G=0B=G=0, the conditionalppth moment of every oracle error is zero\. Hence every oracle call made by the algorithm equals∇V\\nabla Valmost surely\. The robust\-estimation stage is then redundant\. Using one exact gradient per iteration and the schedule in Theorem[A\.4](https://arxiv.org/html/2610.12865#A1.Thmtheorem4)gives the adaptive exact\-gradient method and its sharper bounds\. WhenL1=0L\_\{1\}=0as well, Corollary[A\.5](https://arxiv.org/html/2610.12865#A1.Thmtheorem5)gives the classical globally smooth complexity𝒪⁡\(L02​d2/ε2\)\\mathcal\{O\}\(L\_\{0\}^\{2\}d^\{2\}/\\varepsilon^\{2\}\)\.

## Appendix BAdditional Experimental Details

### B\.1Synthetic Experiments

#### B\.1\.1Experimental Setting

The potentials used in the synthetic experiments and their geometry is provided in Table[B\.1](https://arxiv.org/html/2610.12865#A2.T1)\. All three corresponding target distributions are non\-log\-concave\.

Table B\.1:Target potentials used in the experiments of Fig\.[1](https://arxiv.org/html/2610.12865#S3.F1)and the supplementary figures\. We note that the two radial potentials are not globally smooth\.For the experiments with state\-dependent heavy\-tailed noise oracle, we define the oracle as follows

𝒢⁡\(x,ξ\)=∇V​\(x\)\+sp​\(x\)​ξ,sp​\(x\)=\(8\+8​‖x‖p\)1/p,p=1\.2,ν=1\.4\.\{\\mathcal\{G\}\}\(x,\\xi\)=\\nabla V\(x\)\+s\_\{p\}\(x\)\\xi,\\qquad s\_\{p\}\(x\)=\\bigl\(8\+8\\left\\lVert x\\right\\rVert^\{p\}\\bigr\)^\{1/p\},\\qquad p=1\.2,\\quad\\nu=1\.4\.Hereξ\\xiis a normalized spherical Student random vector\. Specifically, ifZ∼𝒩⁡\(0,I2\)Z\\sim\\mathcal\{N\}\(0,I\_\{2\}\)andW∼χν2W\\sim\\chi^\{2\}\_\{\\nu\}are independent, then

ξ=ZW/ν​mp,ν1/p,mp,ν=νp/2​Γ⁡\(1\+p/2\)​Γ​\(\(ν−p\)/2\)Γ⁡\(ν/2\),\\xi=\\frac\{Z\}\{\\sqrt\{W/\\nu\}\\,m\_\{p,\\nu\}^\{1/p\}\},\\qquad m\_\{p,\\nu\}=\\nu^\{p/2\}\\frac\{\\Gamma\(1\+p/2\)\\Gamma\(\(\\nu\-p\)/2\)\}\{\\Gamma\(\\nu/2\)\},whereΓ⁡\(x\)=∫0∞tx−1​e−1​𝑑t\\Gamma\(x\)=\\int\_\{0\}^\{\\infty\}t^\{x\-1\}e^\{\-1\}dtis gamma function defined forx\>0x\>0andχν2\\chi^\{2\}\_\{\\nu\}is chi\-squared distribution withν\\nudegrees of freedom\. Consequently,𝔼​ξ=0\\mathbb\{E\}\\xi=0,𝔼​‖ξ‖p=1\\mathbb\{E\}\\left\\lVert\\xi\\right\\rVert^\{p\}=1, and𝔼​‖Y⁡\(x\)−∇V​\(x\)‖p=8\+8​‖x‖p\\mathbb\{E\}\\left\\lVert Y\(x\)\-\\nabla V\(x\)\\right\\rVert^\{p\}=8\+8\\left\\lVert x\\right\\rVert^\{p\}\. Since1<p<ν<21<p<\\nu<2, the oracle is unbiased with a finitepp\-th moment and infinite variance, and satisfies the BG\-ppcondition withB=G=8B=G=8andxref=0x\_\{\\mathrm\{ref\}\}=0\.

LMC uses one exact\-gradient evaluation per update\. Both SG\-LMC and R\-LMC use1616independent oracle queries per update\. SG\-LMC averages all1616queries\. R\-LMC averagesb=4b=4robust replicates: each replicate uses the geometric median of three queries as its center and clips the residual of a fourth query at radiussp​\(x\)​41/ps\_\{p\}\(x\)4^\{1/p\}\. Thus the raw and robust stochastic methods have the same per\-update oracle budget, although the adaptive methods may require different numbers of updates to reach a given physical time\. We use a fixed time horizonTTfor the primary comparison since it matches our theoreticalFI\\mathrm\{FI\}guarantees for the time\-averaged lawμ¯T\\bar\{\\mu\}\_\{T\}\. We also report time\-averaged FI after a fixed numberNNof updates as a complementary comparison at equal iteration budgets\.

For Figure[1](https://arxiv.org/html/2610.12865#S3.F1), we useh=0\.02h=0\.02in panels \(a\) and \(b\), andh=0\.16h=0\.16in panel \(c\)\. Herehhis the step size for fixed\-step methods and the base step\-size parameter for adaptive methods\. We run the experiments in this section on CPU\.

An update with Langevin forceFkF\_\{k\}isXk\+1=Xk\+Hk​Fk\+2​Hk​ZkX\_\{k\+1\}=X\_\{k\}\+H\_\{k\}F\_\{k\}\+\\sqrt\{2H\_\{k\}\}\\,Z\_\{k\}\. The state\-dependent\-heavy\-tail setting compares LMC, SG\-LMC, and R\-LMC, all with fixed step\-sizeHk=hH\_\{k\}=h\. The generalized\-smoothness setting compares fixed\-step LMC with A\-LMC, using the adaptive step\-size

Hk=h1\+‖∇V​\(Xk\)‖2/3\.H\_\{k\}=\\frac\{h\}\{1\+\\left\\lVert\\nabla V\(X\_\{k\}\)\\right\\rVert^\{2/3\}\}\.We note that this is random variable as it depends onXkX\_\{k\}\. The combined setting compares all six methods\. A\-LMC usesHk=h/\(1\+‖∇V​\(Xk\)‖1/2\)H\_\{k\}=h/\(1\+\\left\\lVert\\nabla V\(X\_\{k\}\)\\right\\rVert^\{1/2\}\), whereas A\-SG\-LMC and RA\-LMC use

Hk=h1\+‖Fk‖1/2\+κ​sp​\(Xk\)2,κ=11\+82/p=133,H\_\{k\}=\\frac\{h\}\{1\+\\left\\lVert F\_\{k\}\\right\\rVert^\{1/2\}\+\\kappa s\_\{p\}\(X\_\{k\}\)^\{2\}\},\\qquad\\kappa=\\frac\{1\}\{1\+8^\{2/p\}\}=\\frac\{1\}\{33\},which follows \([8](https://arxiv.org/html/2610.12865#S3.E8)\) forp=1\.2p=1\.2\. A\-SG\-LMC uses the negative of the1616\-query minibatch average asFkF\_\{k\}, whereas RA\-LMC uses the negative of the robust estimator with batch sizeb=4b=4\.

Figure B\.1:Fisher information of the time\-averaged lawsμ¯τN\\bar\{\\mu\}\_\{\\tau\_\{N\}\}afterNNupdates for the three settings and step sizes of Figure[1](https://arxiv.org/html/2610.12865#S3.F1)\. For each trajectory, we sample a time uniformly from\[0,τN\]\[0,\\tau\_\{N\}\], whereτN\\tau\_\{N\}is the time reached after itsNNth update and may vary across trajectories for adaptive methods\.
#### B\.1\.2Calculation of Fisher information

At a deterministic timeTT, the time\-averaged lawμ¯T\\bar\{\\mu\}\_\{T\}is the law of interpolation at an independently sampled uniform time in\[0,T\]\[0,T\], andμT\\mu\_\{T\}is the terminal law atTT\. In our theoretical results, we prove nonasmyptotic convergence guarantees to achieveFI\(μ¯T∥π\)≤ε\\mathrm\{FI\}\(\\bar\{\\mu\}\_\{T\}\\\|\\pi\)\\leq\\varepsilon\. Therefore our primary results show comparison ofFI\\mathrm\{FI\}convergence of different LMC methods with respect toμ¯T\\bar\{\\mu\}\_\{T\}\. To estimateμ¯T\\bar\{\\mu\}\_\{T\}, for each 8192 trajectory generated by an LMC method after timeTT, we randomly samplettuniformly from\[0,T\]\[0,T\], and find the interval such thatt∈\[tk,tk\+1\)t\\in\[t\_\{k\},t\_\{k\+1\}\)fork∈\{0,…,N\}k\\in\\\{0,\\ldots,N\\\}\. Then we use the interpolation argument to obtain

Xt=Xtk\+u​Fk\+2​\(Wtk\+u−Wtk\)\.X\_\{t\}=X\_\{t\_\{k\}\}\+uF\_\{k\}\+\\sqrt\{2\}\(W\_\{t\_\{k\}\+u\}\-W\_\{t\_\{k\}\}\)\.Conditional onXkX\_\{k\}anduu,Xtk\+u∼𝒩⁡\(Xk\+u​Fk,2​u​I2\)X\_\{t\_\{k\}\+u\}\\sim\{\\mathcal\{N\}\}\(X\_\{k\}\+uF\_\{k\},2uI\_\{2\}\)\. Then eachXtX\_\{t\}sampled from a trajectory provides a Gaussian component with a known mean and covariance\. Using this fact, the density ofμ¯T\\bar\{\\mu\}\_\{T\}is approximated by the following density

p⁡\(x\)=18192​∑i=18192𝒩⁡\(x,mi,2​ui​I2\),mi=Xki\+ui​Fki,p\(x\)=\\frac\{1\}\{8192\}\\sum\_\{i=1\}^\{8192\}\{\\mathcal\{N\}\}\(x;m\_\{i\},2u\_\{i\}I\_\{2\}\),\\quad m\_\{i\}=X\_\{k\_\{i\}\}\+u\_\{i\}F\_\{k\_\{i\}\},\(123\)wherekik\_\{i\}denotes the iteration index corresponding to the intervalti∈\[tki,tki\+1\)t\_\{i\}\\in\[t\_\{k\_\{i\}\},t\_\{k\_\{i\}\+1\}\)andtit\_\{i\}is a sample uniformly drawn from\[0,T\]\[0,T\]forii\-th trajectory\. To ensure numerical stability for the narrow finite\-mixture components, we smooth both the simulated law and the target law with𝒩⁡\(0,0\.08​I2\)\{\\mathcal\{N\}\}\(0,0\.08I\_\{2\}\)and compute the Fisher information at each iteration or timeTTaccording to \([123](https://arxiv.org/html/2610.12865#A2.E123)\) and the target density\. For each LMC method, we apply this procedure at timesT∈\{0\.25,0\.5,1,2,4,8,16,32\}T\\in\\\{0\.25,0\.5,1,2,4,8,16,32\\\}\. For plots against iteration count, we useN∈\{32,64,128,256,512,1024,2048,4096,8192\}N\\in\\\{32,64,128,256,512,1024,2048,4096,8192\\\}\. The point atT=0T=0shows the initial FI with common Gaussian initialization\.

#### B\.1\.3Additional Results

Each curve shows the median FI across ten seeds, with the shaded band indicating the interquartile range\. Some curves end early because at least one seed does not yield a valid FI estimate at later checkpoints due to numerical instability, a simulation limit, or a failed FI calculation\. Figure[1](https://arxiv.org/html/2610.12865#S3.F1)compares methods at the same physical timeTT, whereas Figure[B\.1](https://arxiv.org/html/2610.12865#A2.F1)compares them after the same numberNNof updates withh=0\.02,0\.02,0\.16h=0\.02,0\.02,0\.16\. At fixedNN, adaptive methods may reach different physical times, but SG\-LMC and R\-LMC have equal oracle\-query budgets because they use the same number of queries per update\. In panel \(a\), R\-LMC closely follows exact\-gradient LMC, while SG\-LMC has substantially higher FI at largerNN\. Panel \(b\) considers exact\-gradient sampling from a target whose curvature can grow without bound; over the observed horizon, A\-LMC reaches lower FI than fixed\-step LMC\. In panel \(c\), SG\-LMC and R\-LMC diverge due to numerical instability, while A\-SG\-LMC converges to a suboptimal FI value\. RA\-LMC, by contrast, converges to a lower FI than LMC and follows A\-LMC, illustrating the benefit of combining robust estimation with adaptive step sizes under a fixed iteration budget\.

To complement Figures[1](https://arxiv.org/html/2610.12865#S3.F1)and[B\.1](https://arxiv.org/html/2610.12865#A2.F1), we repeat the three experiments with larger step\-size parametersh=0\.04h=0\.04,0\.040\.04, and0\.320\.32in panels \(a\)–\(c\), respectively, while keeping the targets, oracles, methods, seeds, and FI calculations unchanged\. Although A\-LMC, A\-SG\-LMC, and RA\-LMC use adaptive steps, their step sizesHkH\_\{k\}scale withhhat a given state, so increasinghhincreases their steps as well\. Figures[B\.2](https://arxiv.org/html/2610.12865#A2.F2)and[B\.3](https://arxiv.org/html/2610.12865#A2.F3)report the FI of the resulting time\-averaged laws relative to the target against timeTTand iteration countNN, respectively\.

Figure B\.2:Fisher information of the time\-averaged lawsμ¯T\\bar\{\\mu\}\_\{T\}across the three settings of Figure[1](https://arxiv.org/html/2610.12865#S3.F1), using larger step sizesh=0\.04h=0\.04,0\.040\.04, and0\.320\.32in panels \(a\)–\(c\), respectively\. For each trajectory, a time is sampled uniformly from\[0,T\]\[0,T\]to estimate the time\-averaged law\.Figure B\.3:Fisher information of the time\-averaged lawsμ¯τN\\bar\{\\mu\}\_\{\\tau\_\{N\}\}afterNNupdates for the same settings and step sizes as Figure[B\.2](https://arxiv.org/html/2610.12865#A2.F2)\. For each trajectory, a time is sampled uniformly from\[0,τN\]\[0,\\tau\_\{N\}\], whereτN\\tau\_\{N\}is the time reached after itsNNth update and may vary across trajectories for adaptive methods\.Figures[B\.2](https://arxiv.org/html/2610.12865#A2.F2)and[B\.3](https://arxiv.org/html/2610.12865#A2.F3)shows the same trends in Figure[1](https://arxiv.org/html/2610.12865#S3.F1)and Figure[B\.1](https://arxiv.org/html/2610.12865#A2.F1)at larger step sizes\. In the heavy\-tailed setting, SG\-LMC’s estimated FI rises sharply, while R\-LMC remains much closer to exact\-gradient LMC\. Under generalized smoothness, A\-LMC achieves lower FI than fixed\-step LMC at later iterations\. In the combined setting, RA\-LMC remains close to A\-LMC and achieves lower FI than fixed\-step LMC and A\-SG\-LMC, whereas the fixed\-step stochastic LMC methods become numerically unstable, leaving their curves absent or truncated\. The fixed\-NNresults complement the fixed\-TTcomparison by showing that these qualitative advantages persist at matched iteration budgets\.

### B\.2Bayesian Fine\-Tuning Experiments

#### B\.2\.1Experimental Setting

This subsection uses two separate fine\-tuning protocols\. To study the BG\-pporacle, we generate trajectories using minibatch\-mean gradients and AdamW, and make all histogram and moment measurements along these trajectories\. For the baseline comparison, empirical RA\-LMC retains AdamW and the Gaussian perturbation but replaces the minibatch mean with our geometric\-median\-centered, residual\-clipped estimate\. Thus \([124](https://arxiv.org/html/2610.12865#A2.E124)\) defines the trajectories used for the oracle evidence, whereas \([127](https://arxiv.org/html/2610.12865#A2.E127)\) defines the RA\-LMC runs reported in Table[1](https://arxiv.org/html/2610.12865#S4.T1)\.

##### Shared model and optimizer settings\.

We fine\-tune Qwen3\-4B on OpenBookQA and Qwen3\-VL\-4B\-Instruct on the SLAKE yes/no subset, following the model and dataset settings in Bayesian Adaptation Gym\([Samplawski et al\., 2026](https://arxiv.org/html/2610.12865#bib.bib53)\)\. The training splits contain 4,957 and 1,681 examples, respectively\. We freeze each backbone and train rank\-8 LoRA factors onq\_proj,v\_proj, andlm\_head, using LoRA scaling 16 and zero dropout\. The vectorwwconcatenates all trainable LoRA entries, and all distances and gradient norms are unnormalized Euclidean norms over this vector\.

Both protocols use 5,000 AdamW updates with moment coefficients\(0\.9,0\.999\)\(0\.9,0\.999\), numerical stabilizer10−610^\{\-6\}, and zero weight decay\. The scalar learning rateηk\\eta\_\{k\}warms up over 300 updates to10−410^\{\-4\}and then decreases linearly\. We use AdamW\-based updates rather than a fixed step size because generalized smoothness permits the local geometry and gradient scale to vary along a fine\-tuning trajectory\. AdamW provides a practical adaptation mechanism without requiring the smoothness and local noise\-scale quantities in our theoretical step\-size rule\. We run the fine\-tuning and evaluation experiments on single NVIDIA H100 GPU 40GB per run\.

##### Trajectories used for oracle evidence\.

To study the ordinary stochastic\-gradient noise, these runs use minibatch\-mean gradients rather than our robust estimator\.

At updatekk, the dataset is shuffled by epoch and the loss is the average negative log\-likelihood over the current minibatch\. Withb=4b=4, the update is

wk\+1=AdamW⁡\(wk,1b​∑j=1b𝒢⁡\(wk,ξk,j\),ηk\)\+γ​2​ηk​Zk,Zk∼𝒩⁡\(0,I\)\.w\_\{k\+1\}=\\operatorname\{AdamW\}\\\!\\left\(w\_\{k\},\\frac\{1\}\{b\}\\sum\_\{j=1\}^\{b\}\{\\mathcal\{G\}\}\(w\_\{k\},\\xi\_\{k,j\}\);\\eta\_\{k\}\\right\)\+\\gamma\\sqrt\{2\\eta\_\{k\}\}\\,Z\_\{k\},\\qquad Z\_\{k\}\\sim\\mathcal\{N\}\(0,I\)\.\(124\)Hereξk,j\\xi\_\{k,j\}indexes thejjth example in the minibatch and𝒢⁡\(wk,ξk,j\)\{\\mathcal\{G\}\}\(w\_\{k\},\\xi\_\{k,j\}\)is its stochastic gradient\. The noise discount factor is selected asγ=10−5\\gamma=10^\{\-5\}and is selected according to a grid searchγ∈\{10−6,10−5,10−4,10−3\}\\gamma\\in\\\{10^\{\-6\},10^\{\-5\},10^\{\-4\},10^\{\-3\}\\\}over the validation sets, similar to the approach in[Kim and Hospedales \(2023\)](https://arxiv.org/html/2610.12865#bib.bib52)\. The isotropic Gaussian increment is added to the FP32 LoRA parameters after the AdamW update\. Figure[2](https://arxiv.org/html/2610.12865#S3.F2)uses the OpenBookQA fine\-tuning trajectory, while Figure[3](https://arxiv.org/html/2610.12865#S3.F3)uses both fine\-tuning trajectories\.

##### RA\-LMC runs used for the baseline comparison\.

For the empirical RA\-LMC results in Table[1](https://arxiv.org/html/2610.12865#S4.T1), each update draws 4 distinct training examples and evaluates their gradients at the same parameter vector\. Let

gk,j\\displaystyle g\_\{k,j\}:=𝒢\(wk,ξk,j\),j=1,…,4,\\displaystyle:=\{\\mathcal\{G\}\}\(w\_\{k\},\\xi\_\{k,j\}\),\\qquad j=1,\\ldots,4,\(125\)Ck\\displaystyle C\_\{k\}:=GeoMed⁡\(gk,1,gk,2,gk,3\),\\displaystyle:=\\mathrm\{GeoMed\}\(g\_\{k,1\},g\_\{k,2\},g\_\{k,3\}\),λk\\displaystyle\\lambda\_\{k\}:=cclip​sk,\\displaystyle:=c\_\{\\mathrm\{clip\}\}s\_\{k\},g^k\\displaystyle\\widehat\{g\}\_\{k\}:=Ck\+clipλk​\(gk,4−Ck\),\\displaystyle:=C\_\{k\}\+\\mathrm\{clip\}\_\{\\lambda\_\{k\}\}\(g\_\{k,4\}\-C\_\{k\}\),whereclipλ​\(v\)=v​min⁡\{1,λ/‖v‖\}\\mathrm\{clip\}\_\{\\lambda\}\(v\)=v\\min\\\{1,\\lambda/\\\|v\\\|\\\}forv≠0v\\neq 0andclipλ​\(0\)=0\\mathrm\{clip\}\_\{\\lambda\}\(0\)=0\. Thuscclipc\_\{\\mathrm\{clip\}\}is the tuned multiplier, whileλk\\lambda\_\{k\}is the resulting clipping threshold at updatekk\. The geometric median is computed over the entire LoRA\-gradient vectors, rather than coordinatewise\. Because the center uses exactly three gradients, we do not solve the geometric\-median problem iteratively\. Although the gradients are high\-dimensional, any three gradient vectors lie in a common two\-dimensional plane or on a line\. Their geometric median can therefore be computed as the Fermat–Torricelli point of the triangle they form\([Mordukhovich and Nam, 2019](https://arxiv.org/html/2610.12865#bib.bib60)\)\. Letaja\_\{j\}denote the side length opposite the angleθj\\theta\_\{j\}atgk,jg\_\{k,j\}\. If someθj≥2​π/3\\theta\_\{j\}\\geq 2\\pi/3, thengk,jg\_\{k,j\}is the geometric median\. Otherwise, we use the explicit barycentric construction

ωj∝ajsin⁡\(θj\+π/3\),Ck=∑j=13ωj​gk,j,∑j=13ωj=1\.\\omega\_\{j\}\\ \\propto\\ \\frac\{a\_\{j\}\}\{\\sin\(\\theta\_\{j\}\+\\pi/3\)\},\\qquad C\_\{k\}=\\sum\_\{j=1\}^\{3\}\\omega\_\{j\}g\_\{k,j\},\\qquad\\sum\_\{j=1\}^\{3\}\\omega\_\{j\}=1\.ComputingCkC\_\{k\}therefore requires only three pairwise norms and one weighted average\. Its cost is𝒪⁡\(d\)\\mathcal\{O\}\(d\)forddtrainable LoRA parameters, with no numerical optimization loop or stopping tolerance\. The implementation also handles repeated and collinear gradients explicitly\.

The theoretical localpp\-moment noise scale depends on the unknown full gradient and gradient\-error distribution, so evaluating it at every update is prohibitively expensive\. We therefore usesks\_\{k\}as an online empirical dispersion proxy\. We initializes02s\_\{0\}^\{2\}with the Bessel\-corrected dispersion of 128 distinct training\-example gradients, providing a stable initial clipping radius\. After formingg¯k=14​∑j=14gk,j\\bar\{g\}\_\{k\}=\\frac\{1\}\{4\}\\sum\_\{j=1\}^\{4\}g\_\{k,j\}, we update the scale by

sk\+12=max⁡\{10−20,0\.95​sk2\+0\.053​∑j=14‖gk,j−g¯k‖2\}\.s\_\{k\+1\}^\{2\}=\\max\\\!\\left\\\{10^\{\-20\},0\.95s\_\{k\}^\{2\}\+\\frac\{0\.05\}\{3\}\\sum\_\{j=1\}^\{4\}\\\|g\_\{k,j\}\-\\bar\{g\}\_\{k\}\\\|^\{2\}\\right\\\}\.\(126\)This exponential moving average tracks changes in local gradient dispersion while reusing the four gradients already required by the estimator, so it introduces no additional oracle complexity\. The current robust estimate in \([125](https://arxiv.org/html/2610.12865#A2.E125)\) uses the lagged valuesks\_\{k\}, before the scale update, which prevents a current outlier from increasing its own clipping threshold\. The tuned multipliercclipc\_\{\\mathrm\{clip\}\}accounts for the difference between this empirical proxy and the theoretical noise scale\. Empirical RA\-LMC then applies

wk\+1=AdamW⁡\(wk,g^k,ηk\)\+γRA​2​ηk​Zk,Zk∼𝒩⁡\(0,I\)\.w\_\{k\+1\}=\\operatorname\{AdamW\}\(w\_\{k\},\\widehat\{g\}\_\{k\};\\eta\_\{k\}\)\+\\gamma\_\{\\mathrm\{RA\}\}\\sqrt\{2\\eta\_\{k\}\}\\,Z\_\{k\},\\qquad Z\_\{k\}\\sim\\mathcal\{N\}\(0,I\)\.\(127\)The Gaussian increment is again added after AdamW and does not enter its running moments\. OpenBookQA uses\(cclip,γRA\)=\(1\.191,3×10−6\)\(c\_\{\\mathrm\{clip\}\},\\gamma\_\{\\mathrm\{RA\}\}\)=\(1\.191,3\\times 10^\{\-6\}\), while SLAKE uses\(5\.554,3×10−4\)\(5\.554,3\\times 10^\{\-4\}\), which were selected by hyperparameter tuning on the validation sets using grid search\. Since RA\-LMC uses4​b4bstochastic gradients per update, we useb=1b=1for a fair comparison with other baselines\. For each reported random seed, ten independent runs of \([127](https://arxiv.org/html/2610.12865#A2.E127)\) produce ten terminal LoRA adapters whose class probabilities are averaged at test time\. We follow a similar procedure for each baseline\.

We also evaluated otherwise identical runs with a zero\-mean Gaussian prior on the trainable LoRA parameters but observed no meaningful difference in performance, so the reported experiments omit an explicit prior\.

##### Gradient\-error measurements\.

The remainder of this subsection considers only the experiments related to BG\-pporacle with trajectories from \([124](https://arxiv.org/html/2610.12865#A2.E124)\)\. The heavy\-tail and state\-dependence experiments both use the norm of the difference between a minibatch\-mean gradient and the gradient calculated over the full training set at a fixed LoRA parameter vector\. The heavy\-tail experiment studies its distribution across minibatches at selected checkpoints\. The state\-dependence experiment instead studies how its conditional first moment changes along the trajectory\. We first describe this common error measurement and then describe two minibatch\-sampling procedures\.

At each checkpoint, we holdwkw\_\{k\}fixed and place the model in evaluation mode\. Writingℓr​\(w\)\\ell\_\{r\}\(w\)for the classification loss of training examplerr, we compute the gradient over the entire training set and theiith minibatch\-mean gradient as

∇V​\(wk\)\\displaystyle\\nabla V\(w\_\{k\}\)=1n​∑r=1n∇ℓr​\(wk\),\\displaystyle=\\frac\{1\}\{n\}\\sum\_\{r=1\}^\{n\}\\nabla\\ell\_\{r\}\(w\_\{k\}\),\(128\)∇V^i,k\\displaystyle\\widehat\{\\nabla V\}\_\{i,k\}:=1b​∑j=1b𝒢⁡\(wk,ξk,i,j\)=1b​∑j=1b∇ℓξk,i,j​\(wk\)\.\\displaystyle:=\\frac\{1\}\{b\}\\sum\_\{j=1\}^\{b\}\{\\mathcal\{G\}\}\(w\_\{k\},\\xi\_\{k,i,j\}\)=\\frac\{1\}\{b\}\\sum\_\{j=1\}^\{b\}\\nabla\\ell\_\{\\xi\_\{k,i,j\}\}\(w\_\{k\}\)\.Hereξk,i,j\\xi\_\{k,i,j\}indexes thejjth example in theiith measurement minibatch at checkpointkk, andb=4b=4\. In both \([124](https://arxiv.org/html/2610.12865#A2.E124)\) and \([128](https://arxiv.org/html/2610.12865#A2.E128)\),𝒢⁡\(w,ξ\)\{\\mathcal\{G\}\}\(w,\\xi\)denotes one stochastic gradient, as in the preliminaries, while the minibatch estimate is written explicitly as an average\. Training also includes the final singleton\. The reference uses every training example, evaluated in chunks of four\. We weight each chunk’s mean gradient by its number of examples, include the final singleton, accumulate in FP64, and divide bynn\. Thus the reference is computed from the full empirical objective, up to numerical precision\. For each measurement minibatchiiat checkpointkk, we record

Ri,k=‖∇V^i,k−∇V​\(wk\)‖2\.R\_\{i,k\}=\\left\\lVert\\widehat\{\\nabla V\}\_\{i,k\}\-\\nabla V\(w\_\{k\}\)\\right\\rVert\_\{2\}\.\(129\)Both gradients in \([129](https://arxiv.org/html/2610.12865#A2.E129)\) are measured before any optimizer transformation or robustification\. We add no Gaussian noise to these gradients; the Gaussian increments affect only the trajectory at which they are evaluated\. No validation or test examples enter the oracle measurements\. The histogram and moment analyses below use this same error norm, but draw their measurement minibatches in different ways\.

#### B\.2\.2Gradient\-Error Histograms

We examine checkpoints at updates00,12501250,25002500,37503750, and50005000\. At each checkpoint, we partition a deterministic permutation of the training examples into disjoint complete four\-example minibatches\. OpenBookQA provides 1,239 complete minibatches from 4,957 training examples, and SLAKE provides 420 from 1,681 examples\.

To assess whether the observed gradient error histograms resemble a theoretical heavy tailed model with an infinite second moment, at each checkpoint we fit the following positiveβ\\beta\-stable reference model to the squared error norms and compare the empirical norm histograms with synthetic norm histograms generated from the fit:

R2=a\+c​Sβ,𝔼⁡\[e−t​Sβ\]=e−tβ,0<β<1,t≥0,R^\{2\}=a\+cS\_\{\\beta\},\\qquad\\mathbb\{E\}\\bigl\[e^\{\-tS\_\{\\beta\}\}\\bigr\]=e^\{\-t^\{\\beta\}\},\\qquad 0<\\beta<1,\\quad t\\geq 0,\(130\)whereSβS\_\{\\beta\}is a standard positiveβ\\beta\-stable random variable, equivalently a one\-sided Lévy\-stable random variable under the normalization in \([130](https://arxiv.org/html/2610.12865#A2.E130)\)\. Since𝔼⁡\[Sβ\]=∞\\mathbb\{E\}\[S\_\{\\beta\}\]=\\inftyfor0<β<10<\\beta<1, the reference model satisfies𝔼⁡\[R2\]=∞\\mathbb\{E\}\[R^\{2\}\]=\\infty\. The parametersaaandccare a nonnegative location and a positive scale, respectively\. The fit minimizes the normalized 80\-bin conditional multinomial negative log\-likelihood plus ten times the Bernoulli KL divergence between the observed and fitted probabilities of the tallest observed bin\. Thus the objective gives additional weight to matching the histogram peak\. We use L\-BFGS\-B with multiple initializations, constrainβ∈\[0\.2,0\.98\]\\beta\\in\[0\.2,0\.98\]andc∈\[10−12,105\]c\\in\[10^\{\-12\},10^\{5\}\], and boundaaabove by0\.9990\.999times the smallest retained squared norm\.

The synthetic panel drawsR=a\+c​SβR=\\sqrt\{a\+cS\_\{\\beta\}\}from the fitted model and retains draws in the same interval until their number equals the retained empirical count\. Both histograms use the same 80 equal\-width bins and divide bin counts by the retained count times bin width\. AlthoughR2=a\+c​SβR^\{2\}=a\+cS\_\{\\beta\}follows a shifted and scaled positiveβ\\beta\-stable distribution, the plotted synthetic quantity isR=a\+c​SβR=\\sqrt\{a\+cS\_\{\\beta\}\}\. Thus, the synthetic error norms are generated from a fitted positiveβ\\beta\-stable reference model for the squared error norms but are not themselves stable random variables\. For Figure[2](https://arxiv.org/html/2610.12865#S3.F2), the fitted parameters areβ=0\.2\\beta=0\.2,a=0\.0729963a=0\.0729963, andc=1\.3290073c=1\.3290073withβ\\betaat its lower search bound\.

#### B\.2\.3State Dependence of Gradient\-Error Moments

For Figure[3](https://arxiv.org/html/2610.12865#S3.F3), we save 4,000 approximately equally spaced checkpoints across the 5,000 update trajectory and compute their distances‖wk−w5000‖2\\left\\lVert w\_\{k\}\-w\_\{5000\}\\right\\rVert\_\{2\}from the final LoRA iterate, which is denoted asω5000\\omega\_\{5000\}\. We place 50 equally spaced target distances between zero and the largest observed distance, then select 50 distinct saved checkpoints by minimizing the sum of absolute differences between their distances and these targets\. This one\-to\-one assignment gives approximately uniform coverage in distance\. Figure[3](https://arxiv.org/html/2610.12865#S3.F3)includes the final iterate for both datasets\. The selected states farthest from the final iterate correspond to updates 1 and 3 for OpenBookQA and SLAKE, respectively\.

At each selected checkpoint, we approximate the exact gradient by computing the gradients across the entire training set and compute 256 independent minibatch gradients\. Each minibatch consists of four examples sampled uniformly from the dataset\. We partition the resulting error norms in \([129](https://arxiv.org/html/2610.12865#A2.E129)\) into eight disjoint groups of 32 and plot

\(‖wk−w5000‖2,m^1,k,j\),m^1,k,j=132​∑i=32​\(j−1\)\+132​jRi,k,j=1,…,8\.\\left\(\\left\\lVert w\_\{k\}\-w\_\{5000\}\\right\\rVert\_\{2\},\\ \\widehat\{m\}\_\{1,k,j\}\\right\),\\qquad\\widehat\{m\}\_\{1,k,j\}=\\frac\{1\}\{32\}\\sum\_\{i=32\(j\-1\)\+1\}^\{32j\}R\_\{i,k\},\\quad j=1,\\ldots,8\.\(131\)This process was repeated for 50 checkpoints\. The eight estimates at a given horizontal coordinate quantify variation across independently sampled minibatch groups\. Each vertical coordinate averages error norms, rather than taking the norm of an averaged error vector\. The results in Figure[3](https://arxiv.org/html/2610.12865#S3.F3)shows these estimates as scatter plots for OpenBookQA and SLAKE datasets\. We report the first moment because it is well defined under every BG\-ppmodel withp\>1p\>1and is less sensitive to extreme gradient errors than higher empirical moments\. This provides a stable and interpretable view of how the typical gradient\-error magnitude changes along the finite\-tuning trajectory\. The resulting plots in Figure[3](https://arxiv.org/html/2610.12865#S3.F3)provide evidence of state dependence of the oracle noise\.

#### B\.2\.4Additional Results

##### Histograms across checkpoints\.

Figures[B\.4](https://arxiv.org/html/2610.12865#A2.F4)and[B\.5](https://arxiv.org/html/2610.12865#A2.F5)compare empirical and synthetic histograms at five checkpoints during Qwen3\-4B fine tuning on OpenBookQA and Qwen3\-VL\-4B\-Instruct fine tuning on SLAKE, respectively\. We use the procedure described in Subsection[B\.2\.2](https://arxiv.org/html/2610.12865#A2.SS2.SSS2)\. In both settings, the empirical error norm distributions closely resemble the synthetic distributions generated from this fitted reference model\. This agreement becomes more pronounced at and after step 1,250\. These results provide further empirical support for the observation in Figure[2](https://arxiv.org/html/2610.12865#S3.F2)that stochastic gradient errors exhibit heavy tailed behavior\.

Figure B\.4:Minibatch gradient\-error norm histograms across Qwen3\-4B LoRA fine\-tuning on OpenBookQA\. Columns show steps00,12501250,25002500,37503750, and50005000\. The top row contains empirical norms and the bottom row contains samples generated from the fitted positiveβ\\beta\-stable reference model in \([130](https://arxiv.org/html/2610.12865#A2.E130)\)\. Both rows use the same 80 bins and axes, and fittedβ\\betavalues are\(0\.200,0\.200,0\.200,0\.607,0\.837\)\(0\.200,0\.200,0\.200,0\.607,0\.837\)\.Figure B\.5:Minibatch gradient\-error norm histograms across Qwen3\-VL\-4B\-Instruct LoRA fine\-tuning on SLAKE\. The rows, columns, bins, and density normalization follow Figure[B\.4](https://arxiv.org/html/2610.12865#A2.F4)\. Fittedβ\\betavalues are\(0\.287,0\.200,0\.578,0\.730,0\.972\)\(0\.287,0\.200,0\.578,0\.730,0\.972\)\.
##### Gradient\-error moments of different orders\.

We additionally repeat the state\-dependence calculation forp∈\{1,1\.25,1\.5,1\.75,2\}p\\in\\\{1,1\.25,1\.5,1\.75,2\\\}\. Using the same checkpoints and measurement minibatches as in \([131](https://arxiv.org/html/2610.12865#A2.E131)\), we compute

m^p,k,j=132∑i=32​\(j−1\)\+132​jRi,kp,j=1,…,8\.\\widehat\{m\}\_\{p,k,j\}=\\frac\{1\}\{32\}\\sum\_\{i=32\(j\-1\)\+1\}^\{32j\}R\_\{i,k\}^\{p\},\\qquad j=1,\\ldots,8\.\(132\)Figures[B\.6](https://arxiv.org/html/2610.12865#A2.F6)and[B\.7](https://arxiv.org/html/2610.12865#A2.F7)plot these estimates against the distance from the final LoRA iterate\. Across the displayed moment orders, the empirical moments generally increase with distance, supporting a state\-dependent noise scale beyond the first\-moment results in Figure[3](https://arxiv.org/html/2610.12865#S3.F3)\.

Figure B\.6:Empiricalpp\-th moments of minibatch gradient\-error norms versus distance from the final LoRA iterate during Qwen3\-4B fine tuning on OpenBookQA\. Panels reportp∈\{1,1\.25,1\.5,1\.75,2\}p\\in\\\{1,1\.25,1\.5,1\.75,2\\\}\. Each panel contains eight estimates at each of 50 checkpoints, and every estimate averagesRi,kpR\_\{i,k\}^\{p\}over 32 independently sampled minibatch estimates with batch size of 4 as in \([132](https://arxiv.org/html/2610.12865#A2.E132)\)\.Figure B\.7:Empiricalpp\-th moments of minibatch gradient\-error norms versus distance from the final LoRA iterate during Qwen3\-VL\-4B\-Instruct fine tuning on SLAKE\. The moment orders, checkpoints, and grouped estimates follow Figure[B\.6](https://arxiv.org/html/2610.12865#A2.F6)\.

## Appendix CRelated Work

### C\.1Non\-Log\-Concave Langevin Sampling

Earlier non\-log\-concave sampling results often seek convergence in a distributional metric such as KL divergence, total variation, or a Wasserstein distance\. Such guarantees describe global mixing, which relative Fisher information \(FI\) alone does not ensure: a law can have small FI while assigning incorrect mass to well\-separated modes\. The stronger conclusion generally comes with assumptions on the target\. For example, analyses based on a log\-Sobolev or Poincaré inequality use that inequality to turn local dissipation into global convergence\([Vempala and Wibisono, 2019](https://arxiv.org/html/2610.12865#bib.bib9);[Chewi et al\., 2022](https://arxiv.org/html/2610.12865#bib.bib12)\)\. Other finite\-time analyses assume dissipativity or convexity outside a compact set\([Durmus and Moulines, 2017](https://arxiv.org/html/2610.12865#bib.bib13);[Cheng et al\., 2018](https://arxiv.org/html/2610.12865#bib.bib14);[Majka et al\., 2020](https://arxiv.org/html/2610.12865#bib.bib15)\), or exploit an explicit mixture structure\([Lee et al\., 2018](https://arxiv.org/html/2610.12865#bib.bib17)\)\. These distributional guarantees are not weaker versions of FI guarantees; they address a different, more global question under additional structure\.

Global gradient Lipschitzness is not universal in this literature\. For example,[Erdogdu and Hosseinzadeh \(2021\)](https://arxiv.org/html/2610.12865#bib.bib16)derive finite\-time KL bounds with Hölder gradients, together with prescribed tail growth\. Tamed Langevin methods accommodate faster\-growing drifts:[Brosse et al\. \(2019\)](https://arxiv.org/html/2610.12865#bib.bib38)impose polynomial local regularity and an outward\-pointing gradient at infinity, obtaining finite\-time weighted total\-variation bounds; their Wasserstein result additionally assumes global strong convexity\.[Lytras and Mertikopoulos \(2024\)](https://arxiv.org/html/2610.12865#bib.bib21)obtain finite\-time distributional bounds under dissipativity and a Poincaré or log\-Sobolev inequality\. For a neural\-network objective,[Lovas et al\. \(2023\)](https://arxiv.org/html/2610.12865#bib.bib42)use polynomial regularity and a superquadratic regularizer to enforce dissipativity, together with high\-order data and initialization moments\. Their guarantees concern the Gibbs law of the regularized objective\. Thus, relaxing global smoothness has generally required other control of the tails or the target geometry\.

Uniformly bounded stochastic\-gradient variance is not assumed by every non\-convex sampling analysis either\.[Raginsky et al\. \(2017\)](https://arxiv.org/html/2610.12865#bib.bib40)allow the conditional variance to grow quadratically with the distance from the origin and prove non\-asymptotic Wasserstein and excess\-risk bounds for SGLD\. Their component losses nonetheless have globally Lipschitz gradients, the potential is dissipative, and the oracle has a finite second moment at every state\. Their result therefore covers state dependence of a BG\-0 type, but not the possibly infinite variance permitted by BG\-ppwhenp<2p<2\. In a different robustness model,[Bhatia et al\. \(2019\)](https://arxiv.org/html/2610.12865#bib.bib41)use robust gradients to handle adversarially contaminated data under global gradient Lipschitzness and strong convexity\. These are finite\-time results, so the distinction from our setting is in the assumptions and convergence criterion, not simply asymptotic versus non\-asymptotic analysis\.

The FI framework of[Balasubramanian et al\. \(2022\)](https://arxiv.org/html/2610.12865#bib.bib1)avoids a functional inequality and dissipativity by bounding stationarity of a time\-averaged law\. Its direct LMC analysis assumes a globally Lipschitz gradient, while its standard SG\-LMC theorem uses a state\-uniform bound on squared gradient error\. The same work gives two important qualifications: Gaussian smoothing handles globally Hölder gradients, and a finite\-sum PAGE construction obtains an FI guarantee without a state\-uniform variance assumption when component gradients are globally Lipschitz\. Subsequent work studies the meaning and complexity of FI guarantees, as well as other algorithms in this framework\([Chewi et al\., 2023](https://arxiv.org/html/2610.12865#bib.bib4);[Cheng et al\., 2023](https://arxiv.org/html/2610.12865#bib.bib6);[Chewi and Wibisono, 2026](https://arxiv.org/html/2610.12865#bib.bib5);[Sahin et al\., 2026b](https://arxiv.org/html/2610.12865#bib.bib2);[Sahin et al\., 2026a](https://arxiv.org/html/2610.12865#bib.bib3)\)\. These results establish non\-asymptotic progress without requiring full mixing, but the aforementioned extensions do not provide a direct SG\-LMC FI bound under both gradient\-dependent curvature and a state\-dependent finite\-pp\-moment oracle\. Our theorems address that setting while retaining the more limited FI conclusion\.

### C\.2Generalized Smoothness and Heavy\-Tailed Noise in Optimization

Generalized smoothness lets the local change in a gradient depend on the gradient magnitude rather than a single global Lipschitz constant\. Under a symmetric generalized\-smoothness condition,[Chen et al\. \(2023\)](https://arxiv.org/html/2610.12865#bib.bib30)derive descent bounds and use normalized gradient descent for non\-convex objectives; they analyze a variance\-reduced stochastic method as well\.[Li et al\. \(2023\)](https://arxiv.org/html/2610.12865#bib.bib31)instead control gradients along the optimization trajectory, allowing standard gradient methods to recover familiar rates under their generalized\-smoothness conditions\. Their stochastic results permit heavy\-tailed distributions with finite variance, which is distinct from assuming only a finitepp\-th moment forp<2p<2\.

For noise with a finitepp\-th moment and potentially infinite variance, optimization methods often limit the influence of extreme gradients\.[Cutkosky and Mehta \(2021\)](https://arxiv.org/html/2610.12865#bib.bib32)use clipping, momentum, and normalization to obtain high\-probability stationarity guarantees\. Related clipped\-gradient bounds are given by[Nguyen et al\. \(2023\)](https://arxiv.org/html/2610.12865#bib.bib33), while[Hübler et al\. \(2025\)](https://arxiv.org/html/2610.12865#bib.bib34)analyze normalized SGD under a finite centralpp\-th moment\. State\-dependent second\-moment assumptions have motivated a different set of techniques\.[Alacaoglu et al\. \(2025\)](https://arxiv.org/html/2610.12865#bib.bib35)study algorithms under quadratically growing variance for convex problems, and[Fazla et al\. \(2026\)](https://arxiv.org/html/2610.12865#bib.bib36)use anchoring and regularization for non\-convex problems under BG\-0 noise\. Work addressing more than one relaxation includes clipped SGD for convex\(L0,L1\)\(L\_\{0\},L\_\{1\}\)\-smooth objectives with heavy\-tailed noise\([Chezhegov et al\., 2025](https://arxiv.org/html/2610.12865#bib.bib37)\)\. These papers target objective suboptimality or small gradient norm, not the distribution of Langevin iterates\.

This change of objective matters when importing an optimization update into a sampler\. The overdamped Langevin diffusion has drift−∇V\-\\nabla Vand stationary densityπ∝e−V\\pi\\propto e^\{\-V\}\. If one replaces that drift by a clipped or normalized vector fieldbbbut leaves the Brownian term unchanged, the adjoint of the new generator acts on the target density as

ℒb∗π=−∇⋅\(π\(b\+∇V\)\),\\mathcal\{L\}\_\{b\}^\{\*\}\\pi=\-\\nabla\\\!\\cdot\\\!\\bigl\(\\pi\(b\+\\nabla V\)\\bigr\),\(133\)which generally does not vanish\. A descent argument for an optimizer therefore does not show that its modified drift samples fromπ\\pi\. Tamed and adaptive sampling methods account for the target law explicitly\([Brosse et al\., 2019](https://arxiv.org/html/2610.12865#bib.bib38);[Leroy et al\., 2024](https://arxiv.org/html/2610.12865#bib.bib39)\)\.

There are two further obstacles for our FI objective\. First, the KL\-dissipation calculation involves a time integral of the*squared*error in the Langevin drift\. An unbiased oracle with only a finitepp\-th moment may have infinite squared error\. Clipping the entire stochastic gradient would control its norm but would distort the target drift, whereas centering and clipping its residual allow us to control estimation bias and squared error separately\. Second, a step chosen from a noisy gradient is correlated with its estimation error\. FI bounds then require control of the error*weighted by the duration of that step*; an ordinary optimization descent bound does not provide this control\. Our adaptive algorithms use physical elapsed time, with the drift and Brownian increment scaled by the same step duration\. This is also why an average over physical time must be distinguished from an average over iteration indices\.

### C\.3Bayesian Fine\-Tuning and Experimental Baselines

Low\-rank adaptation \(LoRA\) fine\-tunes a model through trainable low\-rank factors while freezing the backbone\([Hu et al\., 2022](https://arxiv.org/html/2610.12865#bib.bib44)\)\. Bayesian LoRA methods aim to represent uncertainty in these additional parameters without maintaining a posterior over all backbone weights\. We use the reference implementations of nine methods from Bayesian Adaptation Gym \(BAG\)\([Samplawski et al\., 2026](https://arxiv.org/html/2610.12865#bib.bib53)\)in Table[1](https://arxiv.org/html/2610.12865#S4.T1)\. For methods that produce multiple predictions, we average their class probabilities before evaluating predictive metrics\.

The*MLE*baseline minimizes classification negative log\-likelihood and returns one LoRA adapter\([Hu et al\., 2022](https://arxiv.org/html/2610.12865#bib.bib44)\)\. The BAG*MAP*baseline likewise returns one adapter but trains it with AdamW weight decay of0\.010\.01\([Loshchilov and Hutter, 2019](https://arxiv.org/html/2610.12865#bib.bib68);[Samplawski et al\., 2026](https://arxiv.org/html/2610.12865#bib.bib53)\)\. Neither procedure represents a posterior over adapter parameters\.*Temperature scaling*changes the logits of an MLE adapter by a fitted scalar without retraining the adapter\([Guo et al\., 2017](https://arxiv.org/html/2610.12865#bib.bib45)\)\.*Monte Carlo dropout*instead keeps LoRA dropout active at prediction time and averages ten stochastic forward passes; the BAG implementation uses a dropout probability of0\.10\.1\([Gal and Ghahramani, 2016](https://arxiv.org/html/2610.12865#bib.bib46)\)\.

*Deep Ensemble*trains ten LoRA adapters independently with the MLE objective and averages their predictions\([Lakshminarayanan et al\., 2017](https://arxiv.org/html/2610.12865#bib.bib47)\)\.*Laplace LoRA*starts from a fine\-tuned adapter, fits a local Gaussian approximation using a Kronecker\-factored curvature estimate, and samples approximate logits for prediction\([Yang et al\., 2024](https://arxiv.org/html/2610.12865#bib.bib48)\)\. These methods obtain predictive diversity by independent optimization or a local posterior approximation rather than by following a Langevin chain\.

Other baselines learn or construct distributions over low\-rank updates\.*BLoB*jointly learns a variational mean and covariance during fine\-tuning, using a diagonal Gaussian approximation for a LoRA factor\([Wang et al\., 2024](https://arxiv.org/html/2610.12865#bib.bib49)\)\.*ScalaBL*performs variational inference in a latent space whose dimension is the LoRA rank and maps latent samples through the LoRA factors\([Samplawski et al\., 2025](https://arxiv.org/html/2610.12865#bib.bib50)\)\.*Training\-Free Bayesianization*introduces Gaussian uncertainty around a trained adapter and searches for an acceptable noise level without further adapter training\([Shi et al\., 2025](https://arxiv.org/html/2610.12865#bib.bib51)\)\. The BAG implementations average ten predictive samples for each of these three methods\.

Our additional*SG\-LMC*comparator uses minibatch\-averaged gradients, a fixed step size, and discounted Gaussian perturbations, following the stochastic Langevin principle\([Welling and Teh, 2011](https://arxiv.org/html/2610.12865#bib.bib43)\)\. It differs from RA\-LMC in its gradient estimator and step\-size choice\. Related fine\-tuning work also uses Langevin sampling for a different purpose:*BayesTune*samples sparsity scale variables under a hierarchical prior and uses them to select weights for subsequent sparse fine\-tuning\([Kim and Hospedales, 2023](https://arxiv.org/html/2610.12865#bib.bib52)\)\. Its Langevin iterates are not posterior samples of the final fine\-tuned LoRA adapter, so BayesTune addresses parameter selection rather than the predictive\-sampling comparison studied here\.