A Zeroth-Order Deep Learning Method for Fully Nonlinear Parabolic Partial Differential Equations with Unknown Coefficients

# A Zeroth-Order Deep Learning Method for Fully Nonlinear Parabolic Partial Differential Equations with Unknown Coefficients
Source: [https://arxiv.org/html/2606.24999](https://arxiv.org/html/2606.24999)
\\nameYanwei Jia\\emailyanweijia@cuhk\.edu\.hk \\addrDepartment of Systems Engineering and Engineering Management The Chinese University of Hong Kong Shatin, New Territories, Hong Kong SAR\\nameDu Ouyang\\emailduouyang99@outlook\.com \\addrDepartment of Mathematical Sciences Tsinghua University Beijing, Beijing 100084, China\\nameHuyên Pham\\emailhuyen\.pham@polytechnique\.edu \\addrCMAP Ecole Polytechnique Route de Saclay, Palaiseau 91128, France\\nameXun Yu Zhou\\emailxz2574@columbia\.edu \\addrDepartment of Industrial Engineering and Operations Research & The Data Science Institute Columbia University New York, NY 10027, USA

###### Abstract

High\-dimensional partial differential equations \(PDEs\) with unknown coefficients arise widely in scientific machine learning, including continuous\-time reinforcement learning, yet solving them efficiently in a data\-driven way remains challenging\. Existing deep learning solvers often rely on repeated automatic differentiation to evaluate differential operators, which can cause instability and amplify derivative errors in high dimensions, while probabilistic methods based on stochastic representations require explicit knowledge of the data\-generating dynamics and therefore do not apply to black\-box environments\. We introduce two types of simulators as data\-generating mechanisms, and take a “representing\-then\-learning” approach that learns the solutions and their derivatives under settings where the underlying PDE operators are accessible only through simulations and pointwise evaluations\. Our representation of derivatives relies on the zeroth\-order derivative \(ZOD\) estimators derived from perturbed Monte Carlo trajectories\. This fully model\-free approach generates targets for the gradient and Hessian networks using only function evaluations\. We provide a statistical learning analysis of the proposed approach, including a bias–variance tradeoff for ZODs\. Assuming a standard contraction property of the underlying operator, we establish a non\-asymptotic error bound that decomposes the total error into discretization error, approximation error, statistical error, and ZOD bias\. Crucially, we derive the sample complexity of the learned representations in \(weighted\) Sobolev space, characterizing the error up to second\-order derivatives\. Numerical experiments illustrate the competitive performance of the method in moderate and high dimensions\.

Keywords:Nonlinear PDEs, zeroth\-order derivative estimators, model\-free methods, deep learning, sample complexity in Sobolev space

## 1Introduction

High\-dimensional nonlinear partial differential equations \(PDEs\) are ubiquitous and fundamental in various fields of science and engineering, ranging from physics and fluid dynamics to mathematical finance and stochastic control\. Recent progress in scientific machine learning employs deep neural networks \(DNNs\) for high\-dimensional function approximation to solve PDEs, aiming to overcome the curse of dimensionality where the traditional numerical methods fall apart\. While various deep learning methods have been proposed and have achieved promising numerical results, there are still limitations in their stability, scope of applications, and availability of theoretical foundations\.

Broadly speaking, existing methods fall into two main categories\. The first category consists of the residual\-based methods, such as the deep Galerkin method \(cf\.Sirignano and Spiliopoulos[2018](https://arxiv.org/html/2606.24999#bib.bib23)\), and physics\-informed neural networks \(PINNs; cf\.Raissiet al\.[2019](https://arxiv.org/html/2606.24999#bib.bib24)\)\. These approaches approximate solutions by minimizing the PDE residuals over sampled points in space–time, and are applicable to a wide class of PDEs\. However, they rely on repeated evaluation of differential operators through automatic differentiation and may suffer from optimization difficulties and stability issues, particularly in high dimensions or when second\-order derivatives are involved\. The second category consists of probabilistic methods that exploit stochastic representations of PDEs, including in particular methods based on backward stochastic differential equations \(BSDEs\) such as deep BSDEs\(Eet al\.,[2017](https://arxiv.org/html/2606.24999#bib.bib27)\)and deep backward dynamic programming \(DBDP; cf\.Huréet al\.[2020](https://arxiv.org/html/2606.24999#bib.bib25)\)\. However, these methods are primarily tailored to the special type of nonlinear PDEs where BSDE representations are available based on Peng’s nonlinear Feynman–Kac formula\(Peng,[1992](https://arxiv.org/html/2606.24999#bib.bib3)\); in particular, these PDEs are linear in the Hessians\. Forfullynonlinear PDEs, which are nonlinear in the unknown functions up to the second\-order derivatives, such representations generally do not exist, limiting their applicability\.111We refer toBecket al\.\([2023](https://arxiv.org/html/2606.24999#bib.bib28)\); Germainet al\.\([2023](https://arxiv.org/html/2606.24999#bib.bib29)\)for an overview, and toZhanget al\.\([2026](https://arxiv.org/html/2606.24999#bib.bib22)\); Doumècheet al\.\([2025](https://arxiv.org/html/2606.24999#bib.bib42)\); Gazouliset al\.\([2025](https://arxiv.org/html/2606.24999#bib.bib43)\); Bonitoet al\.\([2026](https://arxiv.org/html/2606.24999#bib.bib44)\); Wanget al\.\([2022](https://arxiv.org/html/2606.24999#bib.bib67)\)for more recent progress on these two types of methods\.

Notably, most of the existing literature on numerically solving PDEs, including the aforementioned papers, are “model\-based”, namely, the studied PDEs are completely specified with their coefficient functions assumed to be known and given \(hence so are the derivatives of those coefficients\)\. However, PDEs that modelcomplexsystems can be impossible or impractical to specify, leading to equations without knowledge of the underlying operators and the derivatives of their coefficients\. These PDEs operate like black\-boxes for which we can only observe certain input–output data streams\. Solving those black\-box PDEs is motivated by ample real\-life applications such as fluid turbulence, climate control, and the stabilization of plasma fusion reactors; see e\.g\.Liet al\.\([2020](https://arxiv.org/html/2606.24999#bib.bib4)\); Jiaoet al\.\([2021](https://arxiv.org/html/2606.24999#bib.bib5)\); Lagergrenet al\.\([2020](https://arxiv.org/html/2606.24999#bib.bib6)\)and the references therein for more motivating applications\. It is also related to the recently developed continuous\-time reinforcement learning\(Wanget al\.,[2020](https://arxiv.org/html/2606.24999#bib.bib2); Jia and Zhou,[2022a](https://arxiv.org/html/2606.24999#bib.bib12),[b](https://arxiv.org/html/2606.24999#bib.bib13),[2023](https://arxiv.org/html/2606.24999#bib.bib10)\), where the Hamilton–Jacobi–Bellman \(HJB\) equation becomes a black\-box PDE when the parameters of the underlying control problem are unknown\.

Another important aspect is that, besides obtaining the function approximation to the solution to a PDE, one often needs to find its derivatives for the downstream tasks, e\.g\., executing a control in a robotic system and hedging in finance, which concern the sensitivity of the solution with respect to the spatial variable\. Existing methods generally treat derivatives implicitly\. For example, PINNs and deep Galerkin follow a “learning\-then\-differentiating” approach: They train a neural network to minimize a value\-based loss, which nonetheless does not theoretically guarantee that its derivatives will converge to the true derivatives\. Consequently, when higher\-order derivatives are computed via automatic differentiation on an imperfectly trained network, approximation errors compound, sometimes catastrophically\. On the other hand, deep BSDE and DBDP require the equation coefficients to be known and given and embed gradients within model\-specific stochastic formulations\.

In this paper, we propose a new, data\-driven approach to learning the solutions to a large class offullynonlinear, black\-box parabolic PDEs with strong theoretical guarantees in \(weighted\) Sobolev space up to second\-order spatial derivatives\. The PDEs consist of linear second\-order parabolic operators and fully nonlinear source \(nonhomogeneous\) terms depending on the unknown solutions up to their second\-order derivatives\. Instead of recovering derivative information indirectly from a learned value approximation or from model\-specific sensitivity representations, we adopt a representing\-then\-learning paradigm, where derivative quantities are first represented through stochastic estimators and then learned directly\. This approach kills two birds with one stone: it deals with the unknown PDEs and learns the derivatives at the same time\.

One of the main contributions of the paper is to definepreciselywhat “data” and “data\-driven” mean in our context of solving PDEs\. Instead of the knowledge of the functional forms of PDE coefficients, we only require access to a “simulator” that, given an initial state, produces stochastic trajectories \(i\.e\., “data”\)\. These trajectories are diffusion processes, or equivalently solutions to the stochastic differential equations \(SDEs\) whose infinitesimal generator coincides with the aforementioned parabolic PDE operators\. Moreover, the nonlinear source term and the function in the terminal condition are also black\-box functions that can only output their function values when inputs are given\. Our approach is applicable under two simulator settings: a “weak simulator” that takes an initial state as input and generates an independently and identically distributed \(i\.i\.d\.\) random trajectory at each query, and a “strong simulator” that returns afamilyof trajectories indexed by a given set of initial states and driven by thesameunderlying randomness \(Brownian motion in our setting\), while remaining independent across queries\. The notions of weak and strong simulators align with the classical definitions of weak and strong solutions to SDEs and capture the key distinction between them: a strong solution is defined with respect to a fixed Brownian motion, whereas in a weak solution the Brownian motion is itself a part of the solution \(cf\.Karatzas and Shreve[2014](https://arxiv.org/html/2606.24999#bib.bib46), Chapter 5, Definitions 2\.1 and 3\.1\)\. Clearly, a strong simulator is automatically a weak one \(but notvice versa\) for which the “random seed” is also an input, so that querying the same initial state with the same seed produces the same trajectory\. Importantly, the availability of the strong simulator covers model\-based PDEs as a special case\. It is also valid in applications such as finance where the “random seed” represents the realized asset price movements and inventory systems where it represents exogenous demands\. Note that in both cases, the simulator remains a black box, and we neither know nor attempt to learn or infer its internal structure\.222There is a line of research called operator learning that aims to learn PDE solution operators by input–output data and DNNs \(cf\.Luet al\.[2021a](https://arxiv.org/html/2606.24999#bib.bib1)\), which is ultimately a model\-based approach\. By contrast, our approach is “model\-free” \(up to the form of the PDEs we study\) in that we learn the solutions directly without learning the PDEs themselves\.

Our method combines: \(i\) approximate value iteration, to reduce a nonlinear problem to a sequence of linear problems, each effectively a Feynman–Kac\-type equation \(equivalent to the Bellman equation for policy evaluation in reinforcement learning\); and \(ii\) zeroth\-order derivative \(ZOD\) estimators, to approximate gradients and Hessians via function values at random perturbed points\. At each iteration, wejointlylearn the value, gradient, and Hessian through a composite loss function, ensuring consistency across all quantities\. This loss function consists of two parts: the value loss and the derivative loss\. The value loss is in the same spirit as the martingale loss function\(Jia and Zhou,[2022a](https://arxiv.org/html/2606.24999#bib.bib12)\)for policy evaluation in continuous\-time reinforcement learning, which captures the mean\-squared error of the function value\. The derivative loss captures the distance between the gradient/Hessian networks and their stochastic ZOD targets\. Crucially, ZOD estimators require only function evaluations at perturbed trajectories and do not rely on model derivatives or automatic differentiation\.

The main contributions of this paper are as follows:

- •We introduce the notions of weak and strong simulators, and develop a zeroth\-order deep learning method to solve a wide class of fully nonlinear black\-box PDEs\. At each iteration, we query the available simulator with randomly generated initial states based on a specially designed distribution and collect corresponding trajectories\. In addition, we only rely on the point\-wise evaluation of the nonlinear source term in a PDE, which is a general function of the state, function value, gradient, and Hessian\. Thus, our method leads to amodel\-freealgorithm applicable in black\-box environments, in particular without attempting to estimate the PDE coefficients\.
- •To ensure convergence in \(weighted\) Sobolev space, we introduce a novel use of ZOD estimators as targets for approximating gradients and Hessians in the context of solving PDEs\. In general, ZOD estimators are biased and noisy, and we characterize their unique bias–variance tradeoff through their bandwidth\. In particular, the conventional variance reduction techniques for ZOD in optimization \(e\.g\., the multi\-point scheme\) are not sufficient in our setting, and we show that the variance of ZOD can only be bounded with the strong simulator but not the weak one\. Furthermore, using ZODs, we essentially first derive a \(biased\) stochastic representation for the gradient and Hessian and then fit two neural networks for them respectively\. This representing\-then\-learning paradigm is therefore fundamentally different from approaches that recover derivatives indirectly from value approximations or through model\-specific sensitivity representations\. Notably, no automatic differentiation or the knowledge of the operator is required to derive ZOD\.
- •Most importantly, we establish convergence and sample complexity of the proposed learning method in the \(weighted\) Sobolev space up to second\-order derivatives\. To our best knowledge, these are the first theoretical results on model\-free learning for PDEs with an error analysis in second\-order Sobolev space for fully nonlinear PDEs\. Our theoretical analysis is built on several standard assumptions on PDEs and statistical learning theory for DNNs\. On the one hand, we require a contraction property of the solution operator to an associated linear PDE\. This property is typical to ensure the existence and uniqueness of the solution to the original fully nonlinear PDE, and is often necessary for most value iteration algorithms\. On the other hand, we exclude the optimization error from our analysis and assume that the class of neural networks is sufficiently smooth and regular\. Under these assumptions, we obtain an error bound characterizing the trade\-offs among approximation error, sampling error, and ZOD bias\. The overall error consists of an exponentially decaying initialization error \(in terms of the number of approximate value iterations\) and a geometrically weighted accumulation of one\-step learning errors \(Theorem[6](https://arxiv.org/html/2606.24999#Thmtheorem6)\)\. When thedd\-dimensional state process is observed on a uniform time grid withNNgrid points and the sample size isBB, our results show that, besides the neural\-network optimization and approximation errors, choosing the ZOD bandwidth optimally yields a one\-step high\-probability error bound of orderO​\(\(N−1\+𝒫​\(d\)​B−1/2\)1/4\)O\(\(N^\{\-1\}\+\\mathcal\{P\}\(d\)B^\{\-1/2\}\)^\{1/4\}\)under the weak simulator with one\-point ZOD estimators, and of orderO​\(\(N−1\+𝒫​\(d\)​B−1/2\)1/2\)O\(\(N^\{\-1\}\+\\mathcal\{P\}\(d\)B^\{\-1/2\}\)^\{1/2\}\)under the strong simulator with multi\-point ZOD estimators, up to logarithmic factors, where𝒫​\(d\)\\mathcal\{P\}\(d\)is a constant that depends polynomially on the dimensiondd\(Theorem[9](https://arxiv.org/html/2606.24999#Thmtheorem9)\)\. We also experimentally demonstrate accurate approximation of both solutions and derivatives in moderate\- and high\-dimensional problems\.

### Related Literature

As a key ingredient of our method, ZOD estimators have been widely used in \(black\-box\) stochastic optimization and reinforcement learning, where gradient information is unavailable; see, e\.g\.,Nesterov and Spokoiny \([2017](https://arxiv.org/html/2606.24999#bib.bib73)\); Salimanset al\.\([2017](https://arxiv.org/html/2606.24999#bib.bib39)\)among many others, but not yet in the context of PDE solvers to our best knowledge\.333ZOD has been used to replace auto\-differentiation\. For instance,Heet al\.\([2023](https://arxiv.org/html/2606.24999#bib.bib68)\); Shiet al\.\([2024](https://arxiv.org/html/2606.24999#bib.bib70)\); Huet al\.\([2025](https://arxiv.org/html/2606.24999#bib.bib69)\)use ZOD to approximate the differential operators of a neural network in PINN\. The role of ZOD is fundamentally different in our approach\.Due to the unique context of solving parabolic PDEs, we rely on the celebrated Feynman–Kac formula to obtain a stochastic representation of the function value\. Moreover, our ZOD estimator differs from the traditional one in optimization in two aspects\. First, at every iteration, we query the simulator to generate a stochastic trajectory with the given initial state, which is then used to compute a functional value using the current function value, gradient and Hessian\. Second, the ZOD estimator is not used for optimization directly; instead, it is used as a target to train other networks\. Therefore, the bias–variance tradeoff of ZOD and its impact on the entire iterative procedure are both new problems in our context\.

Another key ingredient of our method is the approximate value iteration, which is a classical approach adopted in reinforcement learning to solve the Bellman equations or HJB equations in continuous\-time optimal controls, a well\-known class of nonlinear equations; see, e\.g\.,Bertsekas and Tsitsiklis \([1996](https://arxiv.org/html/2606.24999#bib.bib47)\); Sutton and Barto \([1998](https://arxiv.org/html/2606.24999#bib.bib48)\)for discrete\-time Bellman equations andYong and Zhou \([1999](https://arxiv.org/html/2606.24999#bib.bib49)\); Fleming and Soner \([2006](https://arxiv.org/html/2606.24999#bib.bib51)\); Pham \([2009](https://arxiv.org/html/2606.24999#bib.bib50)\)for continuous\-time HJB equations\. The core idea for the approximate value iteration is to reduce a fully nonlinear equation into successive linear equations\. At every step, it is sufficient to solve the linear equation approximately, and the total error would be accumulated and controlled under suitable regularity conditions\. We borrow this idea and solve a linear PDE at each iteration\. Moreover, this linear PDE can be interpreted as a Feynman–Kac \(Bellman\) equation for policy evaluation in reinforcement learning and solvable in a model\-free way\(Jia and Zhou,[2022a](https://arxiv.org/html/2606.24999#bib.bib12)\)\. However,Jia and Zhou \([2022a](https://arxiv.org/html/2606.24999#bib.bib12)\)only show the consistency of the value function, not its derivatives\. In the current PDE setting, learning derivatives is equally crucial for the iteration to converge to the true solution because the nonlinear source term of the PDE depends explicitly on the gradient and Hessian\.

In supervised regression, recovering derivatives by automatic differentiation of a network trained only on function values is known to be statistically problematic, despite the availability of the well\-known universal approximation theorem of neural networks that can be employed to approximate derivatives\(Horniket al\.,[1990](https://arxiv.org/html/2606.24999#bib.bib45); Siegel,[2023](https://arxiv.org/html/2606.24999#bib.bib56)\)\. Using the typical mean\-squared error as the loss function, the theoretical guarantee is often on the function value \(e\.g\.,Barron[1994](https://arxiv.org/html/2606.24999#bib.bib52)\)\. To discipline the behavior of the learned derivatives, a commonly used trick is to regularize and penalize large derivatives of the neural network; see, e\.g\.,Kim and Gu \([2004](https://arxiv.org/html/2606.24999#bib.bib54)\); Fischer and Steinwart \([2020](https://arxiv.org/html/2606.24999#bib.bib55)\)among many others\. One can then establish statistical errors in the Sobolev space\. By contrast, we go down a different path due to our access to a simulator and aim to solve a PDE rather than simply fitting a function to a given, exogenous data set\. Indeed, we can simulate stochastic trajectoriesstrategicallytocreatetraining data\. In other words, in our approach, data sets are also endogenously designed to learn a function in an “optimal” way in the Sobolev space\.

Therefore, our approach, representing\-then\-learning, should be regarded as a \(black\-box/model\-free\) simulation\-based learning method, and the first “representing” stage is essentially a sampling stage\. From this perspective, our approach is comparable to the sensitivity analysis associated with mathematical expectations in the simulation literature, such as the likelihood ratio method \(cf\.Glasserman[2004](https://arxiv.org/html/2606.24999#bib.bib21)\) and the Malliavin calculus approach \(cf\.Ma and Zhang[2002](https://arxiv.org/html/2606.24999#bib.bib7)\)\. In a similar context of deep learning for solving PDEs,Hanet al\.\([2026](https://arxiv.org/html/2606.24999#bib.bib30)\)propose the method of deep Picard iteration \(DPI\), where neural networks are trained using Monte Carlo estimates of the solution and its derivatives\. Picard iteration coincides with the approximate value iteration in our context\. The main difference between DPI \(and its extensions\) and our approach is that DPI employs the exact stochastic representations for derivatives based on Malliavin calculus \(Bismut–Elworthy–Li formulas; cf\.Baños[2018](https://arxiv.org/html/2606.24999#bib.bib57)\), which is unbiased but requires the knowledge of the coefficients of the differential operator and their derivatives, and thus operates in a model\-based setting\. Another line of work is the differential learning framework as inLefebvreet al\.\([2023](https://arxiv.org/html/2606.24999#bib.bib31)\), which proposes to jointly learn the solution and its derivatives for fully nonlinear PDEs with convex Hamiltonians\. Their approach leverages stochastic control representations and incorporates derivative information through Malliavin\-type sensitivities\. By contrast, our approach is applicable in a model\-free context at the cost of inevitably incurring more bias\. More importantly, we have theoretically justified the convergence of our resulting algorithm, showing that the bias introduced by ZOD can be controlled\.

This paper also contributes broadly to the literature on solving PDEs by machine learning\. In addition to the above\-mentioned methods such as the deep Galerkin method\(Sirignano and Spiliopoulos,[2018](https://arxiv.org/html/2606.24999#bib.bib23)\), PINNs\(Raissiet al\.,[2019](https://arxiv.org/html/2606.24999#bib.bib24)\), deep BSDEs\(Eet al\.,[2017](https://arxiv.org/html/2606.24999#bib.bib27)\), DBDP\(Huréet al\.,[2020](https://arxiv.org/html/2606.24999#bib.bib25)\), and deep Picard iteration\(Hanet al\.,[2026](https://arxiv.org/html/2606.24999#bib.bib30)\), there are also specifically designed methods for special classes of PDEs, such as deep Ritz methods for variational problems\(E and Yu,[2018](https://arxiv.org/html/2606.24999#bib.bib66); Luet al\.,[2021b](https://arxiv.org/html/2606.24999#bib.bib65)\), actor–critic methods for HJB equations\(Zhouet al\.,[2021](https://arxiv.org/html/2606.24999#bib.bib63); Jia and Zhou,[2022b](https://arxiv.org/html/2606.24999#bib.bib13); Zhuet al\.,[2025](https://arxiv.org/html/2606.24999#bib.bib62)\), and the high\-order scheme for policy evaluation\(Mou and Zhu,[2025](https://arxiv.org/html/2606.24999#bib.bib59)\)\. However, few of them analyze statistical errors and sample complexity in the presence of neural network\-based function approximation\.Mou \([2025](https://arxiv.org/html/2606.24999#bib.bib60)\)analyzes errors in the Sobolev space of a least\-squares temporal\-difference algorithm arising from continuous\-time policy evaluation\.Luet al\.\([2022](https://arxiv.org/html/2606.24999#bib.bib64)\)analyze two particular algorithms to solve linear PDEs\. Little is known about fully nonlinear PDEs, especially in a black\-box setting\. The present paper not only develops a new deep learning method that is applicable to a general class of nonlinear unknown PDEs, but also theoretically establishes the sample complexity\.

The remainder of the paper is organized as follows\. In Section 2, we introduce the problem formulation, put forward the two simulators, and present the approximate value iteration framework for fully nonlinear PDEs\. Section 3 develops the probabilistic representation of derivatives based on zeroth\-order estimators and introduces the representing\-then\-learning paradigm\. In Section 4, we develop the proposed deep learning algorithm, including the loss construction and implementation details\. Section 5 provides the theoretical analysis, establishing convergence of the method and quantifying the impact of discretization, approximation, sampling, and perturbation errors\. Section 6 presents numerical experiments that validate the performance of the proposed approach in both low\- and high\-dimensional settings\. Section[7](https://arxiv.org/html/2606.24999#S7)concludes\. Supplementary materials and proofs of statements are placed in the Appendix\.

## 2Problem Description and Preliminaries

In this paper, we study how to effectively solve a class of nonlinear parabolic PDEs:

\{∂tv\+ℒ​v\+f​\(x,v,∇v,∇2v\)=0,\(t,x\)∈\[0,T\)×ℝd,v​\(T,x\)=g​\(x\),x∈ℝd,\\begin\{cases\}\\partial\_\{t\}v\+\\mathcal\{L\}v\+f\(x,v,\\nabla v,\\nabla^\{2\}v\)=0,\\quad\(t,x\)\\in\[0,T\)\\times\\mathbb\{R\}^\{d\},\\\\ v\(T,x\)=g\(x\),\\quad x\\in\\mathbb\{R\}^\{d\},\\end\{cases\}\(2\.1\)where∇v∈ℝd\\nabla v\\in\\mathbb\{R\}^\{d\}and∇2v∈ℝd×d\\nabla^\{2\}v\\in\\mathbb\{R\}^\{d\\times d\}are respectively the gradient and Hessian ofvvwith respect to the spatial variables, and the second order differential operatorℒ\\mathcal\{L\}is

ℒ​φ:=∑i,j=1dai​j​\(x\)​∂i​jφ\+∑i=1dbi​\(x\)​∂iφ,\\mathcal\{L\}\\varphi:=\\sum\_\{i,j=1\}^\{d\}a\_\{ij\}\(x\)\\partial\_\{ij\}\\varphi\+\\sum\_\{i=1\}^\{d\}b\_\{i\}\(x\)\\partial\_\{i\}\\varphi,with∂t,∂i\\partial\_\{t\},\\partial\_\{i\}being respectively the partial derivative in the time variablettand theii\-th entry of the spatial variablexix\_\{i\}, and∂i​j\\partial\_\{ij\}the corresponding second\-order derivative\. Leta​\(x\)≡\(ai​j​\(x\)\)∈𝕊\+da\(x\)\\equiv\(a\_\{ij\}\(x\)\)\\in\\mathbb\{S\}^\{d\}\_\{\+\}, assumed to be a positive semidefinite matrix\.

The operatorℒ\\mathcal\{L\}can therefore be viewed as the infinitesimal generator of a time\-homogeneous diffusion process driven by add\-dimensional Brownian motionWW, satisfying the following stochastic differential equation \(SDE\):

d​Xs=b​\(Xs\)​d​s\+2​σ​\(Xs\)​d​Ws,\\mathrm\{d\}X\_\{s\}=b\(X\_\{s\}\)\\mathrm\{d\}s\+\\sqrt\{2\}\\sigma\(X\_\{s\}\)\\mathrm\{d\}W\_\{s\},\(2\.2\)whereσ​\(x\)​σ​\(x\)⊤=a​\(x\)\\sigma\(x\)\\sigma\(x\)^\{\\top\}=a\(x\)\.444For ease of presentation, in this paper we consider only the case whereℒ\\mathcal\{L\}andffdo not depend on time, i\.e\. time\-homogeneous problems\. All the conclusions can be readily extended to the time inhomogeneous case\.We further denote byXt,xX^\{t,x\}the solution to the SDE with initial conditionXt=xX\_\{t\}=x\.

We first reiterate the following important consideration and goal in our study:

1. \(1\)The dimension of the spatial variableddcan be potentially high, and we want to avoid the curse of dimensionality arising from the conventional grid\-based numerical methods \(such as finite\-difference and finite\-element methods\)\.
2. \(2\)The PDE can be potentially described in a black\-box manner and we aim to develop model\-free, data\-driven methods\. A black\-box environment means \(i\) the functionsg,fg,fare black\-box functions that can only output their values if inputs are queried; and \(ii\) the operatorℒ\\mathcal\{L\}is not known and only its associated processes/trajectories\{Xst,x\}t≤s≤T\\\{X\_\{s\}^\{t,x\}\\\}\_\{t\\leq s\\leq T\}can be observed upon querying asimulator\. Thus, “data” here means the trajectories that are returned upon a query\.

We now formalize the “simulators” considered in this paper\. Note that any simulator defined below remains unknown to us, being a device outputting data upon a query, and we do not attempt to learn its underlying structure\. We consider two settings\.

###### Condition 1

A weak simulator returns a trajectory\{Xst,x\}t≤s≤T\\\{X\_\{s\}^\{t,x\}\\\}\_\{t\\leq s\\leq T\}for each query with input\(t,x\)\(t,x\), such that

1. \(i\)\{Xst,x\}t≤s≤T\\\{X\_\{s\}^\{t,x\}\\\}\_\{t\\leq s\\leq T\}is the weak solution to \([2\.2](https://arxiv.org/html/2606.24999#S2.E2)\);
2. \(ii\)the random sources \(Brownian motions\) in any two different queries are independent of each other\.

###### Condition 2

A strong simulator returns a collection of trajectories\{\{Xst,xk\}t≤s≤T,k=1,⋯,K\}\\left\\\{\\\{X\_\{s\}^\{t,x\_\{k\}\}\\\}\_\{t\\leq s\\leq T\},\\;k=1,\\cdots,K\\right\\\}for each query whose input is a finite set of initial states\{\(t,x1\),⋯,\(t,xK\)\}\\\{\(t,x\_\{1\}\),\\cdots,\(t,x\_\{K\}\)\\\}, such that

1. \(i\)upon each query, every trajectory of\{Xst,xk\}t≤s≤T\\\{X\_\{s\}^\{t,x\_\{k\}\}\\\}\_\{t\\leq s\\leq T\}is the strong solution to \([2\.2](https://arxiv.org/html/2606.24999#S2.E2)\) with respect to the same Brownian motion\{Ws\}t≤s≤T\\\{W\_\{s\}\\\}\_\{t\\leq s\\leq T\}with the initial states\(t,xk\)\(t,x\_\{k\}\);
2. \(ii\)the random sources \(Brownian motions\) in any two different queries are independent of each other\.

Evidently, a weak simulator is implied by a strong simulator, which is further implied by a PDE with known coefficients\. Neither converse implication, however, is true\. The notions of weak and strong simulators coincide with the classical definitions of weak and strong solutions to SDEs and capture the subtle distinction between them: a strong solution is defined with respect to a fixed Brownian motion, whereas in a weak solution the Brownian motion is a part of the solution\. The weak simulator is the typical setting in which each query produces an i\.i\.d\. sample path\. By contrast, the strong simulator not only generates multiple trajectories per query, but also keeps the underlying randomness fixed within that query, much like fixing a “random seed”\. Mathematically, querying a strong simulator with a set of initial states\{\(t,x1\),⋯,\(t,xK\)\}\\\{\(t,x\_\{1\}\),\\cdots,\(t,x\_\{K\}\)\\\}means thatKKtrajectories\{Xst,xk\}t≤s≤T\\\{X^\{t,x\_\{k\}\}\_\{s\}\\\}\_\{t\\leq s\\leq T\}are generated satisfying

Xτt,xk=xk\+∫tτb​\(Xst,xk\)​ds\+∫tτ2​σ​\(Xst,xk\)​dWs,X\_\{\\tau\}^\{t,x\_\{k\}\}=x\_\{k\}\+\\int\_\{t\}^\{\\tau\}b\(X\_\{s\}^\{t,x\_\{k\}\}\)\\mathrm\{d\}s\+\\int\_\{t\}^\{\\tau\}\\sqrt\{2\}\\sigma\(X\_\{s\}^\{t,x\_\{k\}\}\)\\mathrm\{d\}W\_\{s\},for allτ∈\[t,T\]\\tau\\in\[t,T\]and allk=1,⋯,Kk=1,\\cdots,K, almost surely, where the same Brownian motionWWdrives all theKKtrajectories \(butWWis not the output of the simulator and hence is “hidden”\)\. Upon another query, another hidden independent Brownian motion will be used to generate these trajectories\.

These precise definitions of the simulators also give precise meanings to the terms “model\-free” and “data\-driven”, and it is possible to extend them to settings other than PDEs\. Our approach and results in this paper involve both settings, but the theoretical guarantees differ, depending on which simulator is available\. In short, access to a strong simulator leads to substantially sharper results than access only to a weak simulator because the former facilitates variance reduction; see Proposition[5](https://arxiv.org/html/2606.24999#Thmtheorem5)and Theorem[9](https://arxiv.org/html/2606.24999#Thmtheorem9)\.

Next, we discuss a few important building blocks of our approach and highlight the challenges of the existing approaches\. As they are both well\-known in the literature, we omit to specify needed regularity conditions to avoid unnecessary technicality in the discussion\. Rigorous conditions and analysis will be presented in Section[5](https://arxiv.org/html/2606.24999#S5)\.

### 2\.1Value Iteration of the Nonlinear PDEs

We consider the value iteration for solving the nonlinear PDE \([2\.1](https://arxiv.org/html/2606.24999#S2.E1)\): Define the solution mapping of a linear parabolic equation𝒮:v↦u\\mathcal\{S\}:v\\mapsto u, such thatuusolves

\{∂tu\+ℒ​u\+f​\(x,v,∇v,∇2v\)=0,\(t,x\)∈\[0,T\)×ℝd,u​\(T,x\)=g​\(x\),x∈ℝd\.\\begin\{cases\}\\partial\_\{t\}u\+\\mathcal\{L\}u\+f\(x,v,\\nabla v,\\nabla^\{2\}v\)=0,\\quad\(t,x\)\\in\[0,T\)\\times\\mathbb\{R\}^\{d\},\\\\ u\(T,x\)=g\(x\),\\quad x\\in\\mathbb\{R\}^\{d\}\.\\end\{cases\}\(2\.3\)
By assuming suitable regularity conditions onℒ\\mathcal\{L\}andf,gf,g,𝒮\\mathcal\{S\}is a well\-defined mapping from a suitable space to itself\. For any given initial valuev0v\_\{0\}, we can therefore define the value iteration sequence byv1=𝒮​v0,⋯,vn\+1=𝒮​vn,⋯v\_\{1\}=\\mathcal\{S\}v\_\{0\},\\cdots,v\_\{n\+1\}=\\mathcal\{S\}v\_\{n\},\\cdots\. Based on the Feynman–Kac formula \(cf\.Karatzas and Shreve[2014](https://arxiv.org/html/2606.24999#bib.bib46)\), the solution mapping𝒮\\mathcal\{S\}admits a probabilistic representation:

𝒮​v​\(t,x\)=u​\(t,x\)=𝔼​\[g​\(XTt,x\)\+∫tTf​\(Xst,x,v​\(s,Xst,x\),∇v​\(s,Xst,x\),∇2v​\(s,Xst,x\)\)​ds\]\.\\mathcal\{S\}v\(t,x\)=u\(t,x\)=\\mathbb\{E\}\\left\[g\(X\_\{T\}^\{t,x\}\)\+\\int\_\{t\}^\{T\}f\\left\(X\_\{s\}^\{t,x\},v\(s,X\_\{s\}^\{t,x\}\),\\nabla v\(s,X\_\{s\}^\{t,x\}\),\\nabla^\{2\}v\(s,X\_\{s\}^\{t,x\}\)\\right\)\\mathrm\{d\}s\\right\]\.\(2\.4\)
Hence, correspondingly, the value iteration can be represented by

vn\+1​\(t,x\)\\displaystyle v\_\{n\+1\}\(t,x\)=𝔼​\[g​\(XTt,x\)\+∫tTf​\(Xst,x,vn​\(s,Xst,x\),∇vn​\(s,Xst,x\),∇2vn​\(s,Xst,x\)\)​ds\]\.\\displaystyle=\\mathbb\{E\}\\left\[g\(X\_\{T\}^\{t,x\}\)\+\\int\_\{t\}^\{T\}f\\left\(X\_\{s\}^\{t,x\},v\_\{n\}\(s,X\_\{s\}^\{t,x\}\),\\nabla v\_\{n\}\(s,X\_\{s\}^\{t,x\}\),\\nabla^\{2\}v\_\{n\}\(s,X\_\{s\}^\{t,x\}\)\\right\)\\mathrm\{d\}s\\right\]\.\(2\.5\)
We make three important observations\. First, value iteration reduces the difficulty of solving anonlinearPDE to solvinglinearPDEs successively and, thus, is commonly employed as the first attempt in many numerical procedures\. It is also widely used to establish the well\-posedness of nonlinear PDEs, and is referred to as Picard iteration in the PDE literature \(e\.g\.,Hanet al\.[2026](https://arxiv.org/html/2606.24999#bib.bib30)\)\. In particular, if we assume that the solution mapping𝒮\\mathcal\{S\}is a contraction under a suitable norm, then this iteration converges at an exponential rate\. Such contraction properties are dictated by those ofℒ\\mathcal\{L\}andf,gf,g, and we will present a sufficient condition in Appendix[A](https://arxiv.org/html/2606.24999#A1)for the contraction property\.

Second, besides value iteration, there are alternative ways to solve nonlinear PDEs by iteratively solving linearized PDEs, e\.g\., Newton’s method, which expands the nonlinear termffinto its linear approximation\. Our approach in this paper is a zeroth\-order method: it uses only knowledge offfitself, rather than its first\-order derivative \(which is unknown in the black\-box/model\-free setting\), but can be generalized to Newton’s method if such a form is known\.

Third, the value iteration \([2\.5](https://arxiv.org/html/2606.24999#S2.E5)\) provides a theoretical mechanism to solve the nonlinear PDE by solving a sequence of linear problems\. However, numerical challenges remain, and numerically solving linear \(black\-box\) PDEs in high dimensions is already difficult\. The probabilistic representation of the solution mapping𝒮\\mathcal\{S\}transforms the problem of solving PDEs into evaluating expectations over stochastic trajectories which can in turn be computed using Monte Carlo methods, thus avoiding using grid points in the conventional numerical PDE methods\. Such a probabilistic representation seems to be the only way to remedy the curse of dimensionality in solving PDEs\. Moreover, the probabilistic representation \([2\.4](https://arxiv.org/html/2606.24999#S2.E4)\) only requires the functionvaluesofg,fg,falong the trajectories ofXtX\_\{t\}and not the precise specification ofℒ\\mathcal\{L\}nor the exactfunctional formsofg,fg,f\. Hence it is applicable to a black\-box environment\.555This problem also arises in the context of policy evaluation in reinforcement learning \(see, e\.g\.,Jia and Zhou[2022a](https://arxiv.org/html/2606.24999#bib.bib12)\)\.

### 2\.2Is Deep Neural Network a Savior For All?

To overcome the curse of dimensionality, deep neural networks \(DNNs\) have been widely adopted in recent algorithms based on probabilistic representations such as \([2\.4](https://arxiv.org/html/2606.24999#S2.E4)\) to approximate solutions to PDEs\. DNNs have been shown, both theoretically and empirically, to approximate high\-dimensional functions efficiently without suffering from an exponential dependence on dimension; see e\.g\.Hanet al\.\([2018](https://arxiv.org/html/2606.24999#bib.bib26)\),Becket al\.\([2023](https://arxiv.org/html/2606.24999#bib.bib28)\),Germainet al\.\([2023](https://arxiv.org/html/2606.24999#bib.bib29)\)\. In our context, the value iteration requires successively solving linear PDEs and, hence, DNNs can be used to approximate each solutionvnv\_\{n\}\. Such an approach has been explored byJia and Zhou \([2022a](https://arxiv.org/html/2606.24999#bib.bib12)\)and more broadly in the reinforcement learning literature for policy evaluation\.

Does this, then, mean that DNNs provide a universal remedy for all these problems? The answer is negative\. In value iteration, the solution obtained at the previous step is used as an input to define the next linear PDE\. Therefore, it is crucial to ensure that the one\-step error does not propagate and amplify over iterations\. In particular, for a fully nonlinear PDE like \([2\.3](https://arxiv.org/html/2606.24999#S2.E3)\), the next linear PDE in the value iteration \([2\.5](https://arxiv.org/html/2606.24999#S2.E5)\) depends on the previous solution via its spatialderivativesup to the second order \(i\.e\.,ffdepends on∇vn\\nabla v\_\{n\}and∇2vn\\nabla^\{2\}v\_\{n\}\)\. Consequently, the one\-step approximation error must be controlled not only in the current function iterate, but also in its first\- and second\-order derivatives, which imposes significantly more stringent requirements on the approximation scheme\.

To illustrate the issue, letVnV\_\{n\}be the DNN approximation ofvnv\_\{n\}at stepnn\. To compute the target for stepn\+1n\+1, we must evaluatef​\(⋅,Vn,∇Vn,∇2Vn\)f\(\\cdot,V\_\{n\},\\nabla V\_\{n\},\\nabla^\{2\}V\_\{n\}\)\. A standard approach would be to trainVn\+1V\_\{n\+1\}to minimize theL2L^\{2\}distance to the target value implied by \([2\.5](https://arxiv.org/html/2606.24999#S2.E5)\)\. However, convergence in theL2L^\{2\}norm of the value function does not guarantee convergence of its derivatives \(i\.e\.,‖Vn−v∗‖L2→0\\\|V\_\{n\}\-v^\{\*\}\\\|\_\{L^\{2\}\}\\to 0does not imply‖∇Vn−∇v∗‖L2→0\\\|\\nabla V\_\{n\}\-\\nabla v^\{\*\}\\\|\_\{L^\{2\}\}\\to 0\), particularly in high\-dimensional spaces\. This phenomenon is well\-known in Sobolev training and non\-parametric regression; see e\.g\.Czarneckiet al\.\([2017](https://arxiv.org/html/2606.24999#bib.bib37)\)\.

We emphasize that this error is not due to the numerical error in computing derivatives as in, say, the finite\-difference method\. When using DNNs, derivatives are computed via automatic differentiation \(auto\-diff\) throughbackpropagation, which provides exact derivatives of the network representation\.666The only exception occurs for non\-differentiable functions, e\.g\.v​\(x\)=x\+v\(x\)=x^\{\+\}atx=0x=0\. In practice, this is rarely an issue since neural networks are smooth or almost everywhere differentiable\.Therefore, if∇Vn\\nabla V\_\{n\}and∇2Vn\\nabla^\{2\}V\_\{n\}are computed directly via automatic differentiation of an imperfectly trained networkVnV\_\{n\}, the errors in derivatives may be significantly amplified, destabilizing the source termfffor the next iteration\. Consequently, learning the value function alone is insufficient, and one has to learn the gradient and Hessian, denoted byGn≈∇vnG\_\{n\}\\approx\\nabla v\_\{n\}andHn≈∇2vnH\_\{n\}\\approx\\nabla^\{2\}v\_\{n\}, and update themalongsidewithVnV\_\{n\}\.

Existing approaches approximate derivatives using differential learning techniques in a model\-based setting, often relying on the simulation of tangent processes associated with the underlying stochastic differential equation; seeGlasserman \([2004](https://arxiv.org/html/2606.24999#bib.bib21)\)andHuge and Savine \([2020](https://arxiv.org/html/2606.24999#bib.bib32)\)\. These methods, however, typically require access to derivatives of the coefficients \(e\.g\., the exact form ofℒ\\mathcal\{L\}\) and in addition may introduce additional variance and computational complexity\.

The above discussions motivate us to develop alternative approaches that avoid explicit differentiation while maintaining stability over iterations\.

## 3Approximate Probabilistic Representation of Derivatives

In this section, we detail the proposed numerical scheme\. Based on the discussions in Section[2\.2](https://arxiv.org/html/2606.24999#S2.SS2), the core challenge lies in generating accurate and low\-variance training targets for networks\(G,H\)≈\(∇v,∇2v\)\(G,H\)\\approx\(\\nabla v,\\nabla^\{2\}v\)that can be learned using function values ofvv\. In other words, we will not rely on recovering derivatives from a learned value approximation, whether through automatic differentiation or model\-dependent sensitivity formulas; rather, we need to derive a probabilistic model\-free representation of the derivatives\(∇v,∇2v\)\(\\nabla v,\\nabla^\{2\}v\), i\.e\., forms similar to \([2\.4](https://arxiv.org/html/2606.24999#S2.E4)\), and then train networks\(G,H\)\(G,H\)to approximate them – a “representing\-then\-learning” paradigm\.

To achieve this, we rely on the “zeroth\-order derivative” \(ZOD\) estimators as the fundamental building block of our method\. Since ZOD has not yet been utilized in the context of solving PDEs with DNNs, we first review its properties for the reader’s convenience\.

### 3\.1Classical Zeroth\-Order Derivative Estimators

The basic idea of ZOD is to use the function values at randomly perturbed points,u​\(t,x\+ϵ​Z\)u\(t,x\+\\epsilon Z\), to approximate the derivative∇u\\nabla uand even higher\-order derivatives such as∇2u\\nabla^\{2\}u, whereϵ\>0\\epsilon\>0is the bandwidth of the perturbation andZZfollows a standard, sphere\-symmetric distribution\. In our case, without loss of generality, we takeZ∼𝒩​\(0,I\)Z\\sim\\mathcal\{N\}\(0,I\), add\-dimensional standard normal distribution\. The following Lemmas[1](https://arxiv.org/html/2606.24999#Thmtheorem1)to[3](https://arxiv.org/html/2606.24999#Thmtheorem3)are standard results under slightly different regularity conditions; see, e\.g\.,Nesterov and Spokoiny \([2017](https://arxiv.org/html/2606.24999#bib.bib73)\); Balasubramanian and Ghadimi \([2022](https://arxiv.org/html/2606.24999#bib.bib74)\)\. For completeness and for the reader’s convenience, we provide their proofs in the appendix\. These lemmas construct several well\-known ZOD estimators and present their properties\.

###### Lemma 1

Given functionu​\(⋅,⋅\)u\(\\cdot,\\cdot\),\(t,x\)∈\[0,T\]×ℝd\(t,x\)\\in\[0,T\]\\times\\mathbb\{R\}^\{d\}, andZ∼𝒩​\(0,Id\)Z\\sim\\mathcal\{N\}\(0,I\_\{d\}\)\. Ifu​\(t,⋅\)u\(t,\\cdot\)is three times continuously differentiable and its derivatives satisfy the polynomial growth condition, thenZϵ​u​\(t,x\+ϵ​Z\)\\frac\{Z\}\{\\epsilon\}u\(t,x\+\\epsilon Z\)is a biased estimator for∇u​\(t,x\)\\nabla u\(t,x\), and its bias and variance are upper bounded by

\|𝔼​\[Zϵ​u​\(t,x\+ϵ​Z\)\]−∇u​\(t,x\)\|≤C​ϵ2,Var⁡\[Zϵ​u​\(t,x\+ϵ​Z\)\]≤C​1ϵ2\.\\left\|\\mathbb\{E\}\\left\[\\frac\{Z\}\{\\epsilon\}u\(t,x\+\\epsilon Z\)\\right\]\-\\nabla u\(t,x\)\\right\|\\leq C\\epsilon^\{2\},\\ \\operatorname\{Var\}\\left\[\\frac\{Z\}\{\\epsilon\}u\(t,x\+\\epsilon Z\)\\right\]\\leq C\\frac\{1\}\{\\epsilon^\{2\}\}\.Furthermore, ifu​\(t,⋅\)u\(t,\\cdot\)is four times continuously differentiable and its derivatives satisfy the polynomial growth condition, thenZ​Z⊤−Idϵ2​u​\(t,x\+ϵ​Z\)\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\}u\(t,x\+\\epsilon Z\)is a biased estimator for∇2u​\(t,x\)\\nabla^\{2\}u\(t,x\), and its bias and variance are upper bounded by

\|𝔼​\[Z​Z⊤−Idϵ2​u​\(t,x\+ϵ​Z\)\]−∇2u​\(t,x\)\|≤C​ϵ2,Var⁡\[Z​Z⊤−Idϵ2​u​\(t,x\+ϵ​Z\)\]≤C​1ϵ4\\left\|\\mathbb\{E\}\\left\[\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\}u\(t,x\+\\epsilon Z\)\\right\]\-\\nabla^\{2\}u\(t,x\)\\right\|\\leq C\\epsilon^\{2\},\\ \\operatorname\{Var\}\\left\[\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\}u\(t,x\+\\epsilon Z\)\\right\]\\leq C\\frac\{1\}\{\\epsilon^\{4\}\}\(3\.1\)

##### Variance Reduction of Multi\-Point ZOD Estimators

As shown in Lemma[1](https://arxiv.org/html/2606.24999#Thmtheorem1), the naïve one\-point estimators suffer from a critical drawback: their variances diverge as the perturbation magnitude,ϵ\\epsilon, approaches zero\. This means that to get an accurate derivative estimate,ϵ\\epsilonneeds to be small to reduce the bias but the number of samples need to increase dramatically to reduce the variance\. This issue can often be mitigated by the symmetric \(or central\) differencing technique, known as the multi\-point zeroth\-order estimators\.

We begin by considering the estimation of the gradient,∇u​\(t,x\)\\nabla u\(t,x\)\. A one\-point estimator, as demonstrated in Lemma[1](https://arxiv.org/html/2606.24999#Thmtheorem1), has variances that scale asO​\(ϵ−2\)O\(\\epsilon^\{\-2\}\), whereas a two\-point estimator exhibitsboundedvariances, as stipulated in Lemma[2](https://arxiv.org/html/2606.24999#Thmtheorem2):

###### Lemma 2

Given functionu​\(⋅,⋅\)u\(\\cdot,\\cdot\),\(t,x\)∈\[0,T\]×ℝd\(t,x\)\\in\[0,T\]\\times\\mathbb\{R\}^\{d\}, andZ∼𝒩​\(0,Id\)Z\\sim\\mathcal\{N\}\(0,I\_\{d\}\)\. Assumeu​\(t,⋅\)u\(t,\\cdot\)is three times continuously differentiable and its derivatives satisfy the polynomial growth condition\. Define

g^​\(t,x;ϵ\)≔u​\(t,x\+ϵ​Z\)−u​\(t,x−ϵ​Z\)2​ϵ​Z\.\\hat\{g\}\(t,x;\\epsilon\)\\coloneqq\\frac\{u\(t,x\+\\epsilon Z\)\-u\(t,x\-\\epsilon Z\)\}\{2\\epsilon\}Z\.\(3\.2\)Then

\|𝔼​\[g^​\(t,x;ϵ\)\]−∇u​\(t,x\)\|≤C​ϵ2,Var⁡\[g^​\(t,x;ϵ\)\]≤C′,\\displaystyle\\left\|\\mathbb\{E\}\\left\[\\hat\{g\}\(t,x;\\epsilon\)\\right\]\-\\nabla u\(t,x\)\\right\|\\leq C\\epsilon^\{2\},\\;\\;\\;\\operatorname\{Var\}\\left\[\\hat\{g\}\(t,x;\\epsilon\)\\right\]\\leq C^\{\\prime\},\(3\.3\)for some constantsC,C′\>0C,C^\{\\prime\}\>0that are independent ofϵ\\epsilon\.

The one\-point estimator for Hessian has even larger variances that diverge at a rate ofO​\(ϵ−4\)O\(\\epsilon^\{\-4\}\)and hence is much more inefficient\. However, with symmetric differencing, we can construct a stable estimator with bounded variances\.

###### Lemma 3

Let the functionu:\[0,T\]×ℝd→ℝu:\[0,T\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}satisfy the conditions of Lemma[2](https://arxiv.org/html/2606.24999#Thmtheorem2), and further assumeu​\(t,⋅\)u\(t,\\cdot\)is four times continuously differentiable and its derivatives satisfy the polynomial growth condition\. Define

H^​\(t,x;ϵ\)≔u​\(t,x\+ϵ​Z\)−2​u​\(t,x\)\+u​\(t,x−ϵ​Z\)2​ϵ2​\(Z​Z⊤−Id\)\.\\hat\{H\}\(t,x;\\epsilon\)\\coloneqq\\frac\{u\(t,x\+\\epsilon Z\)\-2u\(t,x\)\+u\(t,x\-\\epsilon Z\)\}\{2\\epsilon^\{2\}\}\(ZZ^\{\\top\}\-I\_\{d\}\)\.\(3\.4\)Then

\|𝔼​\[H^​\(t,x;ϵ\)\]−∇2u​\(t,x\)\|≤C​ϵ2,Var⁡\[H^​\(t,x;ϵ\)\]≤C′,\\displaystyle\\left\|\\mathbb\{E\}\\left\[\\hat\{H\}\(t,x;\\epsilon\)\\right\]\-\\nabla^\{2\}u\(t,x\)\\right\|\\leq C\\epsilon^\{2\},\\;\\;\\;\\operatorname\{Var\}\\left\[\\hat\{H\}\(t,x;\\epsilon\)\\right\]\\leq C^\{\\prime\},\(3\.5\)for some constantsC,C′\>0C,C^\{\\prime\}\>0that are independent ofϵ\\epsilon\.

### 3\.2Illustrations of Representing\-Then\-Learning Paradigm

To illustrate our “representing\-then\-learning” paradigm, we first use a toy task of regression and contrast it with a conventional routine where the derivative is obtained from differentiating the learned value function\. This simple regression task is indeed equivalent to solving a linear PDE, which will be demonstrated numerically in Subsection[6\.1](https://arxiv.org/html/2606.24999#S6.SS1)\.

The regression task is described as follows: suppose\(X,Y\)∈ℝ2\(X,Y\)\\in\\mathbb\{R\}^\{2\}follows a joint distribution that can be written asπ​\(d​x\)​ρ​\(x,d​y\)\\pi\(\\mathrm\{d\}x\)\\rho\(x,\\mathrm\{d\}y\), whereπ​\(d​x\)\\pi\(\\mathrm\{d\}x\)is the marginal distribution ofXXandρ​\(x,d​y\)\\rho\(x,\\mathrm\{d\}y\)is the conditional distribution ofY\|X=xY\|X=x\. We can generate samples from bothXXandY\|X=xY\|X=x, but do not know the functionsπ\\piandρ\\rho\. We are interested in numerically finding the functionf​\(x\)f\(x\)and its derivativesf′​\(x\),f′′​\(x\)f^\{\\prime\}\(x\),f^\{\\prime\\prime\}\(x\)wheref​\(X\)=𝔼​\[Y\|X\]f\(X\)=\\mathbb\{E\}\[Y\|X\]is the conditional expectation\.

##### Learning\-then\-Differentiating

Conventionally, one can first generate samplesXi∼π​\(⋅\)X\_\{i\}\\sim\\pi\(\\cdot\)andYi∼ρ​\(Xi,⋅\)Y\_\{i\}\\sim\\rho\(X\_\{i\},\\cdot\), and then fit a DNN regression by

V=arg⁡minφ∈ℋ⁡1n​∑i=1n\(Yi−φ​\(Xi\)\)2,V=\\arg\\min\_\{\\varphi\\in\\mathcal\{H\}\}\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\left\(Y\_\{i\}\-\\varphi\(X\_\{i\}\)\\right\)^\{2\},whereℋ\\mathcal\{H\}stands for the functional class of the chosen DNN family\. The resulting function approximationVVis well\-known to satisfy performance guarantee \(ignoring the optimization error\)\. However, differentiating \(equivalent to auto\-diff\)VVto obtainV′,V′′V^\{\\prime\},V^\{\\prime\\prime\}does not have any theoretical guarantee\.777One possible mitigation to this is to consider the regularized regression byminφ∈ℋ⁡1n​∑i=1n\(Yi−φ​\(Xi\)\)2\+λ​\|φ′′​\(Xi\)\|2\\min\_\{\\varphi\\in\\mathcal\{H\}\}\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\left\(Y\_\{i\}\-\\varphi\(X\_\{i\}\)\\right\)^\{2\}\+\\lambda\|\\varphi^\{\\prime\\prime\}\(X\_\{i\}\)\|^\{2\}, with a tuning parameterλ\>0\\lambda\>0that induces smoother function approximation; see e\.g\.Wahba \([1990](https://arxiv.org/html/2606.24999#bib.bib41)\)\.

##### Representing\-then\-Learning

Our approach is based on obtaining a sample approximation for the derivatives directly via ZOD estimators\. Let us illustrate the one\-point ZOD for estimating derivatives first\. Besides the samples\(Xi,Yi\)\(X\_\{i\},Y\_\{i\}\), we further sampleZi∼𝒩​\(0,1\)Z\_\{i\}\\sim\\mathcal\{N\}\(0,1\)and sampleYi\(\+\)∼ρ​\(Xi\+ϵ​Zi,⋅\)Y\_\{i\}^\{\(\+\)\}\\sim\\rho\(X\_\{i\}\+\\epsilon Z\_\{i\},\\cdot\)\. Now consider the newly constructed sampleZiϵ​Yi\(\+\)\\frac\{Z\_\{i\}\}\{\\epsilon\}Y\_\{i\}^\{\(\+\)\}\. It follows from Lemma[1](https://arxiv.org/html/2606.24999#Thmtheorem1)that

𝔼​\[Ziϵ​Yi\(\+\)\|Xi\]=𝔼​\[𝔼​\[Ziϵ​Yi\(\+\)\|Xi,Zi\]\|Xi\]=𝔼​\[Ziϵ​f​\(Xi\+ϵ​Zi\)\|Xi\]≈f′​\(Xi\)\+bias\.\\mathbb\{E\}\\left\[\\frac\{Z\_\{i\}\}\{\\epsilon\}Y\_\{i\}^\{\(\+\)\}\\Big\|X\_\{i\}\\right\]=\\mathbb\{E\}\\left\[\\mathbb\{E\}\\left\[\\frac\{Z\_\{i\}\}\{\\epsilon\}Y\_\{i\}^\{\(\+\)\}\\Big\|X\_\{i\},Z\_\{i\}\\right\]\\Big\|X\_\{i\}\\right\]=\\mathbb\{E\}\\left\[\\frac\{Z\_\{i\}\}\{\\epsilon\}f\(X\_\{i\}\+\\epsilon Z\_\{i\}\)\\Big\|X\_\{i\}\\right\]\\approx f^\{\\prime\}\(X\_\{i\}\)\+\\text\{bias\}\.Thus,Z1ϵ​Y1\(\+\),⋯,Znϵ​Yn\(\+\)\\frac\{Z\_\{1\}\}\{\\epsilon\}Y\_\{1\}^\{\(\+\)\},\\cdots,\\frac\{Z\_\{n\}\}\{\\epsilon\}Y\_\{n\}^\{\(\+\)\}arennindependent, biased samples off′​\(X1\),⋯,f′​\(Xn\)f^\{\\prime\}\(X\_\{1\}\),\\cdots,f^\{\\prime\}\(X\_\{n\}\)respectively with heterogeneous bias and standard errors\. Then, we learn a functionGGby

G=arg⁡minφ∈ℋ⁡1n​∑i=1n\(Ziϵ​Yi\(\+\)−φ​\(Xi\)\)2\.G=\\arg\\min\_\{\\varphi\\in\\mathcal\{H\}\}\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\left\(\\frac\{Z\_\{i\}\}\{\\epsilon\}Y\_\{i\}^\{\(\+\)\}\-\\varphi\(X\_\{i\}\)\\right\)^\{2\}\.Similarly, we can construct samples,Zi2−1ϵ2​Yi\(\+\)\\frac\{Z\_\{i\}^\{2\}\-1\}\{\\epsilon^\{2\}\}Y\_\{i\}^\{\(\+\)\},i=1,⋯,ni=1,\\cdots,n, for the second\-order derivative because

𝔼​\[Zi2−1ϵ2​Yi\(\+\)\|Xi\]=𝔼​\[𝔼​\[Zi2−1ϵ2​Yi\(\+\)\|Xi,Zi\]\|Xi\]\\displaystyle\\mathbb\{E\}\\left\[\\frac\{Z\_\{i\}^\{2\}\-1\}\{\\epsilon^\{2\}\}Y\_\{i\}^\{\(\+\)\}\\Big\|X\_\{i\}\\right\]=\\mathbb\{E\}\\left\[\\mathbb\{E\}\\left\[\\frac\{Z\_\{i\}^\{2\}\-1\}\{\\epsilon^\{2\}\}Y\_\{i\}^\{\(\+\)\}\\Big\|X\_\{i\},Z\_\{i\}\\right\]\\Big\|X\_\{i\}\\right\]=𝔼​\[Zi2−1ϵ2​f​\(Xi\+ϵ​Zi\)\|Xi\]\\displaystyle=\\mathbb\{E\}\\left\[\\frac\{Z\_\{i\}^\{2\}\-1\}\{\\epsilon^\{2\}\}f\(X\_\{i\}\+\\epsilon Z\_\{i\}\)\\Big\|X\_\{i\}\\right\]≈f′′​\(Xi\)\+bias\.\\displaystyle\\approx f^\{\\prime\\prime\}\(X\_\{i\}\)\+\\text\{bias\}\.Hence, we can learn a functionHHby

H=arg⁡minφ∈ℋ⁡1n​∑i=1n\(Zi2−1ϵ2​Yi\(\+\)−φ​\(Xi\)\)2\.H=\\arg\\min\_\{\\varphi\\in\\mathcal\{H\}\}\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\left\(\\frac\{Z\_\{i\}^\{2\}\-1\}\{\\epsilon^\{2\}\}Y\_\{i\}^\{\(\+\)\}\-\\varphi\(X\_\{i\}\)\\right\)^\{2\}\.The learned DNNsG,HG,Hwill then be reasonable approximations tof′,f′′f^\{\\prime\},f^\{\\prime\\prime\}respectively, subject to the optimization error \(controlled by the optimization solvers\), approximation error \(controlled by the functional classℋ\\mathcal\{H\}\), sampling error \(controlled bynnandϵ\\epsilon\), and the bias \(controlled byϵ\\epsilon\)\.

Furthermore, the multi\-point ZOD estimators can be constructed as follows: We sampleYi\(±\)∼ρ​\(Xi±ϵ​Zi,⋅\)Y\_\{i\}^\{\(\\pm\)\}\\sim\\rho\(X\_\{i\}\\pm\\epsilon Z\_\{i\},\\cdot\), and considerYi\(\+\)−Yi\(−\)2​ϵ​Zi\\frac\{Y\_\{i\}^\{\(\+\)\}\-Y\_\{i\}^\{\(\-\)\}\}\{2\\epsilon\}Z\_\{i\}andYi\(\+\)−2​Yi\+Yi\(−\)2​ϵ2​\(Zi2−1\)\\frac\{Y\_\{i\}^\{\(\+\)\}\-2Y\_\{i\}\+Y\_\{i\}^\{\(\-\)\}\}\{2\\epsilon^\{2\}\}\\left\(Z\_\{i\}^\{2\}\-1\\right\)for the biased samples off′​\(Xi\),f′′​\(Xi\)f^\{\\prime\}\(X\_\{i\}\),f^\{\\prime\\prime\}\(X\_\{i\}\)respectively\. In the following, for ease of presentation we only describe the construction under the one\-point ZOD, but multi\-point ZOD can be similarly constructed\.

### 3\.3Estimators for Solutions to Linear PDEs

We now apply the ZOD framework to our specific problem of solving nonlinear PDEs\. Recall that in the value iteration, the target function at the\(n\+1\)\(n\+1\)\-th step is given by the conditional expectation:

V^n\+1​\(t,x\)=𝔼​\[Rt,x​\(𝐔n\)\],\\hat\{V\}\_\{n\+1\}\(t,x\)=\\mathbb\{E\}\\left\[R^\{t,x\}\(\\mathbf\{U\}\_\{n\}\)\\right\],\(3\.6\)where𝐔n=\(Vn,Gn,Hn\)\\mathbf\{U\}\_\{n\}=\(V\_\{n\},G\_\{n\},H\_\{n\}\)represents the networks from the previous iteration\. The stochastic reward functional is defined as:

Rt,x​\(𝐔n\):=g​\(XTt,x\)\+∫tTf​\(Xst,x,Vn​\(s,Xst,x\),Gn​\(s,Xst,x\),Hn​\(s,Xst,x\)\)​ds\.R^\{t,x\}\(\\mathbf\{U\}\_\{n\}\):=g\(X\_\{T\}^\{t,x\}\)\+\\int\_\{t\}^\{T\}f\\left\(X^\{t,x\}\_\{s\},V\_\{n\}\(s,X^\{t,x\}\_\{s\}\),G\_\{n\}\(s,X^\{t,x\}\_\{s\}\),H\_\{n\}\(s,X\_\{s\}^\{t,x\}\)\\right\)\\mathrm\{d\}s\.\(3\.7\)SinceV^n\+1\\hat\{V\}\_\{n\+1\}is not available analytically, we cannot directly apply the ZOD estimators to the functionV^n\+1\\hat\{V\}\_\{n\+1\}itself\. However, we have access to the random variableRt,xR^\{t,x\}, which is anunbiasedestimator ofV^n\+1​\(t,x\)\\hat\{V\}\_\{n\+1\}\(t,x\)\.

This motivates theOne\-Point ZOD Estimator\. Instead of estimatingV^n\+1\\hat\{V\}\_\{n\+1\}and then employing ZOD, we directly use the perturbed realization of the rewardRt,x\+ϵ​ZR^\{t,x\+\\epsilon Z\}to estimate the spatial derivatives\. Taking the first\-order derivative as an example, recall thatZϵ​V^n\+1​\(t,x\+ϵ​Z\)\\frac\{Z\}\{\\epsilon\}\\hat\{V\}\_\{n\+1\}\(t,x\+\\epsilon Z\)is a biased estimator for∇V^n\+1​\(t,x\)\\nabla\\hat\{V\}\_\{n\+1\}\(t,x\), andRt,x\+ϵ​Z​\(𝐔n\)R^\{t,x\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\}\)is an unbiased estimator ofV^n\+1​\(t,x\+ϵ​Z\)\\hat\{V\}\_\{n\+1\}\(t,x\+\\epsilon Z\)\. Thus, we are motivated to takeZϵ​Rt,x\+ϵ​Z​\(𝐔n\)\\frac\{Z\}\{\\epsilon\}R^\{t,x\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\}\)as a one\-point estimator for∇V^n\+1​\(t,x\)\\nabla\\hat\{V\}\_\{n\+1\}\(t,x\), without relying on an intermediary estimate ofV^n\+1\\hat\{V\}\_\{n\+1\}\. In a similar fashion, a one\-point estimator for the second\-order derivative can also be constructed\. The following proposition presents the error bounds of the one\-point ZOD estimators inL2L\_\{2\}norm\.

For integersl,k≥0l,k\\geq 0,Cbl,k​\(\[0,T\]×ℝd\)C\_\{b\}^\{l,k\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\)denotes the class of functions that arelltimes continuously differentiable inttandkktimes continuously differentiable inxx, and whose derivatives are uniformly bounded, andCpl,k​\(\[0,T\]×ℝd\)C\_\{p\}^\{l,k\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\)denotes the corresponding class in which these derivatives have at most polynomial growth inxx\. Similarly,Cbk​\(ℝd\)C\_\{b\}^\{k\}\(\\mathbb\{R\}^\{d\}\)andCpk​\(ℝd\)C\_\{p\}^\{k\}\(\\mathbb\{R\}^\{d\}\)denote the time\-independent counterparts\. These notations are understood componentwise for vector\- or matrix\-valued functions\.

We need the following standing assumption\.

###### Assumption 1

The coefficientsbbandσ\\sigmaare sufficiently regular so that the SDE \([2\.2](https://arxiv.org/html/2606.24999#S2.E2)\) admits a unique strong solution\. Moreover, they satisfy the linear growth and bounded condition, i\.e\., there are constantsLbL\_\{b\}andσ0\\sigma\_\{0\}such that

\|b​\(t,x\)\|≤Lb​\(1\+\|x\|\),\|σ​\(t,x\)\|≤σ0,for all\(t,x\)∈\[0,T\]×ℝd\.\\displaystyle\|b\(t,x\)\|\\leq L\_\{b\}\(1\+\|x\|\),\\quad\|\\sigma\(t,x\)\|\\leq\\sigma\_\{0\},\\quad\\text\{for all\}\\ \\ \(t,x\)\\in\[0,T\]\\times\\mathbb\{R\}^\{d\}\.\(3\.8\)

Assumption[1](https://arxiv.org/html/2606.24999#Thmassumption1)is a standard condition for SDEs, and we require the existence and uniqueness of a strong solution since we consider both the weak and strong simulators\. Furthermore, we impose regularity conditions on the drift and volatility coefficient so that the usual moment estimates for the solution of SDE hold, which is needed in the subsequent analysis\. In particular, we require stronger boundedness condition on the volatility coefficient for us to establish the desired statistical error guarantee later; otherwise, the underlying SDEs may be too volatile, losing the property required for the empirical loss functions used in the training to concentrate around its population counterpart\.

Under Assumption[1](https://arxiv.org/html/2606.24999#Thmassumption1), we present the property of ZOD estimators under our context\.

###### Proposition 4

Suppose Assumption[1](https://arxiv.org/html/2606.24999#Thmassumption1)holds\. Assume also that the coefficientsb,σ,g,f,Vn,Gn,Hnb,\\sigma,g,f,V\_\{n\},G\_\{n\},H\_\{n\}are smooth enough so thatV^n\+1∈Cp0,3​\(\[0,T\]×ℝd\)\\hat\{V\}\_\{n\+1\}\\in C\_\{p\}^\{0,3\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\)and𝐔n,f,g\\mathbf\{U\}\_\{n\},f,gall have polynomial growth inxx, uniformly int∈\[0,T\]t\\in\[0,T\]\. ThenZϵ​Rt,Xt\+ϵ​Z​\(𝐔n\)\\frac\{Z\}\{\\epsilon\}R^\{t,X\_\{t\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\}\)is a biased estimator for∇V^n\+1​\(t,Xt\)\\nabla\\hat\{V\}\_\{n\+1\}\(t,X\_\{t\}\)conditioning onXtX\_\{t\}, and there is a constantC\>0C\>0independent ofϵ\\epsilonsuch that its bias and variance are upper bounded by

𝔼\[∫0T\|𝔼\[ZϵRt,Xt\+ϵ​Z\(𝐔n\)\|Xt\]−∇V^n\+1\(t,Xt\)\|2dt\]≤Cϵ4,\|Var\[ZϵRt,Xt\+ϵ​Z\(𝐔n\)\]\|≤C1ϵ2,\\mathbb\{E\}\\left\[\\int\_\{0\}^\{T\}\\left\|\\mathbb\{E\}\\left\[\\frac\{Z\}\{\\epsilon\}R^\{t,X\_\{t\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\}\)\\Big\|X\_\{t\}\\right\]\-\\nabla\\hat\{V\}\_\{n\+1\}\(t,X\_\{t\}\)\\right\|^\{2\}\\mathrm\{d\}t\\right\]\\leq C\\epsilon^\{4\},\\ \\left\|\\operatorname\{Var\}\\left\[\\frac\{Z\}\{\\epsilon\}R^\{t,X\_\{t\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\}\)\\right\]\\right\|\\leq C\\frac\{1\}\{\\epsilon^\{2\}\},Moreover, ifV^n\+1∈Cp0,4​\(\[0,T\]×ℝd\)\\hat\{V\}\_\{n\+1\}\\in C\_\{p\}^\{0,4\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\), thenZ​Z⊤−Idϵ2​Rt,Xt\+ϵ​Z​\(𝐔n\)\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\}R^\{t,X\_\{t\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\}\)is a biased estimator for∇2V^n\+1​\(t,Xt\)\\nabla^\{2\}\\hat\{V\}\_\{n\+1\}\(t,X\_\{t\}\)conditioning onXtX\_\{t\}, and there is a constantC\>0C\>0independent ofϵ\\epsilonsuch that its bias and variance are upper bounded by

𝔼\[∫0T\|𝔼\[Z​Z⊤−Idϵ2Rt,Xt\+ϵ​Z\(𝐔n\)\|Xt\]−∇2V^n\+1\(t,Xt\)\|2dt\]\\displaystyle\\mathbb\{E\}\\left\[\\int\_\{0\}^\{T\}\\left\|\\mathbb\{E\}\\left\[\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\}R^\{t,X\_\{t\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\}\)\\Big\|X\_\{t\}\\right\]\-\\nabla^\{2\}\\hat\{V\}\_\{n\+1\}\(t,X\_\{t\}\)\\right\|^\{2\}\\mathrm\{d\}t\\right\]≤C​ϵ4,\\displaystyle\\leq C\\epsilon^\{4\},\(3\.9\)\|Var⁡\[Z​Z⊤−Idϵ2​Rt,Xt\+ϵ​Z​\(𝐔n\)\]\|\\displaystyle\\left\|\\operatorname\{Var\}\\left\[\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\}R^\{t,X\_\{t\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\}\)\\right\]\\right\|≤C​1ϵ4\.\\displaystyle\\leq C\\frac\{1\}\{\\epsilon^\{4\}\}\.

Proposition[4](https://arxiv.org/html/2606.24999#Thmtheorem4)describes the one\-point ZOD estimators for the derivatives of the solution to a linear PDE and confirms that their biases are small and controlled byϵ2\\epsilon^\{2\}\. A significant limitation is that the variances of the estimators are generally unbounded, and in fact have the orders ofϵ−2\\epsilon^\{\-2\}andϵ−4\\epsilon^\{\-4\}for the gradient and Hessian respectively, similarly to the classical ZOD in Lemma[1](https://arxiv.org/html/2606.24999#Thmtheorem1)\. We refer to them as “ZOD\-1” in our subsequent numerical experiment, and will demonstrate that they usually produce worse results in learning derivatives\.

##### Variance Reduction with Strong Simulators

Theoretically, it is natural to investigate the symmetric versions \(analogous to Lemma[2](https://arxiv.org/html/2606.24999#Thmtheorem2)and Lemma[3](https://arxiv.org/html/2606.24999#Thmtheorem3)\) of ZOD estimators for the gradient and Hessian of the solution to this linear PDE and inquire whether this variance reduction technique is helpful in the current PDE setting as well\. It turns out that the answer depends on which type of simulators is accessible to us, i\.e\., when simulating multiple trajectories starting fromdifferentinitial statesxx,x\+ϵ​Zx\+\\epsilon Z, andx−ϵ​Zx\-\\epsilon Z, whether we can control the noises in the environment in a way that the corresponding rewardsRt,x,Rt,x\+ϵ​Z,Rt,x−ϵ​ZR^\{t,x\},R^\{t,x\+\\epsilon Z\},R^\{t,x\-\\epsilon Z\}are highly correlated and therefore cancel the noises off to great extent\.888This idea resembles the classical simulation techniques ofcontrol variatesandantithetic variates\(cf\.Glasserman[2004](https://arxiv.org/html/2606.24999#bib.bib21), Chapter 4\) to use correlated samples for variance reduction\.With a weak simulator, we need to query the simulatorseparately, one at a time, to generate the trajectoriesXt,x,Xt,x\+ϵ​Z,Xt,x−ϵ​ZX^\{t,x\},X^\{t,x\+\\epsilon Z\},X^\{t,x\-\\epsilon Z\}\. Because the Brownian motions in the underlying stochastic processes are independent, these trajectories, as well as the associated rewardsRt,x,Rt,x\+ϵ​Z,Rt,x−ϵ​ZR^\{t,x\},R^\{t,x\+\\epsilon Z\},R^\{t,x\-\\epsilon Z\}, are conditionally independent \(conditioned on the realizedZZ\), and hence we cannot expect the benefit of variance reduction\. However, with a strong simulator, we are presented with trajectories ofXt,x,Xt,x\+ϵ​Z,Xt,x−ϵ​ZX^\{t,x\},X^\{t,x\+\\epsilon Z\},X^\{t,x\-\\epsilon Z\}driven by thesameBrownian path upon one single query , andRt,x,Rt,x\+ϵ​Z,Rt,x−ϵ​ZR^\{t,x\},R^\{t,x\+\\epsilon Z\},R^\{t,x\-\\epsilon Z\}are expected to be highly correlated, leading to additional noise cancellation inRt,x\+ϵ​Z−Rt,x−ϵ​ZR^\{t,x\+\\epsilon Z\}\-R^\{t,x\-\\epsilon Z\}andRt,x\+ϵ​Z\+Rt,x−ϵ​Z−2​Rt,xR^\{t,x\+\\epsilon Z\}\+R^\{t,x\-\\epsilon Z\}\-2R^\{t,x\}\. The next Proposition[5](https://arxiv.org/html/2606.24999#Thmtheorem5)formalizes and indeed validates this idea\.

###### Proposition 5

Suppose Assumption[1](https://arxiv.org/html/2606.24999#Thmassumption1)holds, and𝐔n,f,g\\mathbf\{U\}\_\{n\},f,gall have polynomial growth inxx, uniformly int∈\[0,T\]t\\in\[0,T\]\. Construct

g^\\displaystyle\\hat\{g\}:=Z2​ϵ​\(Rt,Xt\+ϵ​Z​\(𝐔n\)−Rt,Xt−ϵ​Z​\(𝐔n\)\),\\displaystyle=\\frac\{Z\}\{2\\epsilon\}\\left\(R^\{t,X\_\{t\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\}\)\-R^\{t,X\_\{t\}\-\\epsilon Z\}\(\\mathbf\{U\}\_\{n\}\)\\right\),\(3\.10\)h^\\displaystyle\\hat\{h\}:=Z​Z⊤−Id2​ϵ2​\(Rt,Xt\+ϵ​Z​\(𝐔n\)\+Rt,Xt−ϵ​Z​\(𝐔n\)−2​Rt,Xt​\(𝐔n\)\)\.\\displaystyle=\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{2\\epsilon^\{2\}\}\\left\(R^\{t,X\_\{t\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\}\)\+R^\{t,X\_\{t\}\-\\epsilon Z\}\(\\mathbf\{U\}\_\{n\}\)\-2R^\{t,X\_\{t\}\}\(\\mathbf\{U\}\_\{n\}\)\\right\)\.1. \(a\)With a weak simulator as specified in Condition[1](https://arxiv.org/html/2606.24999#Thmcond1), if the coefficientsb,σ,g,f,Vn,Gn,Hnb,\\sigma,g,f,V\_\{n\},G\_\{n\},H\_\{n\}are smooth enough so thatV^n\+1∈Cp0,3​\(\[0,T\]×ℝd\)\\hat\{V\}\_\{n\+1\}\\in C\_\{p\}^\{0,3\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\), theng^\\hat\{g\}is a biased estimator for∇V^n\+1​\(t,Xt\)\\nabla\\hat\{V\}\_\{n\+1\}\(t,X\_\{t\}\)conditioning onXtX\_\{t\}, and its bias and variance are upper bounded by 𝔼\[∫0T\|𝔼\[g^∣Xt\]−∇V^n\+1\(t,Xt\)\|2dt\]≤Cϵ4,\|Var\[g^\]\|≤C1ϵ2\.\\mathbb\{E\}\\left\[\\int\_\{0\}^\{T\}\\left\|\\mathbb\{E\}\\left\[\\hat\{g\}\\mid X\_\{t\}\\right\]\-\\nabla\\hat\{V\}\_\{n\+1\}\(t,X\_\{t\}\)\\right\|^\{2\}\\mathrm\{d\}t\\right\]\\leq C\\epsilon^\{4\},\\ \\left\|\\operatorname\{Var\}\[\\hat\{g\}\]\\right\|\\leq C\\frac\{1\}\{\\epsilon^\{2\}\}\.\(3\.11\)Furthermore, ifV^n\+1∈Cp0,4​\(\[0,T\]×ℝd\)\\hat\{V\}\_\{n\+1\}\\in C\_\{p\}^\{0,4\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\), thenh^\\hat\{h\}is a biased estimator for∇2V^n\+1​\(t,Xt\)\\nabla^\{2\}\\hat\{V\}\_\{n\+1\}\(t,X\_\{t\}\)conditioning onXtX\_\{t\}, and its bias and variance are upper bounded by 𝔼\[∫0T\|𝔼\[h^∣Xt\]−∇2V^n\+1\(t,Xt\)\|2dt\]≤Cϵ4,\|Var\[h^\]\|≤C1ϵ4\.\\displaystyle\\mathbb\{E\}\\left\[\\int\_\{0\}^\{T\}\\left\|\\mathbb\{E\}\\left\[\\hat\{h\}\\mid X\_\{t\}\\right\]\-\\nabla^\{2\}\\hat\{V\}\_\{n\+1\}\(t,X\_\{t\}\)\\right\|^\{2\}\\mathrm\{d\}t\\right\]\\leq C\\epsilon^\{4\},\\ \\left\|\\operatorname\{Var\}\[\\hat\{h\}\]\\right\|\\leq C\\frac\{1\}\{\\epsilon^\{4\}\}\.\(3\.12\)
2. \(b\)With a strong simulator as specified in Condition[2](https://arxiv.org/html/2606.24999#Thmcond2), ifb,σ∈Cb0,4​\(\[0,T\]×ℝd\)b,\\sigma\\in C\_\{b\}^\{0,4\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\),f~∈Cp0,3​\(\[0,T\]×ℝd\)\\tilde\{f\}\\in C\_\{p\}^\{0,3\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\)wheref~​\(t,x\):=f​\(t,x,𝐔n​\(t,x\)\)\\tilde\{f\}\(t,x\):=f\(t,x,\\mathbf\{U\}\_\{n\}\(t,x\)\)andg∈Cp3​\(ℝd\)g\\in C\_\{p\}^\{3\}\(\\mathbb\{R\}^\{d\}\), theng^\\hat\{g\}has the same order of biases as the one\-point estimator in Proposition[4](https://arxiv.org/html/2606.24999#Thmtheorem4)but bounded variances asϵ→0\\epsilon\\rightarrow 0\. Furthermore, ifb,σ∈Cb0,5​\(\[0,T\]×ℝd\)b,\\sigma\\in C\_\{b\}^\{0,5\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\),f~∈Cp0,4​\(\[0,T\]×ℝd\)\\tilde\{f\}\\in C\_\{p\}^\{0,4\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\)andg∈Cp4​\(ℝd\)g\\in C\_\{p\}^\{4\}\(\\mathbb\{R\}^\{d\}\), thenh^\\hat\{h\}has the same order of biases as the one\-point estimator in Proposition[4](https://arxiv.org/html/2606.24999#Thmtheorem4)but bounded variances asϵ→0\\epsilon\\rightarrow 0\.

Proposition[5](https://arxiv.org/html/2606.24999#Thmtheorem5)reveals a unique bias–variance tradeoff associated with the multi\-point ZOD estimators in our particular context and the critical difference between the weak and strong simulators\. It turns out that the conventional bounded variance in Lemmas[2](https://arxiv.org/html/2606.24999#Thmtheorem2)and[3](https://arxiv.org/html/2606.24999#Thmtheorem3)no longer holds for the weak simulator and only holds for the strong simulator\. In particular, a multi\-point ZOD estimator does not reduce variance with a weak simulator, and its variance still has the same order as the one\-point ZOD in Proposition[4](https://arxiv.org/html/2606.24999#Thmtheorem4)\. Hence, in general, there is no benefit to apply multi\-point ZOD when only a weak simulator is accessible\. We refer to the multi\-point ZOD estimator with a strong simulator as “ZOD\-m” in our subsequent numerical experiments, and will demonstrate that it produces much more stable results in learning derivatives while requiring fewer samples\.

## 4The Zeroth\-Order Deep Learning Algorithm

Let𝐔=\(V,G,H\)\\mathbf\{U\}=\(V,G,H\)denote the triplet of functions\. We define the approximating neural networks as:

φθ:=\(φvθ,φgθ,φhθ\),\\varphi^\{\\theta\}:=\(\\varphi^\{\\theta\}\_\{v\},\\varphi^\{\\theta\}\_\{g\},\\varphi^\{\\theta\}\_\{h\}\),whereφvθ:\[0,T\]×ℝd→ℝ\\varphi^\{\\theta\}\_\{v\}:\[0,T\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\},φgθ:\[0,T\]×ℝd→ℝd\\varphi^\{\\theta\}\_\{g\}:\[0,T\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}, andφhθ:\[0,T\]×ℝd→ℝd×d\\varphi^\{\\theta\}\_\{h\}:\[0,T\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\\times d\}approximate the value, gradient, and Hessian, respectively\. The parameters are denoted byθ∈Θ⊂ℝp\\theta\\in\\Theta\\subset\\mathbb\{R\}^\{p\}\.

We define the reward function starting from state\(t,x\)\(t,x\)given a function triplet𝐔\\mathbf\{U\}as:

Rt,x​\(𝐔\):=g​\(XTt,x\)\+∫tTf​\(Xst,x,V​\(s,Xst,x\),G​\(s,Xst,x\),H​\(s,Xst,x\)\)​ds\.R^\{t,x\}\(\\mathbf\{U\}\):=g\(X\_\{T\}^\{t,x\}\)\+\\int\_\{t\}^\{T\}f\\left\(X^\{t,x\}\_\{s\},V\(s,X^\{t,x\}\_\{s\}\),G\(s,X^\{t,x\}\_\{s\}\),H\(s,X\_\{s\}^\{t,x\}\)\\right\)\\mathrm\{d\}s\.\(4\.1\)Crucially, the source termffis evaluated using the explicit outputs ofGGandHH, rather than derivatives computed via auto\-diff\.

We implement the value iteration \([2\.5](https://arxiv.org/html/2606.24999#S2.E5)\) as follows: At iterationnn, let𝐔n=\(Vn,Gn,Hn\)\\mathbf\{U\}\_\{n\}=\(V\_\{n\},G\_\{n\},H\_\{n\}\)be the trained networks from the previous step\. We aim to learn𝐔n\+1\\mathbf\{U\}\_\{n\+1\}such that:

1. 1\.Vn\+1​\(t,x\)≈V^n\+1​\(t,x\):=𝔼​\[Rt,x​\(𝐔n\)\]V\_\{n\+1\}\(t,x\)\\approx\\hat\{V\}\_\{n\+1\}\(t,x\):=\\mathbb\{E\}\[R^\{t,x\}\(\\mathbf\{U\}\_\{n\}\)\]\.
2. 2\.Gn\+1​\(t,x\)≈∇V^n\+1​\(t,x\)G\_\{n\+1\}\(t,x\)\\approx\\nabla\\hat\{V\}\_\{n\+1\}\(t,x\)\.
3. 3\.Hn\+1​\(t,x\)≈∇2V^n\+1​\(t,x\)H\_\{n\+1\}\(t,x\)\\approx\\nabla^\{2\}\\hat\{V\}\_\{n\+1\}\(t,x\)\.

### 4\.1Loss Function Design

To train the candidate networksφ=\(φv,φg,φh\)\\varphi=\(\\varphi\_\{v\},\\varphi\_\{g\},\\varphi\_\{h\}\)at iterationnn, we minimize a composite loss function that enforces consistency between the networks and the stochastic targets derived from𝐔n\\mathbf\{U\}\_\{n\}:

L​\(φ,𝐔n\):=Lv​\(φv,𝐔n\)\+Lg​\(φg,𝐔n\)\+Lh​\(φh,𝐔n\),L\(\\varphi,\\mathbf\{U\}\_\{n\}\):=L\_\{v\}\(\\varphi\_\{v\},\\mathbf\{U\}\_\{n\}\)\+L\_\{g\}\(\\varphi\_\{g\},\\mathbf\{U\}\_\{n\}\)\+L\_\{h\}\(\\varphi\_\{h\},\\mathbf\{U\}\_\{n\}\),\(4\.2\)where each individual loss will be defined below, withβ\>0\\beta\>0some fixed constant\.

#### 4\.1\.1Value Loss

The value networkφvθ\\varphi^\{\\theta\}\_\{v\}is trained to match the conditional expectation of the reward\. The loss is defined as the mean squared error \(MSE\) against the stochastic realization:

Lv​\(φv,𝐔n\):=β​𝔼​\[∫0Teβ​s​\|φv​\(s,Xs\)−Rs,Xs​\(𝐔n\)\|2​ds\]\.L\_\{v\}\(\\varphi\_\{v\},\\mathbf\{U\}\_\{n\}\):=\\beta\\mathbb\{E\}\\left\[\\int\_\{0\}^\{T\}e^\{\\beta s\}\\left\|\\varphi\_\{v\}\(s,X\_\{s\}\)\-R^\{s,X\_\{s\}\}\(\\mathbf\{U\}\_\{n\}\)\\right\|^\{2\}\\mathrm\{d\}s\\right\]\.\(4\.3\)Minimizing this loss impliesφv​\(t,x\)≈𝔼​\[Rt,x​\(𝐔n\)\]\\varphi\_\{v\}\(t,x\)\\approx\\mathbb\{E\}\[R^\{t,x\}\(\\mathbf\{U\}\_\{n\}\)\]\.

#### 4\.1\.2Zeroth\-Order Derivatives Losses

SinceV^n\+1\\hat\{V\}\_\{n\+1\}is defined as an expectation, its derivatives are not directly available\. We employ the one\-point ZOD estimators \(Proposition[4](https://arxiv.org/html/2606.24999#Thmtheorem4)\) to construct biased targets using perturbed trajectories\.999Here we present the one\-point ZOD estimators for illustration\. Multi\-point ZODà laProposition[5](https://arxiv.org/html/2606.24999#Thmtheorem5)can be discussed similarly\.LetZ∼𝒩​\(0,Id\)Z\\sim\\mathcal\{N\}\(0,I\_\{d\}\)be a perturbation vector independent of the trajectoryXX\.

Gradient Loss\.We rely on the relationship∇𝔼​\[Rt,x\]≈𝔼​\[Zϵ​Rt,x\+ϵ​Z\]\\nabla\\mathbb\{E\}\[R^\{t,x\}\]\\approx\\mathbb\{E\}\[\\frac\{Z\}\{\\epsilon\}R^\{t,x\+\\epsilon Z\}\]\. The gradient networkφgθ\\varphi^\{\\theta\}\_\{g\}minimizes:

Lg​\(φgθ,𝐔n\):=β​𝔼​\[∫0Teβ​t​\|φgθ​\(t,Xt\)−Zϵ​Rt,Xt\+ϵ​Z​\(𝐔n\)\|2​dt\]\.L\_\{g\}\(\\varphi^\{\\theta\}\_\{g\},\\mathbf\{U\}\_\{n\}\):=\\beta\\mathbb\{E\}\\left\[\\int\_\{0\}^\{T\}e^\{\\beta t\}\\left\|\\varphi^\{\\theta\}\_\{g\}\(t,X\_\{t\}\)\-\\frac\{Z\}\{\\epsilon\}R^\{t,X\_\{t\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\}\)\\right\|^\{2\}\\mathrm\{d\}t\\right\]\.\(4\.4\)
Hessian Loss\.Similarly, the Hessian networkφhθ\\varphi^\{\\theta\}\_\{h\}minimizes the error against the second\-order ZOD estimator, where the norm stands for the Frobenius norm for matrices:

Lh​\(φhθ,𝐔n\):=𝔼​\[∫0Teβ​t​\|φhθ​\(t,Xt\)−Z​Z⊤−Idϵ2​Rt,Xt\+ϵ​Z​\(𝐔n\)\|2​dt\]\.L\_\{h\}\(\\varphi^\{\\theta\}\_\{h\},\\mathbf\{U\}\_\{n\}\):=\\mathbb\{E\}\\left\[\\int\_\{0\}^\{T\}e^\{\\beta t\}\\left\|\\varphi^\{\\theta\}\_\{h\}\(t,X\_\{t\}\)\-\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\}R^\{t,X\_\{t\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\}\)\\right\|^\{2\}\\mathrm\{d\}t\\right\]\.\(4\.5\)

### 4\.2Discretization and Implementation

In the numerical implementation, expectations are estimated via Monte Carlo sampling\. We define a uniform time grid𝒢:=\{0=t0<⋯<tN=T\}\\mathcal\{G\}:=\\\{0=t\_\{0\}<\\dots<t\_\{N\}=T\\\}with stepsΔ​tj:=tj\+1−tj=TN\\Delta t\_\{j\}:=t\_\{j\+1\}\-t\_\{j\}=\\frac\{T\}\{N\}\. Let\{X\(b\)\}b=1B\\\{X^\{\(b\)\}\\\}\_\{b=1\}^\{B\}denote a batch of base trajectories starting from\(0,x0\)\(0,x\_\{0\}\)\.

For the value loss, the empirical reward along a base trajectory is computed using the frozen networks𝐔n\\mathbf\{U\}\_\{n\}:

R~tj,Xtj\(b\)​\(𝐔n\):=g​\(XT\(b\)\)\+∑i=jN−1f​\(Xti\(b\),Vn​\(ti,Xti\(b\)\),Gn​\(ti,Xti\(b\)\),Hn​\(ti,Xti\(b\)\)\)​Δ​ti\.\\tilde\{R\}^\{t\_\{j\},X^\{\(b\)\}\_\{t\_\{j\}\}\}\(\\mathbf\{U\}\_\{n\}\):=g\(X\_\{T\}^\{\(b\)\}\)\+\\sum\_\{i=j\}^\{N\-1\}f\\left\(X\_\{t\_\{i\}\}^\{\(b\)\},V\_\{n\}\(t\_\{i\},X\_\{t\_\{i\}\}^\{\(b\)\}\),G\_\{n\}\(t\_\{i\},X\_\{t\_\{i\}\}^\{\(b\)\}\),H\_\{n\}\(t\_\{i\},X\_\{t\_\{i\}\}^\{\(b\)\}\)\\right\)\\Delta t\_\{i\}\.\(4\.6\)
To compute the derivative losses, for each samplebband time stepjj, we generate a random perturbationZ\(b\)∼𝒩​\(0,Id\)Z^\{\(b\)\}\\sim\\mathcal\{N\}\(0,I\_\{d\}\)and simulate \(observe\) one perturbed trajectory\{Xtj,Xtj\(b\)\+ϵ​Z\(b\)\}s≥tj\\\{X^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\}\\\}\_\{s\\geq t\_\{j\}\}starting fromXtj\(b\)\+ϵ​Z\(b\)X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\. The reward along this perturbed path is:

R~tj,Xtj\(b\)\+ϵ​Z\(b\)​\(𝐔n\):=g​\(XTtj,Xtj\(b\)\+ϵ​Z\(b\)\)\\displaystyle\\tilde\{R\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\}\(\\mathbf\{U\}\_\{n\}\)=g\(X\_\{T\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\}\)\(4\.7\)\+\\displaystyle\+∑i=jN−1f​\(Xtitj,Xtj\(b\)\+ϵ​Z\(b\),Vn​\(ti,Xtitj,Xtj\(b\)\+ϵ​Z\(b\)\),Gn​\(ti,Xtitj,Xtj\(b\)\+ϵ​Z\(b\)\),Hn​\(ti,Xtitj,Xtj\(b\)\+ϵ​Z\(b\)\)\)​Δ​ti\.\\displaystyle\\sum\_\{i=j\}^\{N\-1\}f\\left\(X\_\{t\_\{i\}\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\},V\_\{n\}\(t\_\{i\},X\_\{t\_\{i\}\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\}\),G\_\{n\}\(t\_\{i\},X\_\{t\_\{i\}\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\}\),H\_\{n\}\(t\_\{i\},X\_\{t\_\{i\}\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\}\)\\right\)\\Delta t\_\{i\}\.
The tractable empirical loss functions are:

L~v\(B\)​\(φv,𝐔n\)\\displaystyle\\tilde\{L\}\_\{v\}^\{\(B\)\}\(\\varphi\_\{v\},\\mathbf\{U\}\_\{n\}\):=β​1B​∑b=1B∑j=0N−1eβ​tj​\(φv​\(tj,Xtj\(b\)\)−R~tj,Xtj\(b\)​\(𝐔n\)\)2​Δ​tj,\\displaystyle=\\beta\\frac\{1\}\{B\}\\sum\_\{b=1\}^\{B\}\\sum\_\{j=0\}^\{N\-1\}e^\{\\beta t\_\{j\}\}\\left\(\\varphi\_\{v\}\(t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\)\-\\tilde\{R\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\}\(\\mathbf\{U\}\_\{n\}\)\\right\)^\{2\}\\Delta t\_\{j\},\(4\.8\)L~g\(B\)​\(φg,𝐔n\)\\displaystyle\\tilde\{L\}\_\{g\}^\{\(B\)\}\(\\varphi\_\{g\},\\mathbf\{U\}\_\{n\}\):=β​1B​∑b=1B∑j=0N−1eβ​tj​\|φg​\(tj,Xtj\(b\)\)−Z\(b\)ϵ​R~tj,Xtj\(b\)\+ϵ​Z\(b\)​\(𝐔n\)\|2​Δ​tj,\\displaystyle=\\beta\\frac\{1\}\{B\}\\sum\_\{b=1\}^\{B\}\\sum\_\{j=0\}^\{N\-1\}e^\{\\beta t\_\{j\}\}\\left\|\\varphi\_\{g\}\(t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\)\-\\frac\{Z^\{\(b\)\}\}\{\\epsilon\}\\tilde\{R\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\}\(\\mathbf\{U\}\_\{n\}\)\\right\|^\{2\}\\Delta t\_\{j\},L~h\(B\)​\(φh,𝐔n\)\\displaystyle\\tilde\{L\}\_\{h\}^\{\(B\)\}\(\\varphi\_\{h\},\\mathbf\{U\}\_\{n\}\):=1B​∑b=1B∑j=0N−1eβ​tj​\|φh​\(tj,Xtj\(b\)\)−Z\(b\)​\(Z\(b\)\)⊤−Idϵ2​R~tj,Xtj\(b\)\+ϵ​Z\(b\)​\(𝐔n\)\|2​Δ​tj\.\\displaystyle=\\frac\{1\}\{B\}\\sum\_\{b=1\}^\{B\}\\sum\_\{j=0\}^\{N\-1\}e^\{\\beta t\_\{j\}\}\\left\|\\varphi\_\{h\}\(t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\)\-\\frac\{Z^\{\(b\)\}\(Z^\{\(b\)\}\)^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\}\\tilde\{R\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\}\(\\mathbf\{U\}\_\{n\}\)\\right\|^\{2\}\\Delta t\_\{j\}\.
The empirical total loss is defined by

L~\(B\)​\(φ,𝐔n\):=L~v\(B\)​\(φv,𝐔n\)\+L~g\(B\)​\(φg,𝐔n\)\+L~h\(B\)​\(φh,𝐔n\)\.\\tilde\{L\}^\{\(B\)\}\(\\varphi,\\mathbf\{U\}\_\{n\}\):=\\tilde\{L\}\_\{v\}^\{\(B\)\}\(\\varphi\_\{v\},\\mathbf\{U\}\_\{n\}\)\+\\tilde\{L\}^\{\(B\)\}\_\{g\}\(\\varphi\_\{g\},\\mathbf\{U\}\_\{n\}\)\+\\tilde\{L\}^\{\(B\)\}\_\{h\}\(\\varphi\_\{h\},\\mathbf\{U\}\_\{n\}\)\.\(4\.9\)The complete procedure is summarized in Algorithm[1](https://arxiv.org/html/2606.24999#alg1)\.

Algorithm 1Zeroth\-Order PDE Solver via Independent Networks1:Input:Function

ff, terminal condition

gg, time horizon

TT, initial state

x0x\_\{0\}, time grid

𝒢=\{0=t0<⋯<tN=T\}\\mathcal\{G\}=\\\{0=t\_\{0\}<\\dots<t\_\{N\}=T\\\}with

Δ​tj=tj\+1−tj\\Delta t\_\{j\}=t\_\{j\+1\}\-t\_\{j\}, batch size

BB, outer iterations

MM, gradient steps per iteration

KK, learning rate

η\\eta, perturbation size

ϵ\\epsilon\.

2:Required Program:A simulator generating sample trajectories whose infinitesimal generator is

ℒ\\mathcal\{L\}\.

3:Initialize:Independent neural networks

𝐔0=\(V0,G0,H0\)\\mathbf\{U\}\_\{0\}=\(V\_\{0\},G\_\{0\},H\_\{0\}\), parameter

θ0\\theta\_\{0\}\.

4:for

n=0n=0to

M−1M\-1do

5:Initialize candidate networks

φθ=\(φvθ,φgθ,φhθ\)\\varphi^\{\\theta\}=\(\\varphi^\{\\theta\}\_\{v\},\\varphi^\{\\theta\}\_\{g\},\\varphi^\{\\theta\}\_\{h\}\)with parameters

θ=θn\\theta=\\theta\_\{n\}
6:for

k=1k=1to

KKdo

7:Observe

BBtrajectories

\{Xs\(b\)\}s∈\[0,T\],b=1,…,B\\\{X^\{\(b\)\}\_\{s\}\\\}\_\{s\\in\[0,T\],b=1,\\dots,B\}starting from

\(0,x0\)\(0,x\_\{0\}\)\.

8:For

0≤j≤N0\\leq j\\leq N,

1≤b≤B1\\leq b\\leq B, evaluate

Gn​\(tj,Xtj\(b\)\)G\_\{n\}\(t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\)and

Hn​\(tj,Xtj\(b\)\)H\_\{n\}\(t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\)using frozen networks\.

9:Compute empirical reward random variable

R~tj,Xtj\(b\)​\(𝐔n\)\\tilde\{R\}^\{t\_\{j\},X^\{\(b\)\}\_\{t\_\{j\}\}\}\(\\mathbf\{U\}\_\{n\}\)by \([4\.6](https://arxiv.org/html/2606.24999#S4.E6)\)\.

10:Compute

L~v\(B\):=β​1B​∑b=1B∑j=0N−1eβ​tj​\(R~tj,Xtj\(b\)​\(𝐔n\)−φvθ​\(tj,Xtj\(b\)\)\)2​Δ​tj\.\\tilde\{L\}^\{\(B\)\}\_\{v\}:=\\beta\\frac\{1\}\{B\}\\sum\_\{b=1\}^\{B\}\\sum\_\{j=0\}^\{N\-1\}e^\{\\beta t\_\{j\}\}\\left\(\\tilde\{R\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\}\(\\mathbf\{U\}\_\{n\}\)\-\\varphi^\{\\theta\}\_\{v\}\(t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\)\\right\)^\{2\}\\Delta t\_\{j\}\.
11:Sample

\{Z\(b\)\}b=1B∼𝒩​\(0,Id\)\\\{Z^\{\(b\)\}\\\}\_\{b=1\}^\{B\}\\sim\\mathcal\{N\}\(0,I\_\{d\}\), for each

0≤j≤N0\\leq j\\leq N,

1≤b≤B1\\leq b\\leq B, observe one trajectory

\{Xttj,Xtj\(b\)\+ϵ​Z\(b\)\}tj≤t≤T\\left\\\{X\_\{t\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\}\\right\\\}\_\{t\_\{j\}\\leq t\\leq T\}starting at

\(tj,Xtj\(b\)\+ϵ​Z\(b\)\)\(t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\)\.

12:For

0≤j≤N0\\leq j\\leq N,

1≤b≤B1\\leq b\\leq B,

j≤i≤Nj\\leq i\\leq N, evaluate

Gn​\(ti,Xtitj,Xtj\(b\)\+ϵ​Z\(b\)\)G\_\{n\}\(t\_\{i\},X\_\{t\_\{i\}\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\}\)and

Hn​\(ti,Xtitj,Xtj\(b\)\+ϵ​Z\(b\)\)H\_\{n\}\(t\_\{i\},X\_\{t\_\{i\}\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\}\)on the perturbed paths\.

13:Compute perturbed reward random variable

R~tj,Xtj\(b\)\+ϵ​Z\(b\)​\(𝐔n\)\\tilde\{R\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\}\(\\mathbf\{U\}\_\{n\}\)by \([4\.7](https://arxiv.org/html/2606.24999#S4.E7)\)\.

14:Compute the derivatives loss by zeroth\-order estimators:

L~g\(B\)\\displaystyle\\tilde\{L\}^\{\(B\)\}\_\{g\}:=β​1B​∑b=1B∑j=0N−1eβ​tj​\(Z\(b\)ϵ​R~tj,Xtj\(b\)\+ϵ​Z\(b\)​\(𝐔n\)−φgθ​\(tj,Xtj\(b\)\)\)2​Δ​tj,\\displaystyle:=\\beta\\frac\{1\}\{B\}\\sum\_\{b=1\}^\{B\}\\sum\_\{j=0\}^\{N\-1\}e^\{\\beta t\_\{j\}\}\\left\(\\frac\{Z^\{\(b\)\}\}\{\\epsilon\}\\tilde\{R\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\}\(\\mathbf\{U\}\_\{n\}\)\-\\varphi^\{\\theta\}\_\{g\}\(t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\)\\right\)^\{2\}\\Delta t\_\{j\},L~h\(B\)\\displaystyle\\tilde\{L\}^\{\(B\)\}\_\{h\}:=1B​∑b=1B∑j=0N−1eβ​tj​\(Z\(b\)​\(Z\(b\)\)⊤−Idϵ2​R~tj,Xtj\(b\)\+ϵ​Z\(b\)​\(𝐔n\)−φhθ​\(tj,Xtj\(b\)\)\)2​Δ​tj\.\\displaystyle:=\\frac\{1\}\{B\}\\sum\_\{b=1\}^\{B\}\\sum\_\{j=0\}^\{N\-1\}e^\{\\beta t\_\{j\}\}\\left\(\\frac\{Z^\{\(b\)\}\(Z^\{\(b\)\}\)^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\}\\tilde\{R\}^\{t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\+\\epsilon Z^\{\(b\)\}\}\(\\mathbf\{U\}\_\{n\}\)\-\\varphi^\{\\theta\}\_\{h\}\(t\_\{j\},X\_\{t\_\{j\}\}^\{\(b\)\}\)\\right\)^\{2\}\\Delta t\_\{j\}\.
15:Total Loss:

L~\(B\):=L~v\(B\)\+L~g\(B\)\+L~h\(B\)\\tilde\{L\}^\{\(B\)\}:=\\tilde\{L\}^\{\(B\)\}\_\{v\}\+\\tilde\{L\}^\{\(B\)\}\_\{g\}\+\\tilde\{L\}^\{\(B\)\}\_\{h\}
16:Update

θ←θ−η​Dθ​L~\(B\)\\theta\\leftarrow\\theta\-\\eta D\_\{\\theta\}\\tilde\{L\}^\{\(B\)\}\.

17:endfor

18:Set

θn\+1←θ\\theta\_\{n\+1\}\\leftarrow\\thetaand

𝐔n\+1←φθn\+1\\mathbf\{U\}\_\{n\+1\}\\leftarrow\\varphi^\{\\theta\_\{n\+1\}\}
19:endfor

20:Output:Trained networks

𝐔M=\(VM,GM,HM\)\\mathbf\{U\}\_\{M\}=\(V\_\{M\},G\_\{M\},H\_\{M\}\)

## 5Convergence Analysis

In this section, we establish the convergence properties of the proposed zeroth\-order based learning algorithm\. Letp​\(t,⋅\)p\(t,\\cdot\)denote the probability density function ofXtX\_\{t\}\. We define the following weighted norms:

‖v‖L2​\(p\)2:=∫ℝd\|v​\(t,x\)\|2​p​\(t,x\)​dx,‖v‖β2:=∫0Teβ​t​∫ℝd\|v​\(t,x\)\|2​p​\(t,x\)​dx​dt,\\\|v\\\|\_\{L\_\{2\}\(p\)\}^\{2\}:=\\int\_\{\\mathbb\{R\}^\{d\}\}\|v\(t,x\)\|^\{2\}p\(t,x\)\\mathrm\{d\}x,\\quad\\\|v\\\|\_\{\\beta\}^\{2\}:=\\int\_\{0\}^\{T\}e^\{\\beta t\}\\int\_\{\\mathbb\{R\}^\{d\}\}\|v\(t,x\)\|^\{2\}p\(t,x\)\\mathrm\{d\}x\\mathrm\{d\}t,\(5\.1\)as well as the mixed derivative norm:

‖v‖β,mix2:=β​‖v‖β2\+β​‖∇v‖β2\+‖∇2v‖β2\.\\\|v\\\|^\{2\}\_\{\\beta,\\text\{mix\}\}:=\\beta\\\|v\\\|\_\{\\beta\}^\{2\}\+\\beta\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\+\\\|\\nabla^\{2\}v\\\|\_\{\\beta\}^\{2\}\.\(5\.2\)For a function triplet𝐔:=\(V,G,H\)\\mathbf\{U\}:=\(V,G,H\), we abuse the notation slightly and define its corresponding norm as:

‖𝐔‖β,mix2:=β​‖V‖β2\+β​‖G‖β2\+‖H‖β2\.\\\|\\mathbf\{U\}\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}:=\\beta\\\|V\\\|\_\{\\beta\}^\{2\}\+\\beta\\\|G\\\|\_\{\\beta\}^\{2\}\+\\\|H\\\|\_\{\\beta\}^\{2\}\.\(5\.3\)We formally define the solution mapping𝒮\\mathcal\{S\}that maps a coefficient triplet to a solution triplet\. Let𝐔=\(V,G,H\)\\mathbf\{U\}=\(V,G,H\)be a given triplet in an appropriate Hilbert space\. Letuube the unique classical solution to the linear parabolic PDE:

\{∂tu\+ℒ​u\+f​\(x,V​\(t,x\),G​\(t,x\),H​\(t,x\)\)=0,\(t,x\)∈\[0,T\)×ℝd,u​\(T,x\)=g​\(x\),x∈ℝd\.\\begin\{cases\}\\partial\_\{t\}u\+\\mathcal\{L\}u\+f\\big\(x,V\(t,x\),G\(t,x\),H\(t,x\)\\big\)=0,&\(t,x\)\\in\[0,T\)\\times\\mathbb\{R\}^\{d\},\\\\ u\(T,x\)=g\(x\),&x\\in\\mathbb\{R\}^\{d\}\.\\end\{cases\}\(5\.4\)The solution mapping𝒮\\mathcal\{S\}is then defined as the operator that returns the solution and its spatial derivatives:

𝒮​\(𝐔\):=\(u,∇u,∇2u\)\.\\mathcal\{S\}\(\\mathbf\{U\}\):=\(u,\\nabla u,\\nabla^\{2\}u\)\.\(5\.5\)Under this definition, the fixed\-point problem𝐮∗=𝒮​\(𝐮∗\)\\mathbf\{u\}^\{\*\}=\\mathcal\{S\}\(\\mathbf\{u\}^\{\*\}\)is equivalent to the original nonlinear PDE \([2\.1](https://arxiv.org/html/2606.24999#S2.E1)\) withv∗v^\{\*\}being the solution of PDE and𝐮∗=\(v∗,∇v∗,∇2v∗\)\\mathbf\{u\}^\{\*\}=\(v^\{\*\},\\nabla v^\{\*\},\\nabla^\{2\}v^\{\*\}\)\.

To proceed with the analysis, we introduce the following contraction assumption on𝒮\\mathcal\{S\}\.

###### Assumption 2\(Contraction\)

There exist constantsβ\>0\\beta\>0andγ<1\\gamma<1, such that for any two triplets𝐔1=\(V1,G1,H1\)\\mathbf\{U\}\_\{1\}=\(V\_\{1\},G\_\{1\},H\_\{1\}\)and𝐔2=\(V2,G2,H2\)\\mathbf\{U\}\_\{2\}=\(V\_\{2\},G\_\{2\},H\_\{2\}\), the following inequality holds

‖𝒮​\(𝐔1\)−𝒮​\(𝐔2\)‖β,mix≤γ​‖𝐔1−𝐔2‖β,mix\.\\\|\\mathcal\{S\}\(\\mathbf\{U\}\_\{1\}\)\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{2\}\)\\\|\_\{\\beta,\\text\{mix\}\}\\leq\\gamma\\\|\\mathbf\{U\}\_\{1\}\-\\mathbf\{U\}\_\{2\}\\\|\_\{\\beta,\\text\{mix\}\}\.\(5\.6\)

In our context, Assumption[2](https://arxiv.org/html/2606.24999#Thmassumption2)represents the fundamental condition required for the well\-posedness and convergence of the underlying value iteration scheme itself, independent of the numerical approximation errors\. While establishing the necessary and sufficient conditions for general nonlinear PDEs to satisfy this contraction property is beyond the scope of this paper, we will provide a rigorous justification for its validity in an important and relevant setting\. Specifically, in Appendix[A](https://arxiv.org/html/2606.24999#A1)we will study the case of Langevin dynamics with a stationary distribution and present a sufficient condition for Assumption[2](https://arxiv.org/html/2606.24999#Thmassumption2)\.

### 5\.1Approximate Value Iteration

In this subsection, we analyze the propagation of learning errors through the iterative process and establish a global error bound for the proposed algorithm\.

Letℋ\\mathcal\{H\}denote the space of neural network triplets\. We consider the empirical risk minimization \(ERM\) problem \(cf\.Shalev\-Shwartz and Ben\-David[2014](https://arxiv.org/html/2606.24999#bib.bib14)\) at iterationnn\. Let𝐔n\\mathbf\{U\}\_\{n\}be the empirical minimizer:

𝐔n∈arg⁡minφ∈ℋ⁡L~\(B\)​\(φ,𝐔n−1\),\\mathbf\{U\}\_\{n\}\\in\\arg\\min\_\{\\varphi\\in\\mathcal\{H\}\}\\tilde\{L\}^\{\(B\)\}\(\\varphi,\\mathbf\{U\}\_\{n\-1\}\),\(5\.7\)whereL~\(B\)\\tilde\{L\}^\{\(B\)\}is the empirical loss defined in \([4\.9](https://arxiv.org/html/2606.24999#S4.E9)\)\. Under the ERM framework, omitting the optimization error, at each iterationn≥1n\\geq 1, the algorithm produces a neural network triplet𝐔n=\(Vn,Gn,Hn\)\\mathbf\{U\}\_\{n\}=\(V\_\{n\},G\_\{n\},H\_\{n\}\), which is the ERM minimizer, intended to approximate the theoretical target𝒮​\(𝐔n−1\)\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\.

We define theone\-step approximation errorat iterationnnas the deviation between the trained network and the exact image of the previous iterate under the solution map𝒮\\mathcal\{S\}:

En:=‖𝐔n−𝒮​\(𝐔n−1\)‖β,mix\.E\_\{n\}:=\\\|\\mathbf\{U\}\_\{n\}\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\\\|\_\{\\beta,\\text\{mix\}\}\.\(5\.8\)The quantityEnE\_\{n\}aggregates two sources of error: the statistical error arising from the Monte Carlo estimation of the reward and its derivatives \(via ZOD estimators\), and the approximation error associated with the neural network hypothesis space \(optimization and statistical error\) to be defined momentarily\.

The following theorem characterizes the convergence of the sequence\{𝐔n\}n≥0\\\{\\mathbf\{U\}\_\{n\}\\\}\_\{n\\geq 0\}to the ground truth𝐮∗\\mathbf\{u\}^\{\*\}\.

###### Theorem 6

Suppose that Assumption[2](https://arxiv.org/html/2606.24999#Thmassumption2)holds with a contraction constantγ∈\[0,1\)\\gamma\\in\[0,1\)\. Let\{𝐔n=\(Vn,Gn,Hn\)\}n≥0\\\{\\mathbf\{U\}\_\{n\}=\(V\_\{n\},G\_\{n\},H\_\{n\}\)\\\}\_\{n\\geq 0\}be the sequence of function triplets, which are ERM minimizers defined by \([5\.7](https://arxiv.org/html/2606.24999#S5.E7)\)\. Then, almost surely, for anyn≥1n\\geq 1, the total error is bounded by:

‖𝐔n−𝐮∗‖β,mix≤γn​‖𝐔0−𝐮∗‖β,mix\+∑k=1nγn−k​Ek\.\\\|\\mathbf\{U\}\_\{n\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}\\leq\\gamma^\{n\}\\\|\\mathbf\{U\}\_\{0\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}\+\\sum\_\{k=1\}^\{n\}\\gamma^\{n\-k\}E\_\{k\}\.\(5\.9\)

Theorem[6](https://arxiv.org/html/2606.24999#Thmtheorem6)presents the convergence result of the proposed method as anapproximate value iteration\. The error bound consists of two distinct components:

1. 1\.Thecontraction termγn​‖𝐔0−𝐮∗‖β,mix\\gamma^\{n\}\\\|\\mathbf\{U\}\_\{0\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}, which decays exponentially\. This term captures the inherent stability of the value iteration for the underlying PDE\.
2. 2\.Theaccumulation term∑k=1nγn−k​Ek\\sum\_\{k=1\}^\{n\}\\gamma^\{n\-k\}E\_\{k\}, which represents the accumulated error\. The magnitude ofEkE\_\{k\}is determined by the expressivity of the neural networks \(approximation error\) and the variance of the ZOD estimators \(statistical error\)\.

This result highlights a critical trade\-off: while the contraction constantγ\\gammais dictated and fixed by the PDE physics, the approximation accuracyEkE\_\{k\}can be controlled by increasing the network capacity and the batch sizeBB, and properly tuning the ZOD perturbationϵ\\epsilon\. In Section 5\.2, we will provide a detailed analysis ofEkE\_\{k\}, explicitly characterizing its dependence on the sample sizeBBand perturbation parameterϵ\\epsilon\.

### 5\.2One\-Step Sample Complexity Analysis

In this subsection, we provide an analysis of the one\-step approximation errorEnE\_\{n\}\. Recall from \([5\.8](https://arxiv.org/html/2606.24999#S5.E8)\) thatEn=‖𝐔n−𝒮​\(𝐔n−1\)‖β,mixE\_\{n\}=\\\|\\mathbf\{U\}\_\{n\}\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\\\|\_\{\\beta,\\text\{mix\}\}, where𝐔n\\mathbf\{U\}\_\{n\}is the trained network triplet \(the ERM minimizer\) and𝒮​\(𝐔n−1\)\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)is the ground truth target defined by the solution mapping\.

To characterize the learning target, we first introduce theZOD solution mapping𝒮ϵ\\mathcal\{S\}^\{\\epsilon\}, which maps the previous iterate to the expected values of the zeroth\-order estimators\. Namely,𝒮ϵ​\(𝐔n−1\):=\(Vϵ,Gϵ,Hϵ\)\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\):=\(V^\{\\epsilon\},G^\{\\epsilon\},H^\{\\epsilon\}\)is the target triplet defined by the conditional expectations:

Vϵ​\(t,x\)\\displaystyle V^\{\\epsilon\}\(t,x\):=𝔼​\[Rt,x​\(𝐔n−1\)\]≡V^n​\(t,x\),\\displaystyle:=\\mathbb\{E\}\\left\[R^\{t,x\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\]\\equiv\\hat\{V\}\_\{n\}\(t,x\),\(5\.10\)Gϵ​\(t,x\)\\displaystyle G^\{\\epsilon\}\(t,x\):=𝔼​\[Zϵ​Rt,x\+ϵ​Z​\(𝐔n−1\)\],\\displaystyle:=\\mathbb\{E\}\\left\[\\frac\{Z\}\{\\epsilon\}R^\{t,x\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\],\(5\.11\)Hϵ​\(t,x\)\\displaystyle H^\{\\epsilon\}\(t,x\):=𝔼​\[Z​Z⊤−Idϵ2​Rt,x\+ϵ​Z​\(𝐔n−1\)\]\.\\displaystyle:=\\mathbb\{E\}\\left\[\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\}R^\{t,x\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\]\.\(5\.12\)Note that𝒮ϵ​\(𝐔n−1\)\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)represents the “smoothed” solution induced by the Gaussian perturbation, which serves as the proxy for the true solution𝒮​\(𝐔n−1\)\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\.

Under the strong simulator in Condition[2](https://arxiv.org/html/2606.24999#Thmcond2), the derivative components of the empirical loss can alternatively be evaluated by the multi\-point ZOD estimatorsg^\\hat\{g\}andh^\\hat\{h\}in Proposition[5](https://arxiv.org/html/2606.24999#Thmtheorem5)\. In this multi\-point construction, the rewards entering each estimator are generated with the same Brownian path upon a single query\. By the symmetry ofZZand the linearity of expectation, these multi\-point ZOD estimators have the same conditional means asGϵG^\{\\epsilon\}andHϵH^\{\\epsilon\}\. Thus the population target remains𝒮ϵ​\(𝐔n−1\)\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\); in other words, the multi\-point construction changes the variance and the empirical\-process analysis, but not the target being learned\.

The following lemma establishes the relationship between the population loss function and this ZOD target\.

###### Lemma 7

For any tripletφ\\varphi, the population loss with respect to the previous iterate𝐔n−1\\mathbf\{U\}\_\{n\-1\}decomposes as:

L​\(φ,𝐔n−1\)=‖φ−𝒮ϵ​\(𝐔n−1\)‖β,mix2\+C,L\(\\varphi,\\mathbf\{U\}\_\{n\-1\}\)=\\\|\\varphi\-\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}\+C,\(5\.13\)whereCCis a constant independent ofφ\\varphi\.

As a direct consequence of Lemma[7](https://arxiv.org/html/2606.24999#Thmtheorem7), the triplet𝒮ϵ​\(𝐔n−1\)\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)is the unique minimizer of the population loss over the space of all measurable functions:

𝒮ϵ​\(𝐔n−1\)∈arg⁡minφ⁡L​\(φ,𝐔n−1\)\.\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\\in\\arg\\min\_\{\\varphi\}L\(\\varphi,\\mathbf\{U\}\_\{n\-1\}\)\.\(5\.14\)Hence, the learning process can be viewed as approximating the ZOD target𝒮ϵ​\(𝐔n−1\)\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)via empirical risk minimization\.

We now define the network architecture and hypothesis space used in this paper\. For a smooth activation functionρ:ℝ→ℝ\\rho:\\mathbb\{R\}\\to\\mathbb\{R\}, we use the same notationρ​\(z\)\\rho\(z\)for its element\-wise application to a vectorzz\.

###### Definition 8

Given input dimensiondi​nd\_\{in\}, output dimensiondo​u​td\_\{out\}, depthLL, and width vector𝐦=\(m0,m1,…,mL\)\\mathbf\{m\}=\(m\_\{0\},m\_\{1\},\\ldots,m\_\{L\}\)withm0=di​nm\_\{0\}=d\_\{in\}andmL=do​u​tm\_\{L\}=d\_\{out\}, we define the multilayer perceptron \(MLP\)Φ​\(⋅;ϑ\):ℝdi​n→ℝdo​u​t\\Phi\(\\cdot;\\vartheta\):\\mathbb\{R\}^\{d\_\{in\}\}\\to\\mathbb\{R\}^\{d\_\{out\}\}by

Φ​\(x;ϑ\):=𝒜L∘ρ∘𝒜L−1∘⋯∘ρ∘𝒜1​\(x\),\\Phi\(x;\\vartheta\):=\\mathcal\{A\}\_\{L\}\\circ\\rho\\circ\\mathcal\{A\}\_\{L\-1\}\\circ\\cdots\\circ\\rho\\circ\\mathcal\{A\}\_\{1\}\(x\),\(5\.15\)where each𝒜l:ℝml−1→ℝml\\mathcal\{A\}\_\{l\}:\\mathbb\{R\}^\{m\_\{l\-1\}\}\\to\\mathbb\{R\}^\{m\_\{l\}\}is the affine map𝒜l​\(z\)=Wl​z\+bl\\mathcal\{A\}\_\{l\}\(z\)=W\_\{l\}z\+b\_\{l\}\. The parameter vectorϑ\\varthetacollects all trainable parameters of this MLP, namelyϑ=\(Wl,bl\)l=1L\\vartheta=\{\(W\_\{l\},b\_\{l\}\)\}\_\{l=1\}^\{L\}\. The hypothesis space is defined by

ℋ:=\{φθ=\(Vθ,Gθ,Hθ\)∣θ∈Θ\},\\mathcal\{H\}:=\\left\\\{\\varphi\_\{\\theta\}=\(V\_\{\\theta\},G\_\{\\theta\},H\_\{\\theta\}\)\\mid\\theta\\in\\Theta\\right\\\},\(5\.16\)whereVθ:\[0,T\]×ℝd→ℝV\_\{\\theta\}:\[0,T\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\},Gθ:\[0,T\]×ℝd→ℝdG\_\{\\theta\}:\[0,T\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}, andHθ:\[0,T\]×ℝd→ℝd×dH\_\{\\theta\}:\[0,T\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\\times d\}are three possibly different MLPs\. The output ofHθH\_\{\\theta\}has dimensiond2d^\{2\}and is reshaped into ad×dd\\times dmatrix\.

Note that we use the same symbolθ\\thetafor all the three networks:θ\\thetadenotes the collection of their trainable parameters, and when appearing in one ofVθV\_\{\\theta\},GθG\_\{\\theta\}, orHθH\_\{\\theta\}it refers to the corresponding subcollection of parameters\.

###### Assumption 3

The activation function satisfiesρ∈Cb5​\(ℝ\)\\rho\\in C\_\{b\}^\{5\}\(\\mathbb\{R\}\)\. The parameter setΘ⊂ℝp\\Theta\\subset\\mathbb\{R\}^\{p\}is compact and satisfies\|θ\|∞≤R\|\\theta\|\_\{\\infty\}\\leq Rfor allθ∈Θ\\theta\\in\\Theta, whereppis the total number of trainable parameters in the three networks\.

Under the above construction and assumption, due to the smoothness ofρ\\rhoand the compactness ofΘ\\Theta, all functions inℋ\\mathcal\{H\}and their partial derivatives up to the fourth order with respect toxxare uniformly bounded\.

Next we introduce the regularity assumption on the expected reward function induced by any candidate in the hypothesis space\. This is crucial for establishing the one\-step sample complexity result\.

For anyφ∈ℋ\\varphi\\in\\mathcal\{H\}, denote

J1​\(t,x;φ\):=𝔼​\[Rt,x​\(φ\)\],J2​\(t,x;φ\):=𝔼​\[\(Rt,x​\(φ\)\)2\],f~​\(t,x;φ\):=f​\(t,x,φ​\(t,x\)\)\.J\_\{1\}\(t,x;\\varphi\):=\\mathbb\{E\}\\left\[R^\{t,x\}\(\\varphi\)\\right\],\\quad J\_\{2\}\(t,x;\\varphi\):=\\mathbb\{E\}\[\(R^\{t,x\}\(\\varphi\)\)^\{2\}\],\\quad\\tilde\{f\}\(t,x;\\varphi\):=f\(t,x,\\varphi\(t,x\)\)\.\(5\.17\)
###### Assumption 4

The functionsJ1​\(⋅,⋅;φ\)∈Cp1,4​\(\[0,T\]×ℝd\),J2​\(⋅,⋅;φ\)∈Cp1,2​\(\[0,T\]×ℝd\),f~∈Cp1,2​\(\[0,T\]×ℝd\)J\_\{1\}\(\\cdot,\\cdot;\\varphi\)\\in C\_\{p\}^\{1,4\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\),J\_\{2\}\(\\cdot,\\cdot;\\varphi\)\\in C\_\{p\}^\{1,2\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\),\\tilde\{f\}\\in C\_\{p\}^\{1,2\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\)uniformly over the hypothesis spaceφ∈ℋ\\varphi\\in\\mathcal\{H\}in the sense that their derivatives have polynomial growth with constants independent ofφ\\varphi\. The terminal functiongghas polynomial growth\.

Assumption[4](https://arxiv.org/html/2606.24999#Thmassumption4)is a regularity condition on the solutions to the linear PDE in the value iterate\. Typical sufficient conditions for it to hold include stronger regularity conditions on the SDE coefficients and/or uniform ellipticity\.

###### Theorem 9

Suppose Assumptions[1](https://arxiv.org/html/2606.24999#Thmassumption1),[3](https://arxiv.org/html/2606.24999#Thmassumption3), and[4](https://arxiv.org/html/2606.24999#Thmassumption4)hold\. Let0<ϵ<10<\\epsilon<1and0<δ<10<\\delta<1, and defineLB,δ:=log⁡B\+log⁡\(1/δ\)L\_\{B,\\delta\}:=\\log B\+\\log\(1/\\delta\)\. Let𝒫\\mathcal\{P\}be a polynomial determined by the architecture ofℋ\\mathcal\{H\},CCbe a constant depending only on the PDE coefficients and the constants in the above assumptions\. Letqqbe a constant depending only on the polynomial growth orders of the functions in Assumption[4](https://arxiv.org/html/2606.24999#Thmassumption4)\.

1. \(a\)With the weak simulator in Condition[1](https://arxiv.org/html/2606.24999#Thmcond1), if the derivative terms in the empirical loss are evaluated using the one\-point ZOD estimators, then, with probability at least1−δ1\-\\delta, En2≤\\displaystyle E\_\{n\}^\{2\}\\leqC​\(1N​ϵ4⏟Discretization Error\+ϵ4⏟ZOD Bias\+\(1\+log⁡\(1/δ\)\)​𝒫​\(d\)​log⁡\(1/ϵ\)​LB,δqB1/2​ϵ4⏟Statistical Error\)\\displaystyle C\\left\(\\underbrace\{\\frac\{1\}\{N\\epsilon^\{4\}\}\}\_\{\\text\{Discretization Error\}\}\+\\underbrace\{\\epsilon^\{4\}\}\_\{\\text\{ZOD Bias\}\}\+\\underbrace\{\\left\(1\+\\sqrt\{\\log\(1/\\delta\)\}\\right\)\\frac\{\\mathcal\{P\}\(d\)\\sqrt\{\\log\(1/\\epsilon\)\}L\_\{B,\\delta\}^\{q\}\}\{B^\{1/2\}\\epsilon^\{4\}\}\}\_\{\\text\{Statistical Error\}\}\\right\)\(5\.18\)\+2​infφ∈ℋ‖φ−𝒮ϵ​\(𝐔n−1\)‖β,mix2⏟Approximation Error\.\\displaystyle\+\\underbrace\{2\\inf\_\{\\varphi\\in\\mathcal\{H\}\}\\left\\\|\\varphi\-\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}\}\_\{\\text\{Approximation Error\}\}\.
2. \(b\)Suppose further thatb,σ∈Cb0,5​\(\[0,T\]×ℝd\)b,\\sigma\\in C^\{0,5\}\_\{b\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\),g∈Cp4​\(ℝd\)g\\in C^\{4\}\_\{p\}\(\\mathbb\{R\}^\{d\}\)andf​\(t,x,φ​\(t,x\)\)∈Cp1,5​\(\[0,T\]×ℝd\)f\(t,x,\\varphi\(t,x\)\)\\in C^\{1,5\}\_\{p\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\)uniformly over the hypothesis spaceφ∈ℋ\\varphi\\in\\mathcal\{H\}in the sense that their derivatives have polynomial growth with constants independent ofφ\\varphi\. With the strong simulator in Condition[2](https://arxiv.org/html/2606.24999#Thmcond2), if the derivative terms in the empirical loss are evaluated using the multi\-point estimators stipulated in Proposition[5](https://arxiv.org/html/2606.24999#Thmtheorem5), then, with probability at least1−δ1\-\\delta, En2≤\\displaystyle E\_\{n\}^\{2\}\\leqC​\(1N⏟Discretization Error\+ϵ4⏟ZOD Bias\+\(1\+log⁡\(1/δ\)\)​𝒫​\(d\)​Γϵ,B,δB1/2⏟Statistical Error\)\\displaystyle C\\left\(\\underbrace\{\\frac\{1\}\{N\}\}\_\{\\text\{Discretization Error\}\}\+\\underbrace\{\\epsilon^\{4\}\}\_\{\\text\{ZOD Bias\}\}\+\\underbrace\{\\left\(1\+\\sqrt\{\\log\(1/\\delta\)\}\\right\)\\frac\{\\mathcal\{P\}\(d\)\\Gamma\_\{\\epsilon,B,\\delta\}\}\{B^\{1/2\}\}\}\_\{\\text\{Statistical Error\}\}\\right\)\(5\.19\)\+2​infφ∈ℋ‖φ−𝒮ϵ​\(𝐔n−1\)‖β,mix2⏟Approximation Error,\\displaystyle\+\\underbrace\{2\\inf\_\{\\varphi\\in\\mathcal\{H\}\}\\left\\\|\\varphi\-\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}\}\_\{\\text\{Approximation Error\}\},whereΓϵ,B,δ:=min⁡\{ϵ−4​log⁡\(1/ϵ\)​LB,δq,exp⁡\(C​LB,δ\)\}\\Gamma\_\{\\epsilon,B,\\delta\}:=\\min\\left\\\{\\epsilon^\{\-4\}\\sqrt\{\\log\(1/\\epsilon\)\}L\_\{B,\\delta\}^\{q\},\\exp\\\!\\left\(C\\sqrt\{L\_\{B,\\delta\}\}\\right\)\\right\\\}\.

Theorem[9](https://arxiv.org/html/2606.24999#Thmtheorem9)provides high\-probability upper bounds on the one\-step error, under both the weak and strong simulators, in terms of four components: the discretization error from approximating the time integral, the ZOD bias from using ZOD\-based derivative representations instead of exact stochastic representations, the statistical error from the empirical loss due to a finite sample size, and the approximation error of the neural\-network family\. Several points are worth noting\. First, the discretization error is controlled by the number of time\-grid points used to compute the integrals\. Since we use the simplest forward Euler scheme, this term is of orderN−1N^\{\-1\}, which can be improved by adopting a higher\-order numerical integration scheme\. Importantly, increasing the number of time\-grid points does not introduce instability into the algorithm or the loss function\. Second, the ZOD bias is controlled by the bandwidth and is of orderϵ4\\epsilon^\{4\}in both simulator settings\. This part of the error follows from Propositions[4](https://arxiv.org/html/2606.24999#Thmtheorem4)and[5](https://arxiv.org/html/2606.24999#Thmtheorem5)\. Third, the statistical error is jointly controlled by the sample size and the complexity of the neural\-network class\. Our statistical bound is conservative: the squared one\-step error contains a sample\-size term of orderB−1/2B^\{\-1/2\}and depends polynomially on the dimension of the state variable; this rate arises from our use of a global Rademacher\-complexity bound for the empirical\-loss term\.101010Sharper localized critical radius analyses are possible in principle, but they require additional technical assumptions and substantially more involved arguments, which are not the focus of this paper; see, e\.g\.,Bartlettet al\.\([2005](https://arxiv.org/html/2606.24999#bib.bib71)\); Wainwright \([2019](https://arxiv.org/html/2606.24999#bib.bib72)\)\.Finally, the approximation error is in terms of the weighted Sobolev norm of the neural network classes, which reflects the quality of the neural network classes chosen\.

Moreover, under the weak simulator, the ZOD estimator has exploding variances\. As a consequence, both the discretization error and the statistical error contain the factorϵ−4\\epsilon^\{\-4\}, which substantially worsens the bound and creates a bias–variance tradeoff against the ZOD bias\. Ignoring logarithmic factors, the dominant terms are of the order\(N−1\+𝒫​\(d\)​B−1/2\)​ϵ−4\+ϵ4\(N^\{\-1\}\+\\mathcal\{P\}\(d\)B^\{\-1/2\}\)\\epsilon^\{\-4\}\+\\epsilon^\{4\}\. This suggests choosingϵ=O​\(\(N−1\+𝒫​\(d\)​B−1/2\)1/8\)\\epsilon=O\(\(N^\{\-1\}\+\\mathcal\{P\}\(d\)B^\{\-1/2\}\)^\{1/8\}\)to yield the squared one\-step errorEn2=O​\(\(N−1\+𝒫​\(d\)​B−1/2\)1/2\)E\_\{n\}^\{2\}=O\(\(N^\{\-1\}\+\\mathcal\{P\}\(d\)B^\{\-1/2\}\)^\{1/2\}\), up to logarithmic factors and the approximation error\.

By contrast, under the strong simulator, the variance of the multi\-point ZOD estimator is bounded uniformly inϵ\\epsilon\. Hence neither the discretization error nor the statistical error suffers from theϵ−4\\epsilon^\{\-4\}blow\-up\. We note, however, that to cancel thisϵ−4\\epsilon^\{\-4\}term, one relies on the tangent process\. Since the global Rademacher complexity argument requires a bounded loss class, a truncation argument is needed\. The tangent process typically has a heavier tail than the original state process, which leads to the factorexp⁡\(C​log⁡B\+log⁡\(1/δ\)\)\\exp\(C\\sqrt\{\\log B\+\\log\(1/\\delta\)\}\)in the statistical error bound\. Although this factor is worse than the logarithmic factor appearing in the weak one\-point estimate, it is still sub\-polynomial in bothBBand1/δ1/\\delta, becauseexp⁡\(C​LB,δ\)=\(B/δ\)o​\(1\)\\exp\(C\\sqrt\{L\_\{B,\\delta\}\}\)=\(B/\\delta\)^\{o\(1\)\}\. Consequently, in the asymptotic sense, choosingϵ=O​\(\(N−1\+𝒫​\(d\)​B−1/2\)1/4\)\\epsilon=O\(\(N^\{\-1\}\+\\mathcal\{P\}\(d\)B^\{\-1/2\}\)^\{1/4\}\)balances the ZOD bias with the discretization and statistical errors, and gives the squared one\-step errorEn2=O​\(N−1\+𝒫​\(d\)​B−1/2\)E\_\{n\}^\{2\}=O\(N^\{\-1\}\+\\mathcal\{P\}\(d\)B^\{\-1/2\}\), up to the approximation error\. To sum, compared with the weak simulator, the strong simulator gives a substantially better asymptotic order by removing the unfavorableϵ−4\\epsilon^\{\-4\}dependence\.

Combining our analysis for the approximate value iteration and one\-step sample complexity results, Theorem[10](https://arxiv.org/html/2606.24999#Thmtheorem10)presents the total error of the algorithm up to thenn\-th value iterate with any fixed bandwidthϵ\\epsilon, the number of time discretization gridsNN, and the sample sizeBBfor training\.

###### Theorem 10

Suppose Assumptions[1](https://arxiv.org/html/2606.24999#Thmassumption1)–[4](https://arxiv.org/html/2606.24999#Thmassumption4)hold\. Let0<ϵ<10<\\epsilon<1and0<δ<10<\\delta<1, and writeLB,δ,n:=log⁡B\+log⁡\(n/δ\)L\_\{B,\\delta,n\}:=\\log B\+\\log\(n/\\delta\)\. Let𝒫\\mathcal\{P\}be a polynomial determined by the architecture ofℋ\\mathcal\{H\},CCbe a constant depending only on the PDE coefficients and the constants in the assumptions\. Letqqbe a constant depending only on the polynomial growth orders of the functions in Assumption[4](https://arxiv.org/html/2606.24999#Thmassumption4)\.

1. \(a\)Under the conditions of Theorem[9](https://arxiv.org/html/2606.24999#Thmtheorem9)\-\(a\), with probability at least1−δ1\-\\delta, we have ‖𝐔n−𝐮∗‖β,mix\\displaystyle\\\|\\mathbf\{U\}\_\{n\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}≤γn​‖𝐔0−𝐮∗‖β,mix\+2​∑k=1nγn−k​infφ∈ℋ‖φ−𝒮ϵ​\(𝐔k−1\)‖β,mix\\displaystyle\\leq\\gamma^\{n\}\\\|\\mathbf\{U\}\_\{0\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}\+\\sqrt\{2\}\\sum\_\{k=1\}^\{n\}\\gamma^\{n\-k\}\\inf\_\{\\varphi\\in\\mathcal\{H\}\}\\\|\\varphi\-\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{k\-1\}\)\\\|\_\{\\beta,\\text\{mix\}\}\(5\.20\)\+C​1−γn1−γ\\displaystyle\+C\\frac\{1\-\\gamma^\{n\}\}\{1\-\\gamma\}\(N−1/2​ϵ−2\+ϵ2\+\(1\+\(log⁡\(n/δ\)\)1/4\)​𝒫​\(d\)​\(log⁡\(1/ϵ\)\)1/4​LB,δ,nq/2B1/4​ϵ2\)\.\\displaystyle\\left\(N^\{\-1/2\}\\epsilon^\{\-2\}\+\\epsilon^\{2\}\+\\left\(1\+\(\\log\(n/\\delta\)\)^\{1/4\}\\right\)\\frac\{\\sqrt\{\\mathcal\{P\}\(d\)\}\(\\log\(1/\\epsilon\)\)^\{1/4\}L\_\{B,\\delta,n\}^\{q/2\}\}\{B^\{1/4\}\\epsilon^\{2\}\}\\right\)\.
2. \(b\)Under the conditions of Theorem[9](https://arxiv.org/html/2606.24999#Thmtheorem9)\-\(b\), with probability at least1−δ1\-\\delta, we have ‖𝐔n−𝐮∗‖β,mix≤\\displaystyle\\\|\\mathbf\{U\}\_\{n\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}\\leqγn​‖𝐔0−𝐮∗‖β,mix\+2​∑k=1nγn−k​infφ∈ℋ‖φ−𝒮ϵ​\(𝐔k−1\)‖β,mix\\displaystyle\\gamma^\{n\}\\\|\\mathbf\{U\}\_\{0\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}\+\\sqrt\{2\}\\sum\_\{k=1\}^\{n\}\\gamma^\{n\-k\}\\inf\_\{\\varphi\\in\\mathcal\{H\}\}\\\|\\varphi\-\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{k\-1\}\)\\\|\_\{\\beta,\\text\{mix\}\}\(5\.21\)\+C​1−γn1−γ​\(N−1/2\+ϵ2\+\(1\+\(log⁡\(n/δ\)\)1/4\)​𝒫​\(d\)​Γϵ,B,δ,nB1/4\),\\displaystyle\+C\\frac\{1\-\\gamma^\{n\}\}\{1\-\\gamma\}\\left\(N^\{\-1/2\}\+\\epsilon^\{2\}\+\\left\(1\+\(\\log\(n/\\delta\)\)^\{1/4\}\\right\)\\frac\{\\sqrt\{\\mathcal\{P\}\(d\)\\Gamma\_\{\\epsilon,B,\\delta,n\}\}\}\{B^\{1/4\}\}\\right\),whereΓϵ,B,δ,n:=min\{ϵ−4log⁡\(1/ϵ\)LB,δ,nq,exp\(CLB,δ,n\)\}\\Gamma\_\{\\epsilon,B,\\delta,n\}:=\\min\\left\\\{\\epsilon^\{\-4\}\\sqrt\{\\log\(1/\\epsilon\)\}L\_\{B,\\delta,n\}^\{q\},\\exp\\\!\\left\(C\\sqrt\{L\_\{B,\\delta,n\}\}\\right\)\\right\\\}\.

## 6Numerical Experiments

In this section, we investigate the numerical performance of the proposed zeroth\-order solver in learning the full solution triplet\(v⋆,∇v⋆,∇2v⋆\)\(v^\{\\star\},\\nabla v^\{\\star\},\\nabla^\{2\}v^\{\\star\}\)\.

Let𝐔n=\(Vn,Gn,Hn\)\\mathbf\{U\}\_\{n\}=\(V\_\{n\},G\_\{n\},H\_\{n\}\)denote the learned value, gradient, and Hessian networks respectively after thenn\-th value iteration\. Throughout the numerical experiments, we report empirical relative root mean squared errors \(rRMSE\) between the learned networks and the ground truth solution by evaluating the errors on a set of randomly generated test samples\. Specifically, given test sampleszi=\(ti,xi\)z\_\{i\}=\(t\_\{i\},x\_\{i\}\)and a target quantityQ⋆Q^\{\\star\}evaluated at these samples, define

rRMSE⁡\(Q^,Q⋆\)=\(∑i=1Ntest\|Q^​\(zi\)−Q⋆​\(zi\)\|2∑i=1Ntest\|Q⋆​\(zi\)\|2\)1/2,\\operatorname\{rRMSE\}\(\\widehat\{Q\},Q^\{\\star\}\)=\\left\(\\frac\{\\sum\_\{i=1\}^\{N\_\{\\mathrm\{test\}\}\}\\left\|\\widehat\{Q\}\(z\_\{i\}\)\-Q^\{\\star\}\(z\_\{i\}\)\\right\|^\{2\}\}\{\\sum\_\{i=1\}^\{N\_\{\\mathrm\{test\}\}\}\\left\|Q^\{\\star\}\(z\_\{i\}\)\\right\|^\{2\}\}\\right\)^\{1/2\},\(6\.1\)with the absolute value for scalar quantities, the Euclidean norm for vectors, and the Frobenius norm for matrices\.111111In our theoretical analysis, we rely on the weighted Sobolev norm that depends on a constantβ\\beta\. This constant exists in theory but is difficult to determine in experiments\. Therefore, in the numerical study, we only calculate the unweighted Sobolev norm in training and present the results for function value, gradient, and Hessian in terms of the relative mean squared errors respectively\.For the solution triplet,Q^\\widehat\{Q\}is taken directly fromVnV\_\{n\},GnG\_\{n\}, orHnH\_\{n\}and compared withv⋆v^\{\\star\},∇v⋆\\nabla v^\{\\star\}, or∇2v⋆\\nabla^\{2\}v^\{\\star\}, respectively\. Note once again that our approach learns the three networks separately, and the derivatives are not obtained by auto\-differentiating the learned value networkVnV\_\{n\}\.

For the PDE\-solving experiments reported below, we use a short initialization stage as a pre\-training stage\. With the initial scalar networkV0V\_\{0\}fixed, the derivative networksG0G\_\{0\}andH0H\_\{0\}are pretrained to fit∇xV0\\nabla\_\{x\}V\_\{0\}and∇x2V0\\nabla\_\{x\}^\{2\}V\_\{0\}, respectively, where these derivatives are computed by automatic differentiation ofV0V\_\{0\}on samples from the corresponding training law\. This pretraining is used only to initialize the derivative networks\. In the subsequent value iterations, the networks are warm\-started: the parameters ofVn\+1,Gn\+1,Hn\+1V\_\{n\+1\},G\_\{n\+1\},H\_\{n\+1\}are initialized from the trained parameters ofVn,Gn,HnV\_\{n\},G\_\{n\},H\_\{n\}rather than from a fresh random initialization\. All numerical experiments were run on a single NVIDIA RTX 3090 GPU with 24 GB memory\. Code is available at[https://github\.com/du\-ouyang/ZO4PDE](https://github.com/du-ouyang/ZO4PDE)\.

In the following subsections, we first present a simple toy example of a linear PDE in Subsection[6\.1](https://arxiv.org/html/2606.24999#S6.SS1), which is effectively a regression problem explained in Subsection[3\.2](https://arxiv.org/html/2606.24999#S3.SS2), to contrast our approach to the existing ones\. In Subsections[6\.2](https://arxiv.org/html/2606.24999#S6.SS2)and[6\.3](https://arxiv.org/html/2606.24999#S6.SS3), we compare our method to the deep Picard iteration \(DPI\) inHanet al\.\([2026](https://arxiv.org/html/2606.24999#bib.bib30)\)on benchmark nonlinear PDEs where the accuracy can be measured against exact solutions\. For a fairer comparison in these DPI benchmarks, we use the multi\-point ZOD estimators \(ZOD\-m\) for gradients and Hessians \(see Proposition[5](https://arxiv.org/html/2606.24999#Thmtheorem5)\)\. This is because the DPI method is model\-based, which assumes knowing the operatorℒ\\mathcal\{L\}, a much stronger assumption including the availability of a strong simulator as a special case\. Moreover, the SDEs in these benchmarks can be exactly simulated without numerical error\. Hence, the time integrals appearing both in the training loss and in the reward random variableRRcan be estimated by samplingttuniformly on\[0,T\]\[0,T\]and then samplingXtX\_\{t\}exactly from its marginal law\.121212More precisely, the time integral in the loss is evaluated by first samplingt∼Unif​\[0,T\]t\\sim\{\\rm Unif\}\[0,T\]and then samplingXtX\_\{t\}exactly from its marginal law, using the identity∫0Teβ​t​𝔼​\[ℓ​\(t,Xt\)\]​dt=T​𝔼​\[eβ​t​ℓ​\(t,Xt\)\]\\int\_\{0\}^\{T\}e^\{\\beta t\}\\mathbb\{E\}\[\\ell\(t,X\_\{t\}\)\]\\,\\mathrm\{d\}t=T\\mathbb\{E\}\[e^\{\\beta t\}\\ell\(t,X\_\{t\}\)\]\. Similarly, for the rewardRt,x=g​\(XTt,x\)\+∫tTf​\(s,Xst,x,𝐔​\(s,Xst,x\)\)​dsR^\{t,x\}=g\(X\_\{T\}^\{t,x\}\)\+\\int\_\{t\}^\{T\}f\(s,X\_\{s\}^\{t,x\},\\mathbf\{U\}\(s,X\_\{s\}^\{t,x\}\)\)\\,\\mathrm\{d\}s, the integral term is estimated by first samplings∼Unif​\[t,T\]s\\sim\{\\rm Unif\}\[t,T\]and then samplingXst,xX\_\{s\}^\{t,x\}exactly from the transition law, so that∫tT𝔼​\[f​\(s,Xst,x,𝐔​\(s,Xst,x\)\)\]​ds=\(T−t\)​𝔼​\[f​\(s,Xst,x,𝐔​\(s,Xst,x\)\)\]\\int\_\{t\}^\{T\}\\mathbb\{E\}\[f\(s,X\_\{s\}^\{t,x\},\\mathbf\{U\}\(s,X\_\{s\}^\{t,x\}\)\)\]\\,\\mathrm\{d\}s=\(T\-t\)\\mathbb\{E\}\[f\(s,X\_\{s\}^\{t,x\},\\mathbf\{U\}\(s,X\_\{s\}^\{t,x\}\)\)\]\.In this way, the reported errors isolate the learning and ZOD\-estimation errors, without having to consider the time discretization errors\.

### 6\.1A Linear PDE: Regression

We consider the following \(one\-dimensional\) linear PDE:

∂tv​\(t,x\)\+12​σ2​∂x​xv​\(t,x\)=0,\(t,x\)∈\[0,T\)×ℝ,\\partial\_\{t\}v\(t,x\)\+\\frac\{1\}\{2\}\\sigma^\{2\}\\partial\_\{xx\}v\(t,x\)=0,\\ \(t,x\)\\in\[0,T\)\\times\\mathbb\{R\},\(6\.2\)with terminal conditionv​\(T,x\)=g​\(x\)v\(T,x\)=g\(x\)\.

We are interested only in the time\-0 value, gradient, and Hessian, namelyu​\(x\)=v​\(0,x\)u\(x\)=v\(0,x\),u′​\(x\)=∂xv​\(0,x\)u^\{\\prime\}\(x\)=\\partial\_\{x\}v\(0,x\), andu′′​\(x\)=∂x​xv​\(0,x\)u^\{\\prime\\prime\}\(x\)=\\partial\_\{xx\}v\(0,x\)\. By the Feynman–Kac formula, this is equivalent to computing the conditional expectation

u​\(x\)=𝔼​\[g​\(XT\)∣X0=x\],u\(x\)=\\mathbb\{E\}\\left\[g\\left\(X\_\{T\}\\right\)\\mid X\_\{0\}=x\\right\],whered​Xt=σ​d​Wt\\mathrm\{d\}X\_\{t\}=\\sigma\\mathrm\{d\}W\_\{t\}\. It is also equivalent to a regression task introduced in Subsection[3\.2](https://arxiv.org/html/2606.24999#S3.SS2), withY=g​\(X\+σ​ξ\)Y=g\(X\+\\sigma\\xi\)\.

Set the terminal functionggas

g​\(y\)=A1​sin⁡\(α​y\)\+A2​cos⁡\(β​y\)\+∑j=1Jaj​exp⁡\(−\(y−cj\)22​ℓj2\),g\(y\)=A\_\{1\}\\sin\(\\alpha y\)\+A\_\{2\}\\cos\(\\beta y\)\+\\sum\_\{j=1\}^\{J\}a\_\{j\}\\exp\\left\(\-\\frac\{\(y\-c\_\{j\}\)^\{2\}\}\{2\\ell\_\{j\}^\{2\}\}\\right\),\(6\.3\)and the ground truth solution is available by Gaussian convolution:

u∗​\(x\)\\displaystyle u^\{\*\}\(x\)=A1​e−α2​σ2/2​sin⁡\(α​x\)\+A2​e−β2​σ2/2​cos⁡\(β​x\)\\displaystyle=A\_\{1\}e^\{\-\\alpha^\{2\}\\sigma^\{2\}/2\}\\sin\(\\alpha x\)\+A\_\{2\}e^\{\-\\beta^\{2\}\\sigma^\{2\}/2\}\\cos\(\\beta x\)\(6\.4\)\+∑j=1Jaj​ℓjsj​exp⁡\(−\(x−cj\)22​sj2\),sj2=ℓj2\+σ2\.\\displaystyle\\quad\+\\sum\_\{j=1\}^\{J\}a\_\{j\}\\frac\{\\ell\_\{j\}\}\{s\_\{j\}\}\\exp\\left\(\-\\frac\{\(x\-c\_\{j\}\)^\{2\}\}\{2s\_\{j\}^\{2\}\}\\right\),\\qquad s\_\{j\}^\{2\}=\\ell\_\{j\}^\{2\}\+\\sigma^\{2\}\.The ground truth derivatives can then be computed analytically from the above expression, denoted byu∗′,u∗′′\{u^\{\*\}\}^\{\\prime\},\{u^\{\*\}\}^\{\\prime\\prime\}\.

For the learning\-then\-differentiating paradigm, we first train a scalar networkVϑV\_\{\\vartheta\}with the least\-squares loss function:

ℒNN​\(ϑ\)=1B​∑i=1B\|Vϑ​\(xi\)−g​\(xi\+σ​ξi\)\|2,\\mathcal\{L\}\_\{\\mathrm\{NN\}\}\(\\vartheta\)=\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}\\left\|V\_\{\\vartheta\}\(x\_\{i\}\)\-g\(x\_\{i\}\+\\sigma\\xi\_\{i\}\)\\right\|^\{2\},\(6\.5\)wherexi∼Unif​\[−2,2\]x\_\{i\}\\sim\\mathrm\{Unif\}\[\-2,2\]andξi∼𝒩​\(0,T\)\\xi\_\{i\}\\sim\\mathcal\{N\}\(0,T\)\. The derivative estimates are then obtained by automatic differentiation:

Δ^NN​\(x\)=∂xVϑ​\(x\),Γ^NN​\(x\)=∂x​xVϑ​\(x\)\.\\widehat\{\\Delta\}\_\{\\mathrm\{NN\}\}\(x\)=\\partial\_\{x\}V\_\{\\vartheta\}\(x\),\\qquad\\widehat\{\\Gamma\}\_\{\\mathrm\{NN\}\}\(x\)=\\partial\_\{xx\}V\_\{\\vartheta\}\(x\)\.
For our ZOD approach, we train three networks\(Vθv,Gθg,Hθh\)\(V\_\{\\theta\_\{v\}\},G\_\{\\theta\_\{g\}\},H\_\{\\theta\_\{h\}\}\), parameterized collectively byθ\\theta\. At each step, we sample\{xi\}i=1B∼Unif​\[−2,2\]\\\{x\_\{i\}\\\}\_\{i=1\}^\{B\}\\sim\\mathrm\{Unif\}\[\-2,2\],\{zi\}i=1B∼𝒩​\(0,1\)\\\{z\_\{i\}\\\}\_\{i=1\}^\{B\}\\sim\\mathcal\{N\}\(0,1\), and\{ξi\}i=1B∼𝒩​\(0,T\)\\\{\\xi\_\{i\}\\\}\_\{i=1\}^\{B\}\\sim\\mathcal\{N\}\(0,T\)\. With the multi\-point ZOD estimator, define

V^i\\displaystyle\\widehat\{V\}\_\{i\}=g​\(xi\+σ​ξi\),\\displaystyle=g\(x\_\{i\}\+\\sigma\\xi\_\{i\}\),\(6\.6\)G^i\\displaystyle\\widehat\{G\}\_\{i\}=zi2​ϵ​\[g​\(xi\+ϵ​zi\+σ​ξi\)−g​\(xi−ϵ​zi\+σ​ξi\)\],\\displaystyle=\\frac\{z\_\{i\}\}\{2\\epsilon\}\\left\[g\(x\_\{i\}\+\\epsilon z\_\{i\}\+\\sigma\\xi\_\{i\}\)\-g\(x\_\{i\}\-\\epsilon z\_\{i\}\+\\sigma\\xi\_\{i\}\)\\right\],\(6\.7\)H^i\\displaystyle\\widehat\{H\}\_\{i\}=zi2−12​ϵ2​\[g​\(xi\+ϵ​zi\+σ​ξi\)\+g​\(xi−ϵ​zi\+σ​ξi\)−2​g​\(xi\+σ​ξi\)\]\.\\displaystyle=\\frac\{z\_\{i\}^\{2\}\-1\}\{2\\epsilon^\{2\}\}\\left\[g\(x\_\{i\}\+\\epsilon z\_\{i\}\+\\sigma\\xi\_\{i\}\)\+g\(x\_\{i\}\-\\epsilon z\_\{i\}\+\\sigma\\xi\_\{i\}\)\-2g\(x\_\{i\}\+\\sigma\\xi\_\{i\}\)\\right\]\.\(6\.8\)The three networks are trained jointly by

ℒZOD\(θ\)=1B∑i=1B\(\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{ZOD\}\}\(\\theta\)=\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}\\Big\(\|Vθv\(xi\)−V^i\|2\+\|Gθg\(xi\)−G^i\|2\+\|Hθh\(xi\)−H^i\|2\)\.\\displaystyle\|V\_\{\\theta\_\{v\}\}\(x\_\{i\}\)\-\\widehat\{V\}\_\{i\}\|^\{2\}\+\|G\_\{\\theta\_\{g\}\}\(x\_\{i\}\)\-\\widehat\{G\}\_\{i\}\|^\{2\}\+\|H\_\{\\theta\_\{h\}\}\(x\_\{i\}\)\-\\widehat\{H\}\_\{i\}\|^\{2\}\\Big\)\.\(6\.9\)Note that the same Brownian noiseξi\\xi\_\{i\}is reused in definition of the multi\-point ZOD estimatorG^i\\widehat\{G\}\_\{i\}andH^i\\widehat\{H\}\_\{i\}, which is available only with the strong simulator\.

In the experiment, we use

σ=0\.02,T=1,A1=0\.22,A2=0\.06,α=1\.3,β=4\.7,J=7,\(cj\)=\(−1\.55,−1\.05,−0\.62,−0\.18,0\.24,0\.71,1\.28\),\(ℓj\)=\(0\.065,0\.055,0\.070,0\.060,0\.055,0\.065,0\.060\),\(aj\)=\(0\.035,−0\.030,0\.032,0\.028,−0\.034,0\.030,−0\.027\)\.\\begin\{gathered\}\\sigma=0\.02,\\quad T=1,\\quad A\_\{1\}=0\.22,\\quad A\_\{2\}=0\.06,\\quad\\alpha=1\.3,\\quad\\beta=4\.7,\\quad J=7,\\\\ \(c\_\{j\}\)=\(\-1\.55,\-1\.05,\-0\.62,\-0\.18,0\.24,0\.71,1\.28\),\\\\ \(\\ell\_\{j\}\)=\(0\.065,0\.055,0\.070,0\.060,0\.055,0\.065,0\.060\),\\\\ \(a\_\{j\}\)=\(0\.035,\-0\.030,0\.032,0\.028,\-0\.034,0\.030,\-0\.027\)\.\\end\{gathered\}Both auto\-differentiation and ZOD methods use MLPs with four hidden layers of width256256andtanh\\tanhactivation\. The rRMSEs are evaluated on10001000uniformly generated initial spatial points in\[−2,2\]\[\-2,2\]\. NN\-autodiff is trained with 10000 gradient steps and batch sizeB=32768B=32768, learning rate3×10−43\\times 10^\{\-4\}\. Our ZOD estimator is trained with 5000 gradient steps, batch sizeB=16384B=16384, learning rate5×10−45\\times 10^\{\-4\}, and ZOD bandwidthϵ=0\.01\\epsilon=0\.01\.

Table 1:rRMSEs for the linear PDE \([6\.2](https://arxiv.org/html/2606.24999#S6.E2)\)\.Table[1](https://arxiv.org/html/2606.24999#S6.T1)summarizes the rRMSEs of both approaches under comparable running times\. Observe that although the value errors are of the same order, our ZOD approach yields substantially smaller derivative errors in both the first and second order and outperforms by at least 3 times\. Judging from the magnitude of these errors, one might conclude that NN\-autodiff, while lagging behind ZOD, could still achieve reasonable accuracy in the derivatives\. However, rRMSE is anaveragedmeasure \(over the initial states\) that may not fully capture the point\-by\-point errors of a learned function from its oracle\. To visualize the latter, we draw diagnostic plots in Figure[1](https://arxiv.org/html/2606.24999#S6.F1), where we put the ground truth solution and the learned functions by the two approaches together as functions ofxx\. In terms of the learned value function, both methods produce very close fit to the true one, whereas the gap becomes significant for the gradient and Hessian\. In particular, the ZOD\-based derivatives follow the oracle solutions closely all the time, capturing almost every peak and trough as well as the shapes of the functions, but the ones by NN\-autodiff smoothly vary over the whole interval and miss most of the peaks and troughs\. Thereby, despite the small average errors in the gradient and Hessian, the learned derivative functions are not close at all with NN\-autodiff\. This example highlights the limitations of auto\-differentiating a learned value function directly when accurate derivative approximation is required, and justifies our method that aims for theoretical guarantee in not only function values, but also in derivatives values\.

![Refer to caption](https://arxiv.org/html/2606.24999v1/x1.png)Figure 1:The learned value function, gradient, and Hessian for example \([6\.2](https://arxiv.org/html/2606.24999#S6.E2)\)\. NN\-autodiff learns the value function by the least\-squares method and obtains derivatives by automatic differentiation\. ZOD\-m trains value, gradient, and Hessian networks directly using multi\-point ZOD estimators\.
### 6\.2A Semilinear PDE

We next consider a semilinear PDE studied inHanet al\.\([2026](https://arxiv.org/html/2606.24999#bib.bib30)\)\.

∂tu​\(t,x\)\+12​Δ​u​\(t,x\)\+x⋅∇u​\(t,x\)−12​\|∇u​\(t,x\)\|2−d=0,\(t,x\)∈\[0,T\)×ℝd,\\partial\_\{t\}u\(t,x\)\+\\frac\{1\}\{2\}\\Delta u\(t,x\)\+x\\cdot\\nabla u\(t,x\)\-\\frac\{1\}\{2\}\|\\nabla u\(t,x\)\|^\{2\}\-d=0,\\qquad\(t,x\)\\in\[0,T\)\\times\\mathbb\{R\}^\{d\},\(6\.10\)with terminal conditionu​\(T,x\)=g​\(x\):=−log⁡p0​\(x\)u\(T,x\)=g\(x\):=\-\\log p\_\{0\}\(x\)\. Herep0p\_\{0\}is the density of a five\-component Gaussian mixture

p0​\(x\)=∑k=15πk​𝒩​\(x;μ0\(k\),Σ0\(k\)\),p\_\{0\}\(x\)=\\sum\_\{k=1\}^\{5\}\\pi\_\{k\}\\,\\mathcal\{N\}\(x;\\mu\_\{0\}^\{\(k\)\},\\Sigma\_\{0\}^\{\(k\)\}\),\(6\.11\)where the meansμ0\(k\)\\mu\_\{0\}^\{\(k\)\}are sampled uniformly from\[−1,1\]d\[\-1,1\]^\{d\}, the covariance matrices are diagonal withΣ0\(k\)=2​Id\\Sigma\_\{0\}^\{\(k\)\}=2I\_\{d\}, and the weightsπk\\pi\_\{k\}are positive and normalized with∑k=15πk=1\\sum\_\{k=1\}^\{5\}\\pi\_\{k\}=1\.

This equation has differential operatorℒ​u=12​Δ​u\\mathcal\{L\}u=\\frac\{1\}\{2\}\\Delta u, which corresponds to the SDE

d​Xt=d​Wt,X0=ξ,\\mathrm\{d\}X\_\{t\}=\\mathrm\{d\}W\_\{t\},\\quad X\_\{0\}=\\xi,\(6\.12\)and the nonlinear source term isf​\(t,x,∇u\)=x⋅∇u​\(t,x\)−12​\|∇u​\(t,x\)\|2−df\(t,x,\\nabla u\)=x\\cdot\\nabla u\(t,x\)\-\\frac\{1\}\{2\}\|\\nabla u\(t,x\)\|^\{2\}\-d\.

Then we can find the exact solutionu∗u^\{\*\}of the PDE \([6\.10](https://arxiv.org/html/2606.24999#S6.E10)\) as131313This is because we know the densityp​\(t,x\)p\(t,x\)ofX^t\\hat\{X\}\_\{t\}, whered​X^t=−X^t​d​t\+d​Wt\\mathrm\{d\}\\hat\{X\}\_\{t\}=\-\\hat\{X\}\_\{t\}\\mathrm\{d\}t\+\\mathrm\{d\}W\_\{t\}, satisfies the Fokker–Planck equation∂tp=∇⋅\(x​p\)\+12​Δ​p\\partial\_\{t\}p=\\nabla\\cdot\(xp\)\+\\frac\{1\}\{2\}\\Delta p\. And we can verify thatu​\(t,x\)=−log⁡p​\(T−t,x\)u\(t,x\)=\-\\log p\(T\-t,x\)satisfies PDE \([6\.10](https://arxiv.org/html/2606.24999#S6.E10)\)\. Hence, ifX^t\\hat\{X\}\_\{t\}has the initial distributionp0​\(x\)p\_\{0\}\(x\), we can deduce its distribution time\-ttdistributionp​\(t,x\)p\(t,x\)\.

u∗​\(t,x\)=−log⁡p​\(T−t,x\),u^\{\\ast\}\(t,x\)=\-\\log p\(T\-t,x\),\(6\.13\)where

p​\(t,x\)=∑k=15πk​𝒩​\(x;μt\(k\),Σt\(k\)\),p\(t,x\)=\\sum\_\{k=1\}^\{5\}\\pi\_\{k\}\\,\\mathcal\{N\}\(x;\\mu\_\{t\}^\{\(k\)\},\\Sigma\_\{t\}^\{\(k\)\}\),\(6\.14\)with

μt\(k\)=e−t​μ0\(k\),Σt\(k\)=2​e−2​t​Id\+1−e−2​t2​Id\.\\mu\_\{t\}^\{\(k\)\}=e^\{\-t\}\\mu\_\{0\}^\{\(k\)\},\\qquad\\Sigma\_\{t\}^\{\(k\)\}=2e^\{\-2t\}I\_\{d\}\+\\frac\{1\-e^\{\-2t\}\}\{2\}I\_\{d\}\.\(6\.15\)
In our experiment, we setd=20,T=1d=20,T=1\. The above configuration is identical to that inHanet al\.\([2026](https://arxiv.org/html/2606.24999#bib.bib30)\)except that we lower the dimension tod=20d=20\. This is because we also need to evaluation and compare the accuracy of the learned Hessian, which has an output dimensiond2d^\{2\}, whereasHanet al\.\([2026](https://arxiv.org/html/2606.24999#bib.bib30)\)only evaluate the gradient whose output dimension isdd\. For the SDE \([6\.12](https://arxiv.org/html/2606.24999#S6.E12)\) used to generate training and test datasets, we setξ∼𝒩​\(0,4​Id\)\\xi\\sim\\mathcal\{N\}\(0,4I\_\{d\}\)as inHanet al\.\([2026](https://arxiv.org/html/2606.24999#bib.bib30)\)\.

For a fair comparison, both ZOD and the DPI baseline use the same neural network framework fromHanet al\.\([2026](https://arxiv.org/html/2606.24999#bib.bib30)\), which explicitly enforces the terminal condition\. Denote

uθ​\(t,x\)=\(rη​\(T−t\)−r0​\(0\)\)​⟨Nγ​\(T−t,x\),x⟩\+\(1−rη​\(T−t\)\+rη​\(0\)\)​g​\(e−\(T−t\)/2​x\),u\_\{\\theta\}\(t,x\)=\(r\_\{\\eta\}\(T\-t\)\-r\_\{0\}\(0\)\)\\,\\langle N\_\{\\gamma\}\(T\-t,x\),x\\rangle\+\\bigl\(1\-r\_\{\\eta\}\(T\-t\)\+r\_\{\\eta\}\(0\)\\bigr\)\\,g\\\!\\left\(e^\{\-\(T\-t\)/2\}x\\right\),\(6\.16\)whereNγ​\(t,x\)∈ℝdN\_\{\\gamma\}\(t,x\)\\in\\mathbb\{R\}^\{d\}andrη​\(t\)∈ℝ\+r\_\{\\eta\}\(t\)\\in\\mathbb\{R\}^\{\+\}are neural networks\. Consequently,uθ​\(T,x\)=g​\(x\)u\_\{\\theta\}\(T,x\)=g\(x\)\.

Note that we compare our ZOD approach with DPI inHanet al\.\([2026](https://arxiv.org/html/2606.24999#bib.bib30)\)on the2020\-dimensional PDE \([6\.10](https://arxiv.org/html/2606.24999#S6.E10)\), using the same Gaussian\-mixture terminal density \([6\.11](https://arxiv.org/html/2606.24999#S6.E11)\) and the same network \([6\.16](https://arxiv.org/html/2606.24999#S6.E16)\) for the value function\. This isolates the main algorithmic difference: how derivative information is represented and controlled over iterations\.

For ZOD, we use the three\-network model, where the value network uses the parameterization \([6\.16](https://arxiv.org/html/2606.24999#S6.E16)\) and the gradient and Hessian networks are 4\-layer MLPs with width512512and ELU activations\. We use the Adam optimizer with a learning rate10−310^\{\-3\}, batch size1638416384,1010value iterations, and512512gradient updating steps per iteration\. The derivative networks are pretrained for50005000steps\. The bandwidth of ZOD isϵ=0\.02\\epsilon=0\.02\.

For DPI, we use the configuration inHanet al\.\([2026](https://arxiv.org/html/2606.24999#bib.bib30)\)\. Specifically, the value network is again \([6\.16](https://arxiv.org/html/2606.24999#S6.E16)\) with the same 4 hidden layers of width512512and ELU activations\. DPI uses1010outer Picard iterations, batch size512512, data size40964096, and1616epochs per outer iteration, with the Adam optimizer and a learning rate10−310^\{\-3\}\. Gradient supervision is enabled with the same weight as inHanet al\.\([2026](https://arxiv.org/html/2606.24999#bib.bib30)\)\. Moreover, second\-order information is obtained by differentiating the learned scalar value approximation\.

Figure[2](https://arxiv.org/html/2606.24999#S6.F2)reports the rRMSE of the learned value, gradient, and Hessian over iterations for the problem \([6\.10](https://arxiv.org/html/2606.24999#S6.E10)\)\. The curves show the means over three random seeds, and the shaded regions indicate the empirical2\.5%2\.5\\%–97\.5%97\.5\\%quantile range across seeds\. We see that our ZOD approach converges faster than DPI in all the three quantities, and eventually achieves slightly smaller errors in terms of the value and gradient after sufficient numbers of iterations\. However, the performance differs dramatically in terms of the Hessian accuracy\. Over iterations, DPI barely reduces the error while ZOD effectively decreases it although the terminal error tends to be larger than its gradient and value counterparts\.

![Refer to caption](https://arxiv.org/html/2606.24999v1/x2.png)Figure 2:rRMSE of the learned value, gradient, and Hessian over iterationsfor the PDE \([6\.10](https://arxiv.org/html/2606.24999#S6.E10)\) with dimensiond=20d=20\. ZOD\-m and DPI use the same network \([6\.16](https://arxiv.org/html/2606.24999#S6.E16)\) for value function\.Table[2](https://arxiv.org/html/2606.24999#S6.T2)summarizes the corresponding mean final errors at the1010value \(Picard\) iterations\. The two methods produce comparable approximations of the value function and its gradient\. ZOD achieves a smaller rRMSE, while the gradient errors of the two are similar\. However, the final Hessian rRMSE of ZOD is1\.252×10−11\.252\\times 10^\{\-1\}, whereas DPI gives1\.4341\.434, differing by more than 10 times\. This discrepancy is a result of how the Hessian is obtained\. In DPI, the Hessian is not learned directly; rather it is obtained by applying automatic differentiation twice on the learned value network\. Since the training loss does not explicitly control the second\-order derivative, the resulting Hessian need not be accurate even when the value function and its gradient are reasonably well approximated\. By contrast, the proposed ZOD method augments the learning procedure with a dedicated Hessian network and a second\-order zeroth\-order objective\. This second\-order training signal directly targets the Hessian, leading to a substantially more accurate approximation\. Finally, it is notable that ZOD spends about 30% less time obtaining these results\.

Table 2:rRMSEs for the semilinear PDE \([6\.10](https://arxiv.org/html/2606.24999#S6.E10)\)\. The results are based on1010value \(Picard\) iterations\.
### 6\.3A Fully Nonlinear PDE

We next consider a fully nonlinear PDE studied inHanet al\.\([2026](https://arxiv.org/html/2606.24999#bib.bib30)\):

∂tu​\(t,x\)\+12​Δ​u​\(t,x\)\+14​∑i=1d\|∂2u∂xi2​\(t,x\)\|−h​\(t,x\)=0,\(t,x\)∈\[0,T\)×ℝd\.\\partial\_\{t\}u\(t,x\)\+\\frac\{1\}\{2\}\\Delta u\(t,x\)\+\\frac\{1\}\{4\}\\sum\_\{i=1\}^\{d\}\\left\|\\frac\{\\partial^\{2\}u\}\{\\partial x\_\{i\}^\{2\}\}\(t,x\)\\right\|\-h\(t,x\)=0,\\qquad\(t,x\)\\in\[0,T\)\\times\\mathbb\{R\}^\{d\}\.\(6\.17\)The form of this PDE is reverse\-engineered from the exact solution

u⋆​\(t,x\)=∑j=1Jvj​sin⁡\(t\+∑i=1dwij​xi\),u^\{\\star\}\(t,x\)=\\sum\_\{j=1\}^\{J\}v\_\{j\}\\sin\\\!\\left\(t\+\\sum\_\{i=1\}^\{d\}w\_\{i\}^\{j\}x\_\{i\}\\right\),\(6\.18\)with

h​\(t,x\)=∂tu⋆​\(t,x\)\+12​Δ​u⋆​\(t,x\)\+14​∑i=1d\|∂2u⋆∂xi2​\(t,x\)\|\.h\(t,x\)=\\partial\_\{t\}u^\{\\star\}\(t,x\)\+\\frac\{1\}\{2\}\\Delta u^\{\\star\}\(t,x\)\+\\frac\{1\}\{4\}\\sum\_\{i=1\}^\{d\}\\left\|\\frac\{\\partial^\{2\}u^\{\\star\}\}\{\\partial x\_\{i\}^\{2\}\}\(t,x\)\\right\|\.\(6\.19\)The PDE \([6\.17](https://arxiv.org/html/2606.24999#S6.E17)\) has differential operatorℒ​u=12​Δ​u\\mathcal\{L\}u=\\frac\{1\}\{2\}\\Delta u, and the source termf​\(t,x,∇2u\)=14​∑i=1d\|∂2u∂xi2​\(t,x\)\|−h​\(t,x\)f\(t,x,\\nabla^\{2\}u\)=\\frac\{1\}\{4\}\\sum\_\{i=1\}^\{d\}\\left\|\\frac\{\\partial^\{2\}u\}\{\\partial x\_\{i\}^\{2\}\}\(t,x\)\\right\|\-h\(t,x\)\. Hence, it corresponds to the SDEd​Xt=d​Wt\\mathrm\{d\}X\_\{t\}=\\mathrm\{d\}W\_\{t\}\.

We used=20d=20,T=1\.0T=1\.0,J=2J=2and setX0=0X\_\{0\}=0\. The parameters in \([6\.18](https://arxiv.org/html/2606.24999#S6.E18)\) are sampled once and then fixed throughout the experiment:

wij∼d−1/2​𝒩​\(0,1\),vj∼𝒩​\(0,1\)\.w\_\{i\}^\{j\}\\sim d^\{\-1/2\}\\mathcal\{N\}\(0,1\),\\qquad v\_\{j\}\\sim\\mathcal\{N\}\(0,1\)\.
For the DPI baseline, we use a configuration with data size40964096, integral sample size10241024, three hidden layers of width128128,1616epochs per iteration, and in total4040iterations\.

For ZOD, we use the three\-network model with33hidden layers of width6464, ELU activations, ZOD bandwidthϵ=0\.05\\epsilon=0\.05\. The derivative networks are pretrained for50005000steps\. To ensure a fair comparison, we control the computational time of ZOD to be comparable to that of DPI\. To this end, in our implementation of ZOD, we use batch size3276832768and40964096gradient steps per value iteration\.

Figure[3](https://arxiv.org/html/2606.24999#S6.F3)compares the time\-matched ZOD run with the4040\-iteration DPI run\. The rapid decay of the ZOD errors in the first few value \(Picard\) iterations is consistent with the contraction estimate in Theorem[6](https://arxiv.org/html/2606.24999#Thmtheorem6)\. After a small number of Picard iterations, the curves approach stable accuracy levels\. This behavior agrees with the structure of \([5\.9](https://arxiv.org/html/2606.24999#S5.E9)\): once the one\-step learning error has been reduced to a fixed floor, the geometrically weighted accumulation term limits the additional gain from further iterations\. Meanwhile, DPI decreases more gradually under the reported configuration\. In particular, directly learning the Hessian through a second\-order ZOD target gives a substantially lower second\-order error than differentiating a learned scalar value approximation\. Table[3](https://arxiv.org/html/2606.24999#S6.T3)further reports the final errors and runtimes\. ZOD is more accurate than DPI for all the three quantities, and the gap is the largest for the Hessian\.

![Refer to caption](https://arxiv.org/html/2606.24999v1/x3.png)Figure 3:rRMSE of the learned value, gradient, and Hessian over iterationsfor the fully nonlinear PDE \([6\.17](https://arxiv.org/html/2606.24999#S6.E17)\) withd=20d=20\. The panels show value, gradient, and Hessian rRMSEs over iterations for the time\-matched ZOD configuration and DPI\. Curves are averaged over three random seeds, and shaded regions indicate empirical2\.5%2\.5\\%–97\.5%97\.5\\%quantile bands\.Table 3:rRMSE for the fully nonlinear PDE \([6\.17](https://arxiv.org/html/2606.24999#S6.E17)\)\. The results of ZOD are based on 10 value iterations, while DPI is based on 40 iterations\.Finally, we examine two controlled ablations for our ZOD approach\. Table[4](https://arxiv.org/html/2606.24999#S6.T4)summarizes the impact of training budget, batch size, and availability of the strong simulator\. In the training\-budget panel, we vary the number of gradient steps while fixing the batch size at3276832768and the bandwidth atϵ=0\.05\\epsilon=0\.05\. In the batch\-size panel, we vary the batch size while fixing the training budget at40964096steps and the bandwidth atϵ=0\.05\\epsilon=0\.05\. These two panels demonstrate the expected monotone improvement as the training budget and batch size increase, consistent with the theoretical decomposition in Theorem[9](https://arxiv.org/html/2606.24999#Thmtheorem9)\. Specifically, increasing the number of gradient steps reduces the optimization error in training the empirical loss \(although we did not include the optimization error in Theorem[9](https://arxiv.org/html/2606.24999#Thmtheorem9)\)\. On the other hand, the batch\-size ablation aligns directly with the theorem: a larger batch size decreases the statistical error term in the one\-step bound, and therefore is expected to reduce the overall error\. These results are also illustrated in Figure[4](https://arxiv.org/html/2606.24999#S6.F4)\.

The last panel of Table[4](https://arxiv.org/html/2606.24999#S6.T4)shows the effect of the strong simulator availability when everything else is the same\. As predicted by the theoretical results, access to the strong simulator boosts performance\. In Appendix[B](https://arxiv.org/html/2606.24999#A2), we further examine the performance of ZOD when only the weak simulator is available\. Overall, ZOD\-1 is slightly worse than ZOD\-m, and a small bandwidth is not always preferred with the former\.

Table 4:ZOD ablations for the fully nonlinear PDE \([6\.17](https://arxiv.org/html/2606.24999#S6.E17)\)\. Each row reports the mean final rRMSE over three random seeds\.![Refer to caption](https://arxiv.org/html/2606.24999v1/x4.png)Figure 4:Ablations for the fully nonlinear PDE \([6\.17](https://arxiv.org/html/2606.24999#S6.E17)\)\.All curves report the mean rRMSE over 3 seeds, and shaded regions indicate one empirical standard deviation\.

## 7Concluding Remarks

In this paper we develop a data\-driven learning method for solving a broad class of potentially high\-dimensional, fully nonlinear black\-box PDEs\. The thrust of our analysis is to represent the derivatives of the solutions via ZODs and then learn them in the same way as learning the solutions, rather than outright differentiating the learned solutions\. Thus, the learning ultimately corresponds to policy evaluation\(Jia and Zhou,[2022a](https://arxiv.org/html/2606.24999#bib.bib12)\)in continuous\-time reinforcement learning \(RL\)\. Indeed, we solve the nonlinear PDEs through iterations each of which is a linear PDE and hence a policy evaluation problem\. This idea further inspires us to formulate learning the gradient and Hessian also as policy evaluation problems\.141414It should be noted that although the PDEs studied in this paper includein formthe class of HJB equations as a special case, our assumption that the value of the nonlinear source term is known for each given input excludes the applicability to HJB in the model\-free RL setting\. However,Jia and Zhou \([2022b](https://arxiv.org/html/2606.24999#bib.bib13),[2023](https://arxiv.org/html/2606.24999#bib.bib10)\)decompose the RL into policy evaluation and policy improvement/q\-learning, and show that the latter can be reformulated as policy evaluation\. In this sense, black\-box HJB equations can also be solved essentially by policy evaluation, albeit in a very different way from this paper\.

The approach developed in this paper hints on an important implication in the study of data\-driven methodologies in machine learning: the design of algorithms should be tailored to the availability of the data\-generating mechanisms or the properties of simulators\. With strong simulators similar to the one introduced in this paper, various variance reduction tricks in simulations may be employed to improve the efficiency of the algorithms\. This opens a gate to further studying its potential in other tasks\.

Acknowledgments and Disclosure of Funding

Huyên Pham is supported by the Société Générale Chair “Risques Financiers”, FiME \(Laboratory of Finance and Energy Markets\), and the EDF–CACIB Chair “Finance and Sustainable Development”\. Yanwei Jia is supported by the University Start\-up Fund at The Chinese University of Hong Kong\. Xun Yu Zhou is supported by the Nie Center for Intelligent Wealth Management at Columbia\. The authors thank Xuefeng Gao, Chun Liu, and Jiale Zha for discussions on the contents of this paper\.

## Appendix AA Sufficient Condition for Assumption[2](https://arxiv.org/html/2606.24999#Thmassumption2)

We provide a concrete sufficient condition for Assumption[2](https://arxiv.org/html/2606.24999#Thmassumption2)\.

Consider the operatorℒ​v:=Δ​v−∇h⋅∇v\\mathcal\{L\}v:=\\Delta v\-\\nabla h\\cdot\\nabla vthat corresponds to the Langevin SDE:

d​Xt=−∇h​\(Xt\)​d​t\+2​d​Wt\.\\mathrm\{d\}X\_\{t\}=\-\\nabla h\(X\_\{t\}\)\\mathrm\{d\}t\+\\sqrt\{2\}\\mathrm\{d\}W\_\{t\}\.\(A\.1\)Consider the norm defined in Section[5](https://arxiv.org/html/2606.24999#S5)with a weighting parameterβ\\betato be specified later:

‖v‖β,mix2:=β​‖v‖β2\+β​‖∇v‖β2\+‖∇2v‖β2\.\\\|v\\\|^\{2\}\_\{\\beta,\\textrm\{mix\}\}:=\\beta\\\|v\\\|\_\{\\beta\}^\{2\}\+\\beta\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\+\\\|\\nabla^\{2\}v\\\|\_\{\\beta\}^\{2\}\.\(A\.2\)For a triplet𝐔=\(V,G,H\)\\mathbf\{U\}=\(V,G,H\), we denote the corresponding norm as

‖𝐔‖β,mix2:=β​‖V‖β2\+β​‖G‖β2\+‖H‖β2,\\\|\\mathbf\{U\}\\\|^\{2\}\_\{\\beta,\\textrm\{mix\}\}:=\\beta\\\|V\\\|\_\{\\beta\}^\{2\}\+\\beta\\\|G\\\|\_\{\\beta\}^\{2\}\+\\\|H\\\|\_\{\\beta\}^\{2\},\(A\.3\)and denote the solution mapping as

𝐔↦v=𝒮​\(𝐔\),such that​∂tv\+ℒ​v\+f​\(x,V,G,H\)=0,v​\(T,x\)=g​\(x\)\.\\mathbf\{U\}\\mapsto v=\\mathcal\{S\}\(\\mathbf\{U\}\),\\text\{ such that \}\\partial\_\{t\}v\+\\mathcal\{L\}v\+f\(x,V,G,H\)=0,\\;\\;v\(T,x\)=g\(x\)\.For any two input triplets𝐔i=\(Vi,Gi,Hi\)\\mathbf\{U\}\_\{i\}=\(V\_\{i\},G\_\{i\},H\_\{i\}\),i=1,2i=1,2\. Denoteδ​f:=f​\(x,𝐔1\)−f​\(x,𝐔2\)\\delta f:=f\(x,\\mathbf\{U\}\_\{1\}\)\-f\(x,\\mathbf\{U\}\_\{2\}\),v:=𝒮​\(𝐔1\)−𝒮​\(𝐔2\)v:=\\mathcal\{S\}\(\\mathbf\{U\}\_\{1\}\)\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{2\}\)\. Thenvvsatisfies the linear PDE

∂tv\+ℒ​v=−δ​f,v​\(T,x\)=0\.\\partial\_\{t\}v\+\\mathcal\{L\}v=\-\\delta f,\\quad v\(T,x\)=0\.\(A\.4\)We aim to prove the contraction property through the bound

‖v‖β,mix≤γ​‖𝐔1−𝐔2‖β,mix\\\|v\\\|\_\{\\beta,\\textrm\{mix\}\}\\leq\\gamma\\\|\\mathbf\{U\}\_\{1\}\-\\mathbf\{U\}\_\{2\}\\\|\_\{\\beta,\\textrm\{mix\}\}\(A\.5\)for some constantγ<1\\gamma<1\. We need the following assumption in this section\.

###### Assumption 5

1. \(i\)h∈Cp2​\(ℝd\)h\\in C^\{2\}\_\{p\}\(\\mathbb\{R\}^\{d\}\), the eigenvalues of∇2h\\nabla^\{2\}hhave a uniform lower bound−k\-kfor somek≥0k\\geq 0, and lim\|x\|→∞h​\(x\)log⁡\|x\|=∞\.\\lim\_\{\|x\|\\to\\infty\}\\frac\{h\(x\)\}\{\\log\|x\|\}=\\infty\.\(A\.6\)
2. \(ii\)The nonlinearityffis Lipschitz in the value, gradient, and Hessian variables: \|f​\(x,V,G,H\)−f​\(x,V′,G′,H′\)\|≤Lv​\|V−V′\|\+Lg​\|G−G′\|\+Lh​\|H−H′\|\.\|f\(x,V,G,H\)\-f\(x,V^\{\\prime\},G^\{\\prime\},H^\{\\prime\}\)\|\\leq L\_\{v\}\|V\-V^\{\\prime\}\|\+L\_\{g\}\|G\-G^\{\\prime\}\|\+L\_\{h\}\|H\-H^\{\\prime\}\|\.\(A\.7\)
3. \(iii\)v:=𝒮​\(𝐔1\)−𝒮​\(𝐔2\)∈Cp1,3​\(\[0,T\]×ℝd\)v:=\\mathcal\{S\}\(\\mathbf\{U\}\_\{1\}\)\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{2\}\)\\in C^\{1,3\}\_\{p\}\(\[0,T\]\\times\\mathbb\{R\}^\{d\}\)\.

Under Assumption[5](https://arxiv.org/html/2606.24999#Thmassumption5)\([i](https://arxiv.org/html/2606.24999#A1.I1.i1)\), the potentialhhis a smooth confining potential, and the Langevin dynamics admits the invariant density, see, e\.g\.,\(Pavliotis,[2014](https://arxiv.org/html/2606.24999#bib.bib58), Proposition 4\.2\),

p​\(x\):=1C​exp⁡\{−h​\(x\)\},p\(x\):=\\frac\{1\}\{C\}\\exp\\left\\\{\-h\(x\)\\right\\\},\(A\.8\)whereC:=∫ℝdexp⁡\{−h​\(x\)\}​dx<∞C:=\\int\_\{\\mathbb\{R\}^\{d\}\}\\exp\\\{\-h\(x\)\\\}\\mathrm\{d\}x<\\infty\.

From now on, we takeX0∼p​\(⋅\)X\_\{0\}\\sim p\(\\cdot\), and thenXt∼p​\(⋅\)X\_\{t\}\\sim p\(\\cdot\)for allt≥0t\\geq 0\. Consequently, the norms have the following explicit form:

‖v‖L2​\(p\)2:=∫ℝd\|v​\(t,x\)\|2​p​\(x\)​dx,‖v‖β2:=∫0Teβ​t​∫ℝd\|v​\(t,x\)\|2​p​\(x\)​dx​dt\.\\\|v\\\|\_\{L\_\{2\}\(p\)\}^\{2\}:=\\int\_\{\\mathbb\{R\}^\{d\}\}\|v\(t,x\)\|^\{2\}p\(x\)\\mathrm\{d\}x,\\quad\\\|v\\\|\_\{\\beta\}^\{2\}:=\\int\_\{0\}^\{T\}e^\{\\beta t\}\\int\_\{\\mathbb\{R\}^\{d\}\}\|v\(t,x\)\|^\{2\}p\(x\)\\mathrm\{d\}x\\mathrm\{d\}t\.\(A\.9\)
The following theorem provides a sufficient condition for the contraction property used in the main text\.

###### Theorem 11

Suppose Assumption[5](https://arxiv.org/html/2606.24999#Thmassumption5)holds\. If12​Lh2<112L\_\{h\}^\{2\}<1, then there exists a sufficiently largeβ≥k\\beta\\geq ksuch that

ηβ:=12​\(\(Lv2\+Lg2\)​β\+1β2\+Lh2​β\+1β\)<1,\\eta\_\{\\beta\}:=12\\left\(\(L\_\{v\}^\{2\}\+L\_\{g\}^\{2\}\)\\frac\{\\beta\+1\}\{\\beta^\{2\}\}\+L\_\{h\}^\{2\}\\frac\{\\beta\+1\}\{\\beta\}\\right\)<1,\(A\.10\)and setγ=ηβ\\gamma=\\sqrt\{\\eta\_\{\\beta\}\}\. Then

‖𝒮​\(𝐔1\)−𝒮​\(𝐔2\)‖β,mix≤γ​‖𝐔1−𝐔2‖β,mix\.\\\|\\mathcal\{S\}\(\\mathbf\{U\}\_\{1\}\)\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{2\}\)\\\|\_\{\\beta,\\textrm\{mix\}\}\\leq\\gamma\\\|\\mathbf\{U\}\_\{1\}\-\\mathbf\{U\}\_\{2\}\\\|\_\{\\beta,\\textrm\{mix\}\}\.\(A\.11\)That is, Assumption[2](https://arxiv.org/html/2606.24999#Thmassumption2)holds\.

The rest of this appendix is devoted to proving Theorem[11](https://arxiv.org/html/2606.24999#Thmtheorem11)\. We first analyze the norm of the value and the gradient\. The following lemma holds in a general diffusion setting, whereℒ\\mathcal\{L\}is the infinitesimal generator of the diffusion,ℒ​v=b⋅∇v\+12​Tr⁡\(σ​σ⊤​∇2v\)\\mathcal\{L\}v=b\\cdot\\nabla v\+\\frac\{1\}\{2\}\\operatorname\{Tr\}\\left\(\\sigma\\sigma^\{\\top\}\\nabla^\{2\}v\\right\)\.

###### Lemma 12

Suppose Assumption[5](https://arxiv.org/html/2606.24999#Thmassumption5)holds\. Assume thatσ​σ⊤\\sigma\\sigma^\{\\top\}is uniformly elliptic, i\.e\., there is a constantλ\>0\\lambda\>0, such thatx⊤​σ​σ⊤​x≥λ​\|x\|2x^\{\\top\}\\sigma\\sigma^\{\\top\}x\\geq\\lambda\|x\|^\{2\}\. Then the value and gradient have the following upper bound:

β2​‖v‖β2\+λ​‖∇v‖β2≤2β​‖δ​f‖β2\.\\frac\{\\beta\}\{2\}\\\|v\\\|\_\{\\beta\}^\{2\}\+\\lambda\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\\leq\\frac\{2\}\{\\beta\}\\\|\\delta f\\\|\_\{\\beta\}^\{2\}\.\(A\.12\)

ProofDenoteYt:=eβ​t​v​\(t,Xt\)2Y\_\{t\}:=e^\{\\beta t\}v\(t,X\_\{t\}\)^\{2\}\. It follows from Ito’s formula that

d​Yt=β​eβ​t​v2​d​t\+2​eβ​t​v​\(∂t\+ℒ\)​v​d​t\+eβ​t​∇v⊤​σ​σ⊤​∇v​d​t\+d​Mt,\\displaystyle\\mathrm\{d\}Y\_\{t\}=\\beta e^\{\\beta t\}v^\{2\}\\mathrm\{d\}t\+2e^\{\\beta t\}v\(\\partial\_\{t\}\+\\mathcal\{L\}\)v\\mathrm\{d\}t\+e^\{\\beta t\}\\nabla v^\{\\top\}\\sigma\\sigma^\{\\top\}\\nabla v\\mathrm\{d\}t\+\\mathrm\{d\}M\_\{t\},\(A\.13\)whereMMis a martingale\. Taking expectation on both sides and notingYT=0,\(∂t\+ℒ\)​v=−δ​fY\_\{T\}=0,\(\\partial\_\{t\}\+\\mathcal\{L\}\)v=\-\\delta f, we have

−𝔼​\[Y0\]=𝔼​\[∫0Tβ​eβ​t​v2−2​eβ​t​v​δ​f\+eβ​t​∇v⊤​σ​σ⊤​∇v​d​t\]\.\-\\mathbb\{E\}\[Y\_\{0\}\]=\\mathbb\{E\}\\left\[\\int\_\{0\}^\{T\}\\beta e^\{\\beta t\}v^\{2\}\-2e^\{\\beta t\}v\\delta f\+e^\{\\beta t\}\\nabla v^\{\\top\}\\sigma\\sigma^\{\\top\}\\nabla v\\mathrm\{d\}t\\right\]\.\(A\.14\)Thus,

β​‖v‖β2\+𝔼​\[∫0Teβ​t​∇v⊤​σ​σ⊤​∇v​d​t\]=𝔼​\[∫0T2​eβ​t​v​δ​f​dt\]−𝔼​\[Y0\]\.\\beta\\\|v\\\|\_\{\\beta\}^\{2\}\+\\mathbb\{E\}\\left\[\\int\_\{0\}^\{T\}e^\{\\beta t\}\\nabla v^\{\\top\}\\sigma\\sigma^\{\\top\}\\nabla v\\mathrm\{d\}t\\right\]=\\mathbb\{E\}\\left\[\\int\_\{0\}^\{T\}2e^\{\\beta t\}v\\delta f\\mathrm\{d\}t\\right\]\-\\mathbb\{E\}\[Y\_\{0\}\]\.\(A\.15\)It follows from the uniform ellipticity ofσ​σ⊤\\sigma\\sigma^\{\\top\}that∇v⊤​σ​σ⊤​∇v≥λ​‖∇v‖2\\nabla v^\{\\top\}\\sigma\\sigma^\{\\top\}\\nabla v\\geq\\lambda\\\|\\nabla v\\\|^\{2\}\. This along with the inequality2​\|v​δ​f\|≤β2​\|v\|2\+2β​\|δ​f\|22\|v\\delta f\|\\leq\\frac\{\\beta\}\{2\}\|v\|^\{2\}\+\\frac\{2\}\{\\beta\}\|\\delta f\|^\{2\}yields

β​‖v‖β2\+λ​‖∇v‖β2≤β2​‖v‖β2\+2β​‖δ​f‖β2−𝔼​\[Y0\]\.\\beta\\\|v\\\|\_\{\\beta\}^\{2\}\+\\lambda\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\\leq\\frac\{\\beta\}\{2\}\\\|v\\\|\_\{\\beta\}^\{2\}\+\\frac\{2\}\{\\beta\}\\\|\\delta f\\\|\_\{\\beta\}^\{2\}\-\\mathbb\{E\}\[Y\_\{0\}\]\.\(A\.16\)However,Y0≥0Y\_\{0\}\\geq 0; thus

β2​‖v‖β2\+λ​‖∇v‖β2≤2β​‖δ​f‖β2\.\\frac\{\\beta\}\{2\}\\\|v\\\|\_\{\\beta\}^\{2\}\+\\lambda\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\\leq\\frac\{2\}\{\\beta\}\\\|\\delta f\\\|\_\{\\beta\}^\{2\}\.\(A\.17\)The desired result then follows\.

The preceding estimate controls the value component and provides an initial control on the gradient\. To handle the Hessian component, we use the reversibility of the Langevin generator with respect to the stationary density\.

###### Lemma 13

Suppose Assumption[5](https://arxiv.org/html/2606.24999#Thmassumption5)\([i](https://arxiv.org/html/2606.24999#A1.I1.i1)\) holds\. Letr∈Cp2​\(ℝd\)r\\in C^\{2\}\_\{p\}\(\\mathbb\{R\}^\{d\}\)andg∈Cp1​\(ℝd\)g\\in C^\{1\}\_\{p\}\(\\mathbb\{R\}^\{d\}\)\. Then

∫ℝd\(ℒ​r\)​g​\(x\)​p​\(x\)​dx=−∫ℝd∇r⋅∇g​p​\(x\)​dx\.\\int\_\{\\mathbb\{R\}^\{d\}\}\(\\mathcal\{L\}r\)g\(x\)p\(x\)\\mathrm\{d\}x=\-\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla r\\cdot\\nabla gp\(x\)\\mathrm\{d\}x\.\(A\.18\)

ProofNote that∇p=1C​exp⁡\{−h​\(x\)\}​\(−∇h\)=−p​∇h\\nabla p=\\frac\{1\}\{C\}\\exp\\left\\\{\-h\(x\)\\right\\\}\(\-\\nabla h\)=\-p\\nabla h\. Thus,

∇p\+p​∇h=0\.\\nabla p\+p\\nabla h=0\.\(A\.19\)It follows from the integration by parts \(the remainder terms vanish due to \([A\.6](https://arxiv.org/html/2606.24999#A1.E6)\)\) that

∫ℝd\(ℒ​r\)​g​\(x\)​p​\(x\)​dx\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}\(\\mathcal\{L\}r\)g\(x\)p\(x\)\\mathrm\{d\}x=∫ℝd\(Δ​r−∇h⋅∇r\)​g​\(x\)​p​\(x\)​dx\\displaystyle=\\int\_\{\\mathbb\{R\}^\{d\}\}\(\\Delta r\-\\nabla h\\cdot\\nabla r\)g\(x\)p\(x\)\\mathrm\{d\}x\(A\.20\)=−∫ℝd∇r⋅∇\(g​p\)⁡d​x−∫ℝd∇h⋅∇r​g​p​d​x\\displaystyle=\-\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla r\\cdot\\nabla\(gp\)\\mathrm\{d\}x\-\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla h\\cdot\\nabla rgp\\mathrm\{d\}x=−∫ℝd∇r⋅∇g​p​d​x−∫ℝd∇r⋅\(∇p\+p​∇h\)​g​dx\\displaystyle=\-\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla r\\cdot\\nabla gp\\mathrm\{d\}x\-\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla r\\cdot\(\\nabla p\+p\\nabla h\)g\\mathrm\{d\}x=−∫ℝd∇r⋅∇g​p​d​x\.\\displaystyle=\-\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla r\\cdot\\nabla gp\\mathrm\{d\}x\.where the last equality follows from the identity \([A\.19](https://arxiv.org/html/2606.24999#A1.E19)\)\. Next, we turn to the Hessian bound\. The following lemma represents the norm of the Hessian∇2v\\nabla^\{2\}v\.

###### Lemma 14

Under Assumption[5](https://arxiv.org/html/2606.24999#Thmassumption5), forℒ​v=Δ​v−∇h⋅∇v\\mathcal\{L\}v=\\Delta v\-\\nabla h\\cdot\\nabla v, we have

12​ℒ​\(\|∇v\|2\)−∇v⋅∇\(ℒ​v\)=\|∇2v\|2\+∇v⊤​∇2h​∇v\.\\frac\{1\}\{2\}\\mathcal\{L\}\(\|\\nabla v\|^\{2\}\)\-\\nabla v\\cdot\\nabla\(\\mathcal\{L\}v\)=\|\\nabla^\{2\}v\|^\{2\}\+\\nabla v^\{\\top\}\\nabla^\{2\}h\\nabla v\.\(A\.21\)

ProofNote that

12​Δ​\(\|∇v\|2\)=12​∑i=1d∂i2∑j=1d\(∂jv\)2=12​∑i,j=1d∂i\(2​∂jv​∂i​j2v\)\\displaystyle\\frac\{1\}\{2\}\\Delta\(\|\\nabla v\|^\{2\}\)=\\frac\{1\}\{2\}\\sum\_\{i=1\}^\{d\}\\partial\_\{i\}^\{2\}\\sum\_\{j=1\}^\{d\}\(\\partial\_\{j\}v\)^\{2\}=\\frac\{1\}\{2\}\\sum\_\{i,j=1\}^\{d\}\\partial\_\{i\}\(2\\partial\_\{j\}v\\partial\_\{ij\}^\{2\}v\)=∑i,j=1d\(∂i​j2v​∂i​j2v\+∂jv​∂i\(∂i​j2v\)\)\\displaystyle=\\sum\_\{i,j=1\}^\{d\}\(\\partial\_\{ij\}^\{2\}v\\partial\_\{ij\}^\{2\}v\+\\partial\_\{j\}v\\partial\_\{i\}\(\\partial\_\{ij\}^\{2\}v\)\)\(A\.22\)=\|∇2v\|2\+∇v⋅∇\(Δ​v\)\.\\displaystyle=\|\\nabla^\{2\}v\|^\{2\}\+\\nabla v\\cdot\\nabla\(\\Delta v\)\.Thus,

12​ℒ​\(\|∇v\|2\)−∇v⋅∇\(ℒ​v\)\\displaystyle\\frac\{1\}\{2\}\\mathcal\{L\}\(\|\\nabla v\|^\{2\}\)\-\\nabla v\\cdot\\nabla\(\\mathcal\{L\}v\)=12​Δ​\(\|∇v\|2\)−12​∇h⋅∇\(\|∇v\|2\)−∇v⋅∇\(Δ​v−∇h⋅∇v\)\\displaystyle=\\frac\{1\}\{2\}\\Delta\(\|\\nabla v\|^\{2\}\)\-\\frac\{1\}\{2\}\\nabla h\\cdot\\nabla\(\|\\nabla v\|^\{2\}\)\-\\nabla v\\cdot\\nabla\(\\Delta v\-\\nabla h\\cdot\\nabla v\)\(A\.23\)=12​Δ​\(\|∇v\|2\)−∇v⋅∇\(Δ​v\)−12​∇h⋅∇\(\|∇v\|2\)\+∇v⋅∇\(∇h⋅∇v\)\\displaystyle=\\frac\{1\}\{2\}\\Delta\(\|\\nabla v\|^\{2\}\)\-\\nabla v\\cdot\\nabla\(\\Delta v\)\-\\frac\{1\}\{2\}\\nabla h\\cdot\\nabla\(\|\\nabla v\|^\{2\}\)\+\\nabla v\\cdot\\nabla\(\\nabla h\\cdot\\nabla v\)=\|∇2v\|2−12​∇h⋅∇\(\|∇v\|2\)\+∇v⋅∇\(∇h⋅∇v\)\.\\displaystyle=\|\\nabla^\{2\}v\|^\{2\}\-\\frac\{1\}\{2\}\\nabla h\\cdot\\nabla\(\|\\nabla v\|^\{2\}\)\+\\nabla v\\cdot\\nabla\(\\nabla h\\cdot\\nabla v\)\.We now compute the second and third terms, respectively\. Indeed,

−12​∇h⋅∇\(\|∇v\|2\)=−12​∑i=1d∂ih​∂i\(∑j=1d\(∂jv\)2\)=−∑i,j=1d∂ih​∂jv​∂i​j2v,\\displaystyle\-\\frac\{1\}\{2\}\\nabla h\\cdot\\nabla\(\|\\nabla v\|^\{2\}\)=\-\\frac\{1\}\{2\}\\sum\_\{i=1\}^\{d\}\\partial\_\{i\}h\\partial\_\{i\}\(\\sum\_\{j=1\}^\{d\}\(\\partial\_\{j\}v\)^\{2\}\)=\-\\sum\_\{i,j=1\}^\{d\}\\partial\_\{i\}h\\partial\_\{j\}v\\partial\_\{ij\}^\{2\}v,\(A\.24\)while

∇v⋅∇\(∇h⋅∇v\)=∑i=1d∂iv​∂i\(∑j=1d∂jh​∂jv\)\\displaystyle\\nabla v\\cdot\\nabla\(\\nabla h\\cdot\\nabla v\)=\\sum\_\{i=1\}^\{d\}\\partial\_\{i\}v\\partial\_\{i\}\(\\sum\_\{j=1\}^\{d\}\\partial\_\{j\}h\\partial\_\{j\}v\)=∑i,j=1d∂iv​∂i​j2h​∂jv\+∑i,j=1d∂ih​∂jv​∂i​j2v\\displaystyle=\\sum\_\{i,j=1\}^\{d\}\\partial\_\{i\}v\\partial\_\{ij\}^\{2\}h\\partial\_\{j\}v\+\\sum\_\{i,j=1\}^\{d\}\\partial\_\{i\}h\\partial\_\{j\}v\\partial\_\{ij\}^\{2\}v\(A\.25\)=∇v⊤​∇2h​∇v\+∑i,j=1d∂ih​∂jv​∂i​j2v\.\\displaystyle=\\nabla v^\{\\top\}\\nabla^\{2\}h\\nabla v\+\\sum\_\{i,j=1\}^\{d\}\\partial\_\{i\}h\\partial\_\{j\}v\\partial\_\{ij\}^\{2\}v\.The desired result follows from \([A\.23](https://arxiv.org/html/2606.24999#A1.E23)\), \([A\.24](https://arxiv.org/html/2606.24999#A1.E24)\) and \([A\.25](https://arxiv.org/html/2606.24999#A1.E25)\)\.

Integrating the above identity against the invariant density converts the local Hessian identity into anL2​\(p\)L\_\{2\}\(p\)estimate\. The integration by parts formula above removes the total\-divergence term\.

###### Lemma 15

Under Assumption[5](https://arxiv.org/html/2606.24999#Thmassumption5), we have

‖∇2v‖β2=‖ℒ​v‖β2−∫0Teβ​t​∫ℝd∇v⊤​∇2h​∇v​p​d​x​d​t\.\\\|\\nabla^\{2\}v\\\|\_\{\\beta\}^\{2\}=\\\|\\mathcal\{L\}v\\\|\_\{\\beta\}^\{2\}\-\\int\_\{0\}^\{T\}e^\{\\beta t\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla v^\{\\top\}\\nabla^\{2\}h\\nabla vp\\mathrm\{d\}x\\mathrm\{d\}t\.\(A\.26\)

ProofIt follows from the formula \([A\.21](https://arxiv.org/html/2606.24999#A1.E21)\) that

∫ℝd12​ℒ​\(\|∇v\|2\)​p​dx−∫ℝd∇v⋅∇\(ℒ​v\)⁡p​d​x=∫ℝd\|∇2v\|2​p​dx\+∫ℝd∇v⊤​∇2h​∇v​p​d​x\.\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}\\frac\{1\}\{2\}\\mathcal\{L\}\(\|\\nabla v\|^\{2\}\)p\\mathrm\{d\}x\-\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla v\\cdot\\nabla\(\\mathcal\{L\}v\)p\\mathrm\{d\}x=\\int\_\{\\mathbb\{R\}^\{d\}\}\|\\nabla^\{2\}v\|^\{2\}p\\mathrm\{d\}x\+\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla v^\{\\top\}\\nabla^\{2\}h\\nabla vp\\mathrm\{d\}x\.\(A\.27\)Lemma[13](https://arxiv.org/html/2606.24999#Thmtheorem13)yields that the first term on the left hand side of the above equation equals0:

∫ℝd12​ℒ​\(\|∇v\|2\)​p​dx=−∫ℝd∇\(\|∇v\|2\)⋅∇\(12\)⁡p​d​x=0\.\\int\_\{\\mathbb\{R\}^\{d\}\}\\frac\{1\}\{2\}\\mathcal\{L\}\(\|\\nabla v\|^\{2\}\)p\\mathrm\{d\}x=\-\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla\(\|\\nabla v\|^\{2\}\)\\cdot\\nabla\\left\(\\frac\{1\}\{2\}\\right\)p\\mathrm\{d\}x=0\.\(A\.28\)The second term can be written as

−∫ℝd∇v⋅∇\(ℒ​v\)⁡p​d​x=∫ℝd\(ℒ​v\)2​p​dx\.\-\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla v\\cdot\\nabla\(\\mathcal\{L\}v\)p\\mathrm\{d\}x=\\int\_\{\\mathbb\{R\}^\{d\}\}\(\\mathcal\{L\}v\)^\{2\}p\\mathrm\{d\}x\.\(A\.29\)Therefore, integrating \([A\.27](https://arxiv.org/html/2606.24999#A1.E27)\) over\[0,T\]\[0,T\]with the weighteβ​te^\{\\beta t\}, we have

‖∇2v‖β2=‖ℒ​v‖β2−∫0Teβ​t​∫ℝd∇v⊤​∇2h​∇v​p​d​x​d​t\.\\\|\\nabla^\{2\}v\\\|\_\{\\beta\}^\{2\}=\\\|\\mathcal\{L\}v\\\|\_\{\\beta\}^\{2\}\-\\int\_\{0\}^\{T\}e^\{\\beta t\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla v^\{\\top\}\\nabla^\{2\}h\\nabla vp\\mathrm\{d\}x\\mathrm\{d\}t\.\(A\.30\)The desired result follows\.

Lemma[15](https://arxiv.org/html/2606.24999#Thmtheorem15)gives the unfinished bound of Hessian∇2v\\nabla^\{2\}v, which contains the norm ofℒ​v\\mathcal\{L\}v\. To boundℒ​v\\mathcal\{L\}v, we have to analyze∂tv\\partial\_\{t\}v\.

###### Lemma 16

Under Assumption[5](https://arxiv.org/html/2606.24999#Thmassumption5), we have

‖∂tv‖β2\+β​‖∇v‖β2≤‖δ​f‖β2\.\\\|\\partial\_\{t\}v\\\|\_\{\\beta\}^\{2\}\+\\beta\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\\leq\\\|\\delta f\\\|\_\{\\beta\}^\{2\}\.\(A\.31\)

ProofMultiply \([A\.4](https://arxiv.org/html/2606.24999#A1.E4)\) by∂tv\\partial\_\{t\}vand integrate overℝd\\mathbb\{R\}^\{d\}with weightppto get

∫ℝd\(∂tv\)2​p​dx\+∫ℝdℒ​v​∂tv​p​d​x=∫ℝd−δ​f​∂tv​p​d​x\.\\displaystyle\\int\_\{\\mathbb\{R\}^\{d\}\}\(\\partial\_\{t\}v\)^\{2\}p\\mathrm\{d\}x\+\\int\_\{\\mathbb\{R\}^\{d\}\}\\mathcal\{L\}v\\partial\_\{t\}vp\\mathrm\{d\}x=\\int\_\{\\mathbb\{R\}^\{d\}\}\-\\delta f\\partial\_\{t\}vp\\mathrm\{d\}x\.\(A\.32\)For the second term on the left hand side of the above equation, by Lemma[13](https://arxiv.org/html/2606.24999#Thmtheorem13),

∫ℝdℒ​v​∂tv​p​d​x=−∫ℝd∇v⋅∇\(∂tv\)⁡p​d​x=−12​dd​t​∫ℝd\|∇v\|2​p​dx\.\\int\_\{\\mathbb\{R\}^\{d\}\}\\mathcal\{L\}v\\partial\_\{t\}vp\\mathrm\{d\}x=\-\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla v\\cdot\\nabla\(\\partial\_\{t\}v\)p\\mathrm\{d\}x=\-\\frac\{1\}\{2\}\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\int\_\{\\mathbb\{R\}^\{d\}\}\|\\nabla v\|^\{2\}p\\mathrm\{d\}x\.\(A\.33\)Integrating over\[0,T\]\[0,T\]with weighteβ​te^\{\\beta t\}and using integration by parts with respect tott, we obtain

∫0Teβ​t−12​dd​t​∫ℝd\|∇v\|2​p​dx​dt\\displaystyle\\int\_\{0\}^\{T\}e^\{\\beta t\}\-\\frac\{1\}\{2\}\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\int\_\{\\mathbb\{R\}^\{d\}\}\|\\nabla v\|^\{2\}p\\mathrm\{d\}x\\mathrm\{d\}t=−12​∫0Tdd​t​\(eβ​t​∫ℝd\|∇v\|2​p​dx\)​dt\\displaystyle=\-\\frac\{1\}\{2\}\\int\_\{0\}^\{T\}\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\left\(e^\{\\beta t\}\\int\_\{\\mathbb\{R\}^\{d\}\}\|\\nabla v\|^\{2\}p\\mathrm\{d\}x\\right\)\\mathrm\{d\}t\(A\.34\)\+β2​∫0Teβ​t​∫ℝd\|∇v\|2​p​dx​dt\\displaystyle\\qquad\+\\frac\{\\beta\}\{2\}\\int\_\{0\}^\{T\}e^\{\\beta t\}\\int\_\{\\mathbb\{R\}^\{d\}\}\|\\nabla v\|^\{2\}p\\mathrm\{d\}x\\mathrm\{d\}t=12​∫ℝd\|∇v​\(0,⋅\)\|2​p​dx\+β2​‖∇v‖β2≥β2​‖∇v‖β2,\\displaystyle=\\frac\{1\}\{2\}\\int\_\{\\mathbb\{R\}^\{d\}\}\|\\nabla v\(0,\\cdot\)\|^\{2\}p\\mathrm\{d\}x\+\\frac\{\\beta\}\{2\}\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\\geq\\frac\{\\beta\}\{2\}\\\|\\nabla v\\\|\_\{\\beta\}^\{2\},where in the second equality, we used the fact thatv​\(T,x\)=0v\(T,x\)=0\. This along with \([A\.32](https://arxiv.org/html/2606.24999#A1.E32)\) yields

‖∂tv‖β2\+β2​‖∇v‖β2≤∫0Teβ​t​∫ℝd\|δ​f​∂tv\|​p​dx​dt≤12​‖δ​f‖β2\+12​‖∂tv‖β2\.\\\|\\partial\_\{t\}v\\\|\_\{\\beta\}^\{2\}\+\\frac\{\\beta\}\{2\}\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\\leq\\int\_\{0\}^\{T\}e^\{\\beta t\}\\int\_\{\\mathbb\{R\}^\{d\}\}\|\\delta f\\partial\_\{t\}v\|p\\mathrm\{d\}x\\mathrm\{d\}t\\leq\\frac\{1\}\{2\}\\\|\\delta f\\\|\_\{\\beta\}^\{2\}\+\\frac\{1\}\{2\}\\\|\\partial\_\{t\}v\\\|\_\{\\beta\}^\{2\}\.\(A\.35\)This proves the desired result\. Combining these lemmas, we have the upper bound for the norm of Hessian∇2v\\nabla^\{2\}v\.

###### Theorem 17

Under Assumption[5](https://arxiv.org/html/2606.24999#Thmassumption5),

‖∇2v‖β2≤4​‖δ​f‖β2−\(2​β−k\)​‖∇v‖β2\.\\\|\\nabla^\{2\}v\\\|\_\{\\beta\}^\{2\}\\leq 4\\\|\\delta f\\\|\_\{\\beta\}^\{2\}\-\(2\\beta\-k\)\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\.\(A\.36\)

ProofIt follows from Lemma[15](https://arxiv.org/html/2606.24999#Thmtheorem15)and Lemma[16](https://arxiv.org/html/2606.24999#Thmtheorem16)that

‖∇2v‖β2\\displaystyle\\\|\\nabla^\{2\}v\\\|\_\{\\beta\}^\{2\}=‖ℒ​v‖β2−∫0Teβ​t​∫ℝd∇v⊤​∇2h​∇v​p​d​x​d​t\\displaystyle=\\\|\\mathcal\{L\}v\\\|\_\{\\beta\}^\{2\}\-\\int\_\{0\}^\{T\}e^\{\\beta t\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla v^\{\\top\}\\nabla^\{2\}h\\nabla vp\\mathrm\{d\}x\\mathrm\{d\}t\(A\.37\)≤2​‖∂tv‖β2\+2​‖δ​f‖β2−∫0Teβ​t​∫ℝd∇v⊤​∇2h​∇v​p​d​x​d​t\\displaystyle\\leq 2\\\|\\partial\_\{t\}v\\\|\_\{\\beta\}^\{2\}\+2\\\|\\delta f\\\|\_\{\\beta\}^\{2\}\-\\int\_\{0\}^\{T\}e^\{\\beta t\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla v^\{\\top\}\\nabla^\{2\}h\\nabla vp\\mathrm\{d\}x\\mathrm\{d\}t≤4​‖δ​f‖β2−2​β​‖∇v‖β2−∫0Teβ​t​∫ℝd∇v⊤​∇2h​∇v​p​d​x​d​t\.\\displaystyle\\leq 4\\\|\\delta f\\\|\_\{\\beta\}^\{2\}\-2\\beta\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\-\\int\_\{0\}^\{T\}e^\{\\beta t\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla v^\{\\top\}\\nabla^\{2\}h\\nabla vp\\mathrm\{d\}x\\mathrm\{d\}t\.Since the eigenvalues of∇2h\\nabla^\{2\}hhave a uniform lower bound−k\-k,∇2h\+k​I\\nabla^\{2\}h\+kIis positive semi\-definite\. Therefore,

∫0Teβ​t​∫ℝd∇v⊤​∇2h​∇v​p​d​x​d​t≥−k​‖∇v‖β2,\\int\_\{0\}^\{T\}e^\{\\beta t\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\nabla v^\{\\top\}\\nabla^\{2\}h\\nabla vp\\mathrm\{d\}x\\mathrm\{d\}t\\geq\-k\\\|\\nabla v\\\|\_\{\\beta\}^\{2\},which yields

‖∇2v‖β2≤4​‖δ​f‖β2−\(2​β−k\)​‖∇v‖β2\.\\\|\\nabla^\{2\}v\\\|\_\{\\beta\}^\{2\}\\leq 4\\\|\\delta f\\\|\_\{\\beta\}^\{2\}\-\(2\\beta\-k\)\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\.\(A\.38\)This proves the desired result\.

We now finish the proof of the sufficient condition by relatingδ​f\\delta fto the distance between the two input triplets\.

Proof\[Proof of Theorem[11](https://arxiv.org/html/2606.24999#Thmtheorem11)\] For the Langevin dynamics considered here, \([A\.12](https://arxiv.org/html/2606.24999#A1.E12)\) holds withλ=2\\lambda=2\. Hence

β​‖v‖β2≤4β​‖δ​f‖β2,β​‖∇v‖β2≤‖δ​f‖β2\.\\beta\\\|v\\\|\_\{\\beta\}^\{2\}\\leq\\frac\{4\}\{\\beta\}\\\|\\delta f\\\|\_\{\\beta\}^\{2\},\\quad\\beta\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\\leq\\\|\\delta f\\\|\_\{\\beta\}^\{2\}\.\(A\.39\)Together with Theorem[17](https://arxiv.org/html/2606.24999#Thmtheorem17), this gives

‖v‖β,mix2\\displaystyle\\\|v\\\|^\{2\}\_\{\\beta,\\textrm\{mix\}\}=β​‖v‖β2\+β​‖∇v‖β2\+‖∇2v‖β2\\displaystyle=\\beta\\\|v\\\|\_\{\\beta\}^\{2\}\+\\beta\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\+\\\|\\nabla^\{2\}v\\\|\_\{\\beta\}^\{2\}\(A\.40\)≤4​‖δ​f‖β2−\(2​β−k\)​‖∇v‖β2\+β​‖∇v‖β2\+4β​‖δ​f‖β2\\displaystyle\\leq 4\\\|\\delta f\\\|\_\{\\beta\}^\{2\}\-\(2\\beta\-k\)\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\+\\beta\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\+\\frac\{4\}\{\\beta\}\\\|\\delta f\\\|\_\{\\beta\}^\{2\}≤4​\(1\+1β\)​3​\(Lv2​‖V1−V2‖β2\+Lg2​‖G1−G2‖β2\+Lh2​‖H1−H2‖β2\)−\(β−k\)​‖∇v‖β2\\displaystyle\\leq 4\(1\+\\frac\{1\}\{\\beta\}\)3\\left\(L\_\{v\}^\{2\}\\\|V\_\{1\}\-V\_\{2\}\\\|\_\{\\beta\}^\{2\}\+L\_\{g\}^\{2\}\\\|G\_\{1\}\-G\_\{2\}\\\|\_\{\\beta\}^\{2\}\+L\_\{h\}^\{2\}\\\|H\_\{1\}\-H\_\{2\}\\\|\_\{\\beta\}^\{2\}\\right\)\-\(\\beta\-k\)\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}≤12​\(\(Lv2\+Lg2\)​β\+1β2\+Lh2​β\+1β\)​‖𝐔1−𝐔2‖β,mix2−\(β−k\)​‖∇v‖β2\.\\displaystyle\\leq 2\\left\(\(L\_\{v\}^\{2\}\+L\_\{g\}^\{2\}\)\\frac\{\\beta\+1\}\{\\beta^\{2\}\}\+L\_\{h\}^\{2\}\\frac\{\\beta\+1\}\{\\beta\}\\right\)\\\|\\mathbf\{U\}\_\{1\}\-\\mathbf\{U\}\_\{2\}\\\|^\{2\}\_\{\\beta,\\textrm\{mix\}\}\-\(\\beta\-k\)\\\|\\nabla v\\\|\_\{\\beta\}^\{2\}\.For anyβ≥k\\beta\\geq k, the last term is nonpositive, and therefore

‖v‖β,mix2≤ηβ​‖𝐔1−𝐔2‖β,mix2,\\\|v\\\|^\{2\}\_\{\\beta,\\textrm\{mix\}\}\\leq\\eta\_\{\\beta\}\\\|\\mathbf\{U\}\_\{1\}\-\\mathbf\{U\}\_\{2\}\\\|^\{2\}\_\{\\beta,\\textrm\{mix\}\},\(A\.41\)whereηβ\\eta\_\{\\beta\}is defined in \([A\.10](https://arxiv.org/html/2606.24999#A1.E10)\)\. Ifηβ<1\\eta\_\{\\beta\}<1, taking square roots gives the contraction inequality withγ=ηβ<1\\gamma=\\sqrt\{\\eta\_\{\\beta\}\}<1\. Finally, if12​Lh2<112L\_\{h\}^\{2\}<1, thenηβ→12​Lh2\\eta\_\{\\beta\}\\to 12L\_\{h\}^\{2\}asβ→∞\\beta\\to\\infty, so such a choice ofβ\\betais available\.

## Appendix BAddition Numerical Results with Weak Simulator

We examine the performance of our approach when only the weak simulator is available on the fully nonlinear PDE example in Subsection[6\.3](https://arxiv.org/html/2606.24999#S6.SS3), as a robustness check\. The results are summarized in Table[5](https://arxiv.org/html/2606.24999#A2.T5)\. Overall, consistent with our theoretical results, the performance of ZOD\-1 is slightly worse than that with ZOD\-m\. Indeed, in this case it is no longer true that the smaller bandwidth the better performance\.

Table 5:Summary of ZOD ablations for the fully nonlinear PDE \([6\.17](https://arxiv.org/html/2606.24999#S6.E17)\)\. Each row reports the mean final rRMSE over three random seeds\.
## Appendix CProof of Statements

### C\.1Proof of Lemma[1](https://arxiv.org/html/2606.24999#Thmtheorem1)

ProofWe prove thed=1d=1case for illustration\. It follows from Taylor’s expansion that

Zϵ​u​\(t,x\+ϵ​Z\)=Zϵ​\(u​\(t,x\)\+∇u​\(t,x\)​ϵ​Z\+12​∇2u​\(t,x\)​ϵ2​Z2\+c​\(t,x,x\+ϵ​Z\)​ϵ3​Z3\)\.\\frac\{Z\}\{\\epsilon\}u\(t,x\+\\epsilon Z\)=\\frac\{Z\}\{\\epsilon\}\\left\(u\(t,x\)\+\\nabla u\(t,x\)\\epsilon Z\+\\frac\{1\}\{2\}\\nabla^\{2\}u\(t,x\)\\epsilon^\{2\}Z^\{2\}\+c\(t,x,x\+\\epsilon Z\)\\epsilon^\{3\}Z^\{3\}\\right\)\.\(C\.1\)Thus,

\|𝔼​\[Zϵ​u​\(t,x\+ϵ​Z\)\]−∇u​\(t,x\)\|\\displaystyle\\left\|\\mathbb\{E\}\\left\[\\frac\{Z\}\{\\epsilon\}u\(t,x\+\\epsilon Z\)\\right\]\-\\nabla u\(t,x\)\\right\|=\|𝔼​\[12​∇2u​\(t,x\)​ϵ​Z3\+c​\(t,x,x\+ϵ​Z\)​ϵ2​Z4\]\|\\displaystyle=\\left\|\\mathbb\{E\}\\left\[\\frac\{1\}\{2\}\\nabla^\{2\}u\(t,x\)\\epsilon Z^\{3\}\+c\(t,x,x\+\\epsilon Z\)\\epsilon^\{2\}Z^\{4\}\\right\]\\right\|\(C\.2\)=\|𝔼​\[c​\(t,x,x\+ϵ​Z\)​Z4\]\|​ϵ2≤C​ϵ2\.\\displaystyle=\\left\|\\mathbb\{E\}\[c\(t,x,x\+\\epsilon Z\)Z^\{4\}\]\\right\|\\epsilon^\{2\}\\leq C\\epsilon^\{2\}\.Similarly, ifuuhas the fourth\-order Taylor expansion, then

Z​Z⊤−Idϵ2​u​\(t,x\+ϵ​Z\)\\displaystyle\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\}u\(t,x\+\\epsilon Z\)\(C\.3\)=Z2−1ϵ2​u​\(t,x\)\+Z3−Zϵ​∇u​\(t,x\)\+12​\(Z4−Z2\)​∇2u​\(t,x\)\\displaystyle=\\frac\{Z^\{2\}\-1\}\{\\epsilon^\{2\}\}u\(t,x\)\+\\frac\{Z^\{3\}\-Z\}\{\\epsilon\}\\nabla u\(t,x\)\+\\frac\{1\}\{2\}\(Z^\{4\}\-Z^\{2\}\)\\nabla^\{2\}u\(t,x\)\+16​ϵ​∇3u​\(t,x\)​\(Z5−Z3\)\+c′​\(t,x,x\+ϵ​Z\)​\(Z6−Z4\)​ϵ2\.\\displaystyle\\quad\\quad\\quad\+\\frac\{1\}\{6\}\\epsilon\\nabla^\{3\}u\(t,x\)\(Z^\{5\}\-Z^\{3\}\)\+c^\{\\prime\}\(t,x,x\+\\epsilon Z\)\(Z^\{6\}\-Z^\{4\}\)\\epsilon^\{2\}\.Consequently,

\|𝔼​\[Z​Z⊤−Idϵ2​u​\(t,x\+ϵ​Z\)\]−∇2u​\(t,x\)\|=\|𝔼​\[c′​\(t,x,x\+ϵ​Z\)​\(Z6−Z4\)\]\|​ϵ2≤C​ϵ2\.\\left\|\\mathbb\{E\}\\left\[\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\}u\(t,x\+\\epsilon Z\)\\right\]\-\\nabla^\{2\}u\(t,x\)\\right\|=\\left\|\\mathbb\{E\}\[c^\{\\prime\}\(t,x,x\+\\epsilon Z\)\(Z^\{6\}\-Z^\{4\}\)\]\\right\|\\epsilon^\{2\}\\leq C\\epsilon^\{2\}\.\(C\.4\)The variance parts follow from \([C\.1](https://arxiv.org/html/2606.24999#A3.E1)\), \([C\.3](https://arxiv.org/html/2606.24999#A3.E3)\) and the fact thatc,c′c,c^\{\\prime\}have polynomial growth\.

### C\.2Proof of Lemma[2](https://arxiv.org/html/2606.24999#Thmtheorem2)

ProofFor notational simplicity, we suppress the variablett\. The Taylor expansions ofu​\(x±ϵ​Z\)u\(x\\pm\\epsilon Z\)are:

u​\(x\+ϵ​Z\)\\displaystyle u\(x\+\\epsilon Z\)=u​\(x\)\+ϵ​∇u⊤​Z\+ϵ22​Z⊤​∇2u​\(x\)​Z\+O​\(ϵ3\),\\displaystyle=u\(x\)\+\\epsilon\\nabla u^\{\\top\}Z\+\\frac\{\\epsilon^\{2\}\}\{2\}Z^\{\\top\}\\nabla^\{2\}u\(x\)Z\+O\(\\epsilon^\{3\}\),u​\(x−ϵ​Z\)\\displaystyle u\(x\-\\epsilon Z\)=u​\(x\)−ϵ​∇u⊤​Z\+ϵ22​Z⊤​∇2u​\(x\)​Z\+O​\(ϵ3\)\.\\displaystyle=u\(x\)\-\\epsilon\\nabla u^\{\\top\}Z\+\\frac\{\\epsilon^\{2\}\}\{2\}Z^\{\\top\}\\nabla^\{2\}u\(x\)Z\+O\(\\epsilon^\{3\}\)\.Subtracting the second expansion from the first cancels all even\-order terms:

u​\(x\+ϵ​Z\)−u​\(x−ϵ​Z\)=2​ϵ​∇u⊤​Z\+O​\(ϵ3\)\.u\(x\+\\epsilon Z\)\-u\(x\-\\epsilon Z\)=2\\epsilon\\nabla u^\{\\top\}Z\+O\(\\epsilon^\{3\}\)\.Substituting this into the estimator definition in Eq\. \([3\.2](https://arxiv.org/html/2606.24999#S3.E2)\):

g^​\(x;ϵ\)=2​ϵ​∇u⊤​Z\+O​\(ϵ3\)2​ϵ​Z=\(∇u⊤​Z\)​Z\+O​\(ϵ2\)\.\\hat\{g\}\(x;\\epsilon\)=\\frac\{2\\epsilon\\nabla u^\{\\top\}Z\+O\(\\epsilon^\{3\}\)\}\{2\\epsilon\}Z=\(\\nabla u^\{\\top\}Z\)Z\+O\(\\epsilon^\{2\}\)\.The expectation is𝔼​\[\(∇u⊤​Z\)​Z\]=∇u\\mathbb\{E\}\[\(\\nabla u^\{\\top\}Z\)Z\]=\\nabla u, since𝔼​\[Z​Z⊤\]=Id\\mathbb\{E\}\[ZZ^\{\\top\}\]=I\_\{d\}\. The bias is therefore dominated by theO​\(ϵ2\)O\(\\epsilon^\{2\}\)term\.

For the variance, observe that the leading term of the estimator,\(∇u⊤​Z\)​Z\(\\nabla u^\{\\top\}Z\)Z, is independent ofϵ\\epsilon\. Its variance is a constant determined by the moments ofZZand the magnitude of∇u\\nabla u:

limϵ→0Var⁡\[g^​\(x;ϵ\)\]=Var⁡\[\(∇u⊤​Z\)​Z\]\.\\lim\_\{\\epsilon\\to 0\}\\operatorname\{Var\}\[\\hat\{g\}\(x;\\epsilon\)\]=\\operatorname\{Var\}\[\(\\nabla u^\{\\top\}Z\)Z\]\.This variance is finite and does not depend onϵ\\epsilon\. Thus, the total variance is bounded\.

### C\.3Proof of Lemma[3](https://arxiv.org/html/2606.24999#Thmtheorem3)

ProofWe suppress the variablett\. Using the Taylor expansions up to the fourth order, the central difference numerator becomes:

u​\(x\+ϵ​Z\)−2​u​\(x\)\+u​\(x−ϵ​Z\)=ϵ2​Z⊤​∇2u​Z\+ϵ412​∑1≤i,j,k,l≤d∂4u​\(x\)∂xi​∂xj​∂xk​∂xl​Zi​Zj​Zk​Zl\+o​\(ϵ4\)\.u\(x\+\\epsilon Z\)\-2u\(x\)\+u\(x\-\\epsilon Z\)=\\epsilon^\{2\}Z^\{\\top\}\\nabla^\{2\}uZ\+\\frac\{\\epsilon^\{4\}\}\{12\}\\sum\_\{1\\leq i,j,k,l\\leq d\}\\frac\{\\partial^\{4\}u\(x\)\}\{\\partial x\_\{i\}\\partial x\_\{j\}\\partial x\_\{k\}\\partial x\_\{l\}\}Z\_\{i\}Z\_\{j\}Z\_\{k\}Z\_\{l\}\+o\(\\epsilon^\{4\}\)\.The symmetric construction cancels all the odd\-order terms\. Substituting this into Eq\. \([3\.4](https://arxiv.org/html/2606.24999#S3.E4)\) yields:

H^​\(x;ϵ\)=\(12​Z⊤​∇2u​Z\+ϵ224​∑1≤i,j,k,l≤d∂4u​\(x\)∂xi​∂xj​∂xk​∂xl​Zi​Zj​Zk​Zl\+o​\(ϵ2\)\)​\(Z​Z⊤−Id\)\.\\hat\{H\}\(x;\\epsilon\)=\\left\(\\frac\{1\}\{2\}Z^\{\\top\}\\nabla^\{2\}uZ\+\\frac\{\\epsilon^\{2\}\}\{24\}\\sum\_\{1\\leq i,j,k,l\\leq d\}\\frac\{\\partial^\{4\}u\(x\)\}\{\\partial x\_\{i\}\\partial x\_\{j\}\\partial x\_\{k\}\\partial x\_\{l\}\}Z\_\{i\}Z\_\{j\}Z\_\{k\}Z\_\{l\}\+o\(\\epsilon^\{2\}\)\\right\)\(ZZ^\{\\top\}\-I\_\{d\}\)\.\(C\.5\)Thus, the variance is bounded whenϵ\\epsilontends to0\. For the bias part, we take the expectation of the leading term\. Using the property of Gaussian moments that𝔼​\[\(Z⊤​∇2u​Z\)​\(Z​Z⊤−Id\)\]=2​H\\mathbb\{E\}\[\(Z^\{\\top\}\\nabla^\{2\}uZ\)\(ZZ^\{\\top\}\-I\_\{d\}\)\]=2Hfor a symmetric matrix∇2u\\nabla^\{2\}u\(derived from Isserlis’ theorem\), we find

𝔼​\[H^​\(x;ϵ\)\]\\displaystyle\\mathbb\{E\}\[\\hat\{H\}\(x;\\epsilon\)\]=12​𝔼​\[\(Z⊤​∇2u​Z\)​\(Z​Z⊤−Id\)\]\+O​\(ϵ2\)\\displaystyle=\\frac\{1\}\{2\}\\mathbb\{E\}\\left\[\(Z^\{\\top\}\\nabla^\{2\}uZ\)\(ZZ^\{\\top\}\-I\_\{d\}\)\\right\]\+O\(\\epsilon^\{2\}\)=12​\(2​H\)\+O​\(ϵ2\)=∇2u\+O​\(ϵ2\)\.\\displaystyle=\\frac\{1\}\{2\}\(2H\)\+O\(\\epsilon^\{2\}\)=\\nabla^\{2\}u\+O\(\\epsilon^\{2\}\)\.The bias is therefore of orderO​\(ϵ2\)O\(\\epsilon^\{2\}\)\.

### C\.4Proof of Proposition[5](https://arxiv.org/html/2606.24999#Thmtheorem5)

ProofThe proof of the results for the weak simulator case is the same as the Proposition[4](https://arxiv.org/html/2606.24999#Thmtheorem4)\. We only prove the strong simulator case\.

For simplicity, we useR​\(t,x\)R\(t,x\)to representRt,x​\(𝐔n\)R^\{t,x\}\(\\mathbf\{U\}\_\{n\}\)\. We only prove the variance result forh^\\hat\{h\}; the other parts are similar\. Due to the regularity conditions ofbbandσ\\sigma, it follows fromProtter \([2005](https://arxiv.org/html/2606.24999#bib.bib34), Theorem 40\)andKunita \([2019](https://arxiv.org/html/2606.24999#bib.bib33), Theorem 3\.4\.2\)thatXst,xX\_\{s\}^\{t,x\}is four times continuously differentiable inxx, and for everyp\>2p\>2, there is a constantCp\>0C\_\{p\}\>0such that for any multi\-index𝒊\\boldsymbol\{i\}with0<\|𝒊\|≤40<\|\\boldsymbol\{i\}\|\\leq 4, it holds that

supx∈ℝd,0≤t≤T𝔼​\[supt≤s≤T\|∂x𝒊Xst,x\|p\]≤Cp\.\\sup\_\{x\\in\\mathbb\{R\}^\{d\},\\ 0\\leq t\\leq T\}\\mathbb\{E\}\\left\[\\sup\_\{t\\leq s\\leq T\}\|\\partial^\{\\boldsymbol\{i\}\}\_\{x\}X\_\{s\}^\{t,x\}\|^\{p\}\\right\]\\leq C\_\{p\}\.\(C\.6\)Therefore,R​\(t,x\)R\(t,x\)is also four times continuously differentiable inxxand admits the Taylor expansion

R​\(t,x\+ϵ​Z\)\\displaystyle R\(t,x\+\\epsilon Z\)=R​\(t,x\)\+ϵ​∇R​\(t,x\)⊤​Z\+ϵ22​Z⊤​∇2R​\(t,x\)​Z\+ϵ36​∇3R​\(t,x\)​\(Z,Z,Z\)\\displaystyle=R\(t,x\)\+\\epsilon\\nabla R\(t,x\)^\{\\top\}Z\+\\frac\{\\epsilon^\{2\}\}\{2\}Z^\{\\top\}\\nabla^\{2\}R\(t,x\)Z\+\\frac\{\\epsilon^\{3\}\}\{6\}\\nabla^\{3\}R\(t,x\)\(Z,Z,Z\)\(C\.7\)\+ϵ424​∇4R​\(t,x\+θ1​ϵ​Z\)​\(Z,Z,Z,Z\),\\displaystyle\\qquad\+\\frac\{\\epsilon^\{4\}\}\{24\}\\nabla^\{4\}R\(t,x\+\\theta\_\{1\}\\epsilon Z\)\(Z,Z,Z,Z\),R​\(t,x−ϵ​Z\)\\displaystyle R\(t,x\-\\epsilon Z\)=R​\(t,x\)−ϵ​∇R​\(t,x\)⊤​Z\+ϵ22​Z⊤​∇2R​\(t,x\)​Z−ϵ36​∇3R​\(t,x\)​\(Z,Z,Z\)\\displaystyle=R\(t,x\)\-\\epsilon\\nabla R\(t,x\)^\{\\top\}Z\+\\frac\{\\epsilon^\{2\}\}\{2\}Z^\{\\top\}\\nabla^\{2\}R\(t,x\)Z\-\\frac\{\\epsilon^\{3\}\}\{6\}\\nabla^\{3\}R\(t,x\)\(Z,Z,Z\)\+ϵ424​∇4R​\(t,x−θ2​ϵ​Z\)​\(Z,Z,Z,Z\),\\displaystyle\\qquad\+\\frac\{\\epsilon^\{4\}\}\{24\}\\nabla^\{4\}R\(t,x\-\\theta\_\{2\}\\epsilon Z\)\(Z,Z,Z,Z\),whereθ1,θ2∈\(0,1\)\\theta\_\{1\},\\theta\_\{2\}\\in\(0,1\)and

∇3R​\(t,x\)​\(Z,Z,Z\)\\displaystyle\\nabla^\{3\}R\(t,x\)\(Z,Z,Z\):=∑1≤i,j,k≤d∂x3R​\(t,x\)∂xi​∂xj​∂xk​Zi​Zj​Zk,\\displaystyle=\\sum\_\{1\\leq i,j,k\\leq d\}\\frac\{\\partial^\{3\}\_\{x\}R\(t,x\)\}\{\\partial x\_\{i\}\\partial x\_\{j\}\\partial x\_\{k\}\}Z\_\{i\}Z\_\{j\}Z\_\{k\},\(C\.8\)∇4R​\(t,x\)​\(Z,Z,Z,Z\)\\displaystyle\\nabla^\{4\}R\(t,x\)\(Z,Z,Z,Z\):=∑1≤i,j,k,l≤d∂x4R​\(t,x\)∂xi​∂xj​∂xk​∂xl​Zi​Zj​Zk​Zl\.\\displaystyle=\\sum\_\{1\\leq i,j,k,l\\leq d\}\\frac\{\\partial^\{4\}\_\{x\}R\(t,x\)\}\{\\partial x\_\{i\}\\partial x\_\{j\}\\partial x\_\{k\}\\partial x\_\{l\}\}Z\_\{i\}Z\_\{j\}Z\_\{k\}Z\_\{l\}\.Denoteh^​\(t,x\):=Z​Z⊤−Id2​ϵ2​\(R​\(t,x\+ϵ​Z\)\+R​\(t,x−ϵ​Z\)−2​R​\(t,x\)\)\\hat\{h\}\(t,x\):=\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{2\\epsilon^\{2\}\}\\left\(R\(t,x\+\\epsilon Z\)\+R\(t,x\-\\epsilon Z\)\-2R\(t,x\)\\right\)\. Then

h^​\(t,x\)=\\displaystyle\\hat\{h\}\(t,x\)=Z​Z⊤−Id2​Z⊤​∇2R​\(t,x\)​Z\\displaystyle\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{2\}Z^\{\\top\}\\nabla^\{2\}R\(t,x\)Z\(C\.9\)\+ϵ248​\(Z​Z⊤−Id\)​\[∇4R​\(t,x\+θ1​ϵ​Z\)\+∇4R​\(t,x−θ2​ϵ​Z\)\]​\(Z,Z,Z,Z\)\.\\displaystyle\+\\frac\{\\epsilon^\{2\}\}\{48\}\(ZZ^\{\\top\}\-I\_\{d\}\)\\left\[\\nabla^\{4\}R\(t,x\+\\theta\_\{1\}\\epsilon Z\)\+\\nabla^\{4\}R\(t,x\-\\theta\_\{2\}\\epsilon Z\)\\right\]\(Z,Z,Z,Z\)\.It follows from the regularity conditions ofg,f~g,\\tilde\{f\}and \([C\.6](https://arxiv.org/html/2606.24999#A3.E6)\) that for everyp\>0p\>0there are constantsCp\>0C\_\{p\}\>0andq\>0q\>0such that for allt,x∈\[0,T\]×ℝdt,x\\in\[0,T\]\\times\\mathbb\{R\}^\{d\},

𝔼​\[\|∇2R​\(t,x\)\|p\]≤Cp​\(1\+\|x\|q\),𝔼​\[\|∇4R​\(t,x\)\|p\]≤C​\(1\+\|x\|q\)\.\\mathbb\{E\}\\left\[\|\\nabla^\{2\}R\(t,x\)\|^\{p\}\\right\]\\leq C\_\{p\}\(1\+\|x\|^\{q\}\),\\quad\\mathbb\{E\}\\left\[\|\\nabla^\{4\}R\(t,x\)\|^\{p\}\\right\]\\leq C\(1\+\|x\|^\{q\}\)\.\(C\.10\)This along with the moments bound ofXtX\_\{t\}yields

𝔼​\[\|h^\|2\]\\displaystyle\\mathbb\{E\}\[\|\\hat\{h\}\|^\{2\}\]≤2​𝔼​\[\|Z​Z⊤−Id2​Z⊤​∇2R​\(t,Xt\)​Z\|2\]\\displaystyle\\leq 2\\mathbb\{E\}\\left\[\\left\|\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{2\}Z^\{\\top\}\\nabla^\{2\}R\(t,X\_\{t\}\)Z\\right\|^\{2\}\\right\]\(C\.11\)\+2​𝔼​\[\|ϵ248​\(Z​Z⊤−Id\)​\[∇4R​\(t,Xt\+θ1​ϵ​Z\)\+∇4R​\(t,Xt−θ2​ϵ​Z\)\]​\(Z,Z,Z,Z\)\|2\]\\displaystyle\\qquad\+2\\mathbb\{E\}\\left\[\\left\|\\frac\{\\epsilon^\{2\}\}\{48\}\(ZZ^\{\\top\}\-I\_\{d\}\)\\left\[\\nabla^\{4\}R\(t,X\_\{t\}\+\\theta\_\{1\}\\epsilon Z\)\+\\nabla^\{4\}R\(t,X\_\{t\}\-\\theta\_\{2\}\\epsilon Z\)\\right\]\(Z,Z,Z,Z\)\\right\|^\{2\}\\right\]≤𝔼​\[\|Z​Z⊤−Id\|2​\|∇2R​\(t,Xt\)\|2​\|Z\|4\]\\displaystyle\\leq\\mathbb\{E\}\\left\[\\left\|ZZ^\{\\top\}\-I\_\{d\}\\right\|^\{2\}\\left\|\\nabla^\{2\}R\(t,X\_\{t\}\)\\right\|^\{2\}\\left\|Z\\right\|^\{4\}\\right\]\+ϵ224​𝔼​\[\|Z​Z⊤−Id\|2​\|∇4R​\(t,Xt\+θ1​ϵ​Z\)\+∇4R​\(t,Xt−θ2​ϵ​Z\)\|2​\|Z\|8\]\\displaystyle\\qquad\+\\frac\{\\epsilon^\{2\}\}\{24\}\\mathbb\{E\}\\left\[\\left\|ZZ^\{\\top\}\-I\_\{d\}\\right\|^\{2\}\\left\|\\nabla^\{4\}R\(t,X\_\{t\}\+\\theta\_\{1\}\\epsilon Z\)\+\\nabla^\{4\}R\(t,X\_\{t\}\-\\theta\_\{2\}\\epsilon Z\)\\right\|^\{2\}\\left\|Z\\right\|^\{8\}\\right\]≤C,\\displaystyle\\leq C,for some constantC\>0C\>0\. This proves the desired result\.

### C\.5Proof of Theorem[6](https://arxiv.org/html/2606.24999#Thmtheorem6)

ProofBy the triangle inequality, we decompose the total error at stepnninto the computational error introduced at the current step and the propagated error from the previous step:

‖𝐔n−𝐮∗‖β,mix≤‖𝐔n−𝒮​\(𝐔n−1\)‖β,mix\+‖𝒮​\(𝐔n−1\)−𝐮∗‖β,mix\.\\\|\\mathbf\{U\}\_\{n\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}\\leq\\\|\\mathbf\{U\}\_\{n\}\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\\\|\_\{\\beta,\\text\{mix\}\}\+\\\|\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}\.\(C\.12\)The first term on the right\-hand side is exactly the definition of the one\-step errorEnE\_\{n\}\. For the second term, using the fixed\-point property𝐮∗=𝒮​\(𝐮∗\)\\mathbf\{u\}^\{\*\}=\\mathcal\{S\}\(\\mathbf\{u\}^\{\*\}\)and the Contraction Assumption[2](https://arxiv.org/html/2606.24999#Thmassumption2), we have:

‖𝒮​\(𝐔n−1\)−𝐮∗‖β,mix=‖𝒮​\(𝐔n−1\)−𝒮​\(𝐮∗\)‖β,mix≤γ​‖𝐔n−1−𝐮∗‖β,mix\.\\\|\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}=\\\|\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\-\\mathcal\{S\}\(\\mathbf\{u\}^\{\*\}\)\\\|\_\{\\beta,\\text\{mix\}\}\\leq\\gamma\\\|\\mathbf\{U\}\_\{n\-1\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}\.\(C\.13\)Combining these inequalities yields the recursive relationship:

‖𝐔n−𝐮∗‖β,mix≤γ​‖𝐔n−1−𝐮∗‖β,mix\+En\.\\\|\\mathbf\{U\}\_\{n\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}\\leq\\gamma\\\|\\mathbf\{U\}\_\{n\-1\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}\+E\_\{n\}\.\(C\.14\)Thus,

‖𝐔n−𝐮∗‖β,mix\\displaystyle\\\|\\mathbf\{U\}\_\{n\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}≤γ​\(γ​‖𝐔n−2−𝐮∗‖β,mix\+En−1\)\+En\\displaystyle\\leq\\gamma\\left\(\\gamma\\\|\\mathbf\{U\}\_\{n\-2\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}\+E\_\{n\-1\}\\right\)\+E\_\{n\}=γ2​‖𝐔n−2−𝐮∗‖β,mix\+γ​En−1\+En\\displaystyle=\\gamma^\{2\}\\\|\\mathbf\{U\}\_\{n\-2\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}\+\\gamma E\_\{n\-1\}\+E\_\{n\}≤γn​‖𝐔0−𝐮∗‖β,mix\+∑k=1nγn−k​Ek\.\\displaystyle\\leq\\gamma^\{n\}\\\|\\mathbf\{U\}\_\{0\}\-\\mathbf\{u\}^\{\*\}\\\|\_\{\\beta,\\text\{mix\}\}\+\\sum\_\{k=1\}^\{n\}\\gamma^\{n\-k\}E\_\{k\}\.This proves the desired result\.

### C\.6Proof of Lemma[7](https://arxiv.org/html/2606.24999#Thmtheorem7)

ProofWe derive the result for the gradient component only; the others follow similarly\. LetYg=Zϵ​Rt,Xt\+ϵ​ZY\_\{g\}=\\frac\{Z\}\{\\epsilon\}R^\{t,X\_\{t\}\+\\epsilon Z\}denote the stochastic gradient target\. The gradient loss is:

Lg​\(φg,𝐔n−1\)=β​𝔼​\[∫0Teβ​t​\|Yg−φg​\(t,Xt\)\|2​dt\]\.L\_\{g\}\(\\varphi\_\{g\},\\mathbf\{U\}\_\{n\-1\}\)=\\beta\\mathbb\{E\}\\left\[\\int\_\{0\}^\{T\}e^\{\\beta t\}\|Y\_\{g\}\-\\varphi\_\{g\}\(t,X\_\{t\}\)\|^\{2\}\\mathrm\{d\}t\\right\]\.\(C\.15\)Note that

𝔼​\[\|φg​\(t,Xt\)−Yg\|2\]=𝔼​\[𝔼​\[\|φg​\(t,Xt\)−Gϵ​\(t,Xt\)\+Gϵ​\(t,Xt\)−Yg\|2\|t,Xt\]\]\.\\displaystyle\\mathbb\{E\}\\left\[\\left\|\\varphi\_\{g\}\(t,X\_\{t\}\)\-Y\_\{g\}\\right\|^\{2\}\\right\]=\\mathbb\{E\}\\left\[\\mathbb\{E\}\\left\[\|\\varphi\_\{g\}\(t,X\_\{t\}\)\-G^\{\\epsilon\}\(t,X\_\{t\}\)\+G^\{\\epsilon\}\(t,X\_\{t\}\)\-Y\_\{g\}\|^\{2\}\\big\|t,X\_\{t\}\\right\]\\right\]\.SinceGϵ​\(t,Xt\)=𝔼​\[Yg\|t,Xt\]G^\{\\epsilon\}\(t,X\_\{t\}\)=\\mathbb\{E\}\[Y\_\{g\}\|t,X\_\{t\}\], the cross\-term in the expansion of the square above vanishes\. Thus

Lg​\(φg,𝐔n−1\)\\displaystyle L\_\{g\}\(\\varphi\_\{g\},\\mathbf\{U\}\_\{n\-1\}\)=β​𝔼​\[∫0Teβ​t​\|φg​\(t,Xt\)−Gϵ​\(t,Xt\)\|2​dt\]\+β​𝔼​\[∫0Teβ​t​\|Gϵ​\(t,Xt\)−Yg\|2​dt\]\\displaystyle=\\beta\\mathbb\{E\}\\left\[\\int\_\{0\}^\{T\}e^\{\\beta t\}\|\\varphi\_\{g\}\(t,X\_\{t\}\)\-G^\{\\epsilon\}\(t,X\_\{t\}\)\|^\{2\}\\mathrm\{d\}t\\right\]\+\\beta\\mathbb\{E\}\\left\[\\int\_\{0\}^\{T\}e^\{\\beta t\}\|G^\{\\epsilon\}\(t,X\_\{t\}\)\-Y\_\{g\}\|^\{2\}\\mathrm\{d\}t\\right\]=:β∥φg−Gϵ∥β2\+C,\\displaystyle=:\\beta\\\|\\varphi\_\{g\}\-G^\{\\epsilon\}\\\|\_\{\\beta\}^\{2\}\+C,whereCCis a constant independent ofφ\\varphi\. Summing the components forV,G,HV,G,Hyields the result\.

### C\.7Proof of Theorem[9](https://arxiv.org/html/2606.24999#Thmtheorem9)

We first prove Theorem[9](https://arxiv.org/html/2606.24999#Thmtheorem9)\-\(a\), the case with the weak simulator\. For simplicity, we suppress the notation𝐔n−1\\mathbf\{U\}\_\{n\-1\}and useL​\(φ\),L~\(B\)​\(φ\)L\(\\varphi\),\\tilde\{L\}^\{\(B\)\}\(\\varphi\)for total loss and empirical total loss, respectively\. The following lemma provides the upper bound ofEnE\_\{n\}in terms of the statistical error, approximation error and the bias due to the ZOD estimator\.

###### Lemma 18

The one step approximation errorEn=‖𝐔n−𝒮​\(𝐔n−1\)‖β,mixE\_\{n\}=\\left\\\|\\mathbf\{U\}\_\{n\}\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}has the upper bound

En2≤4​supφ∈ℋ\|L​\(φ\)−L~\(B\)​\(φ\)\|\+2​infφ∈ℋ‖φ−𝒮ϵ​\(𝐔n−1\)‖β,mix2\+2​‖𝒮ϵ​\(𝐔n−1\)−𝒮​\(𝐔n−1\)‖β,mix2\.E\_\{n\}^\{2\}\\leq 4\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\|L\(\\varphi\)\-\\tilde\{L\}^\{\(B\)\}\(\\varphi\)\|\+2\\inf\_\{\\varphi\\in\\mathcal\{H\}\}\\left\\\|\\varphi\-\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}\+2\\left\\\|\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}\.\(C\.16\)

ProofDenote𝐔nℋ∈arg⁡minφ∈ℋ⁡L​\(φ\)\\mathbf\{U\}^\{\\mathcal\{H\}\}\_\{n\}\\in\\arg\\min\_\{\\varphi\\in\\mathcal\{H\}\}L\(\\varphi\)\. It follows from the decomposition in Lemma[7](https://arxiv.org/html/2606.24999#Thmtheorem7)that

En2\\displaystyle E\_\{n\}^\{2\}=‖𝐔n−𝒮​\(𝐔n−1\)‖β,mix2\\displaystyle=\\left\\\|\\mathbf\{U\}\_\{n\}\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}\(C\.17\)≤2​‖𝐔n−𝒮ϵ​\(𝐔n−1\)‖β,mix2\+2​‖𝒮ϵ​\(𝐔n−1\)−𝒮​\(𝐔n−1\)‖β,mix2\\displaystyle\\leq 2\\left\\\|\\mathbf\{U\}\_\{n\}\-\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}\+2\\left\\\|\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}≤2​\(L​\(𝐔n\)−L​\(𝒮ϵ​\(𝐔n−1\)\)\)\+2​‖𝒮ϵ​\(𝐔n−1\)−𝒮​\(𝐔n−1\)‖β,mix2\\displaystyle\\leq 2\(L\(\\mathbf\{U\}\_\{n\}\)\-L\(\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\)\)\+2\\left\\\|\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}≤2​\(L​\(𝐔n\)−L​\(𝐔nℋ\)\)\+2​\(L​\(𝐔nℋ\)−L​\(𝒮ϵ​\(𝐔n−1\)\)\)\+2​‖𝒮ϵ​\(𝐔n−1\)−𝒮​\(𝐔n−1\)‖β,mix2\\displaystyle\\leq 2\(L\(\\mathbf\{U\}\_\{n\}\)\-L\(\\mathbf\{U\}^\{\\mathcal\{H\}\}\_\{n\}\)\)\+2\(L\(\\mathbf\{U\}^\{\\mathcal\{H\}\}\_\{n\}\)\-L\(\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\)\)\+2\\left\\\|\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}=2​\(L​\(𝐔n\)−L​\(𝐔nℋ\)\)\+2​infφ∈ℋ‖φ−𝒮ϵ​\(𝐔n−1\)‖β,mix2\+2​‖𝒮ϵ​\(𝐔n−1\)−𝒮​\(𝐔n−1\)‖β,mix2,\\displaystyle=2\(L\(\\mathbf\{U\}\_\{n\}\)\-L\(\\mathbf\{U\}^\{\\mathcal\{H\}\}\_\{n\}\)\)\+2\\inf\_\{\\varphi\\in\\mathcal\{H\}\}\\left\\\|\\varphi\-\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}\+2\\left\\\|\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\},where in the last equality we used the factL​\(𝐔nℋ\)−L​\(𝒮ϵ​\(𝐔n−1\)\)=‖𝐔nℋ−𝒮ϵ​\(𝐔n−1\)‖β,mix2=infφ∈ℋ‖φ−𝒮ϵ​\(𝐔n−1\)‖β,mix2L\(\\mathbf\{U\}^\{\\mathcal\{H\}\}\_\{n\}\)\-L\(\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\)=\\left\\\|\\mathbf\{U\}^\{\\mathcal\{H\}\}\_\{n\}\-\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}=\\inf\_\{\\varphi\\in\\mathcal\{H\}\}\\left\\\|\\varphi\-\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}\. For the first term on the right hand side of the above equation, note that

L​\(𝐔n\)−L​\(𝐔nℋ\)\\displaystyle L\(\\mathbf\{U\}\_\{n\}\)\-L\(\\mathbf\{U\}^\{\\mathcal\{H\}\}\_\{n\}\)=L​\(𝐔n\)−L~\(B\)​\(𝐔n\)\+L~\(B\)​\(𝐔n\)−L~\(B\)​\(𝐔nℋ\)\+L~\(B\)​\(𝐔nℋ\)−L​\(𝐔nℋ\)\\displaystyle=L\(\\mathbf\{U\}\_\{n\}\)\-\\tilde\{L\}^\{\(B\)\}\(\\mathbf\{U\}\_\{n\}\)\+\\tilde\{L\}^\{\(B\)\}\(\\mathbf\{U\}\_\{n\}\)\-\\tilde\{L\}^\{\(B\)\}\(\\mathbf\{U\}^\{\\mathcal\{H\}\}\_\{n\}\)\+\\tilde\{L\}^\{\(B\)\}\(\\mathbf\{U\}^\{\\mathcal\{H\}\}\_\{n\}\)\-L\(\\mathbf\{U\}^\{\\mathcal\{H\}\}\_\{n\}\)≤L​\(𝐔n\)−L~\(B\)​\(𝐔n\)\+L~\(B\)​\(𝐔nℋ\)−L​\(𝐔nℋ\)\\displaystyle\\leq L\(\\mathbf\{U\}\_\{n\}\)\-\\tilde\{L\}^\{\(B\)\}\(\\mathbf\{U\}\_\{n\}\)\+\\tilde\{L\}^\{\(B\)\}\(\\mathbf\{U\}^\{\\mathcal\{H\}\}\_\{n\}\)\-L\(\\mathbf\{U\}^\{\\mathcal\{H\}\}\_\{n\}\)≤2​supφ∈ℋ\|L~\(B\)​\(φ\)−L​\(φ\)\|,\\displaystyle\\leq 2\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\|\\tilde\{L\}^\{\(B\)\}\(\\varphi\)\-L\(\\varphi\)\|,where the first inequality follows fromL~\(B\)​\(𝐔n\)−L~\(B\)​\(𝐔nℋ\)≤0\\tilde\{L\}^\{\(B\)\}\(\\mathbf\{U\}\_\{n\}\)\-\\tilde\{L\}^\{\(B\)\}\(\\mathbf\{U\}^\{\\mathcal\{H\}\}\_\{n\}\)\\leq 0due to𝐔n∈arg⁡minφ∈ℋ⁡L~\(B\)​\(φ\)\\mathbf\{U\}\_\{n\}\\in\\arg\\min\_\{\\varphi\\in\\mathcal\{H\}\}\\tilde\{L\}^\{\(B\)\}\(\\varphi\)\. This along with \([C\.17](https://arxiv.org/html/2606.24999#A3.E17)\) yields the desired result\.

##### The ZOD error

It follows from Assumption[4](https://arxiv.org/html/2606.24999#Thmassumption4)and Proposition[4](https://arxiv.org/html/2606.24999#Thmtheorem4)that

‖𝒮ϵ​\(𝐔n−1\)−𝒮​\(𝐔n−1\)‖β,mix2≤C​ϵ4\.\\left\\\|\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}\\leq C\\epsilon^\{4\}\.\(C\.18\)

##### The statistical error

Now we focus on the statistical errorsupφ∈ℋ\|L​\(φ\)−L~\(B\)​\(φ\)\|\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\|L\(\\varphi\)\-\\tilde\{L\}^\{\(B\)\}\(\\varphi\)\|\. Denote the time discretization version of the loss function with respect to the time grid𝒢:=\{0=t0<t1<⋯<tN=T\}\\mathscr\{G\}:=\\left\\\{0=t\_\{0\}<t\_\{1\}<\\cdots<t\_\{N\}=T\\right\\\}as

L𝒢​\(φ\):=Lv𝒢​\(φv\)\+Lg𝒢​\(φg\)\+Lh𝒢​\(φh\)L^\{\\mathscr\{G\}\}\(\\varphi\):=L^\{\\mathscr\{G\}\}\_\{v\}\(\\varphi\_\{v\}\)\+L^\{\\mathscr\{G\}\}\_\{g\}\(\\varphi\_\{g\}\)\+L^\{\\mathscr\{G\}\}\_\{h\}\(\\varphi\_\{h\}\)\(C\.19\)where

Lv𝒢​\(φv\)\\displaystyle L\_\{v\}^\{\\mathscr\{G\}\}\(\\varphi\_\{v\}\):=β​𝔼​\[∑j=0N−1eβ​tj​\(φv​\(tj,Xtj\)−R~tj,Xtj​\(𝐔n−1\)\)2​Δ​tj\],\\displaystyle=\\beta\\mathbb\{E\}\\left\[\\sum\_\{j=0\}^\{N\-1\}e^\{\\beta t\_\{j\}\}\\left\(\\varphi\_\{v\}\(t\_\{j\},X\_\{t\_\{j\}\}\)\-\\tilde\{R\}^\{t\_\{j\},X\_\{t\_\{j\}\}\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\)^\{2\}\\Delta t\_\{j\}\\right\],\(C\.20\)Lg𝒢​\(φg\)\\displaystyle L\_\{g\}^\{\\mathscr\{G\}\}\(\\varphi\_\{g\}\):=β​𝔼​\[∑j=0N−1eβ​tj​\|φg​\(tj,Xtj\)−Zϵ​R~tj,Xtj\+ϵ​Z​\(𝐔n−1\)\|2​Δ​tj\],\\displaystyle=\\beta\\mathbb\{E\}\\left\[\\sum\_\{j=0\}^\{N\-1\}e^\{\\beta t\_\{j\}\}\\left\|\\varphi\_\{g\}\(t\_\{j\},X\_\{t\_\{j\}\}\)\-\\frac\{Z\}\{\\epsilon\}\\tilde\{R\}^\{t\_\{j\},X\_\{t\_\{j\}\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\|^\{2\}\\Delta t\_\{j\}\\right\],Lh𝒢​\(φh\)\\displaystyle L\_\{h\}^\{\\mathscr\{G\}\}\(\\varphi\_\{h\}\):=𝔼​\[∑j=0N−1eβ​tj​\|φh​\(tj,Xtj\)−Z​Z⊤−Idϵ2​R~tj,Xtj\+ϵ​Z​\(𝐔n−1\)\|2​Δ​tj\]\.\\displaystyle=\\mathbb\{E\}\\left\[\\sum\_\{j=0\}^\{N\-1\}e^\{\\beta t\_\{j\}\}\\left\|\\varphi\_\{h\}\(t\_\{j\},X\_\{t\_\{j\}\}\)\-\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\}\\tilde\{R\}^\{t\_\{j\},X\_\{t\_\{j\}\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\|^\{2\}\\Delta t\_\{j\}\\right\]\.Noting that𝔼​\[L~\(B\)​\(φ\)\]=L𝒢​\(φ\)\\mathbb\{E\}\[\\tilde\{L\}^\{\(B\)\}\(\\varphi\)\]=L^\{\\mathscr\{G\}\}\(\\varphi\), we have

supφ∈ℋ\|L​\(φ\)−L~\(B\)​\(φ\)\|≤supφ∈ℋ\|L~\(B\)​\(φ\)−L𝒢​\(φ\)\|\+supφ∈ℋ\|L𝒢​\(φ\)−L​\(φ\)\|\.\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|L\(\\varphi\)\-\\tilde\{L\}^\{\(B\)\}\(\\varphi\)\\right\|\\leq\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|\\tilde\{L\}^\{\(B\)\}\(\\varphi\)\-L^\{\\mathscr\{G\}\}\(\\varphi\)\\right\|\+\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|L^\{\\mathscr\{G\}\}\(\\varphi\)\-L\(\\varphi\)\\right\|\.\(C\.21\)The second term on the right hand side of \([C\.21](https://arxiv.org/html/2606.24999#A3.E21)\) is the time discretization error, which is bounded by the following lemma\.

###### Lemma 19

Under Assumptions[3](https://arxiv.org/html/2606.24999#Thmassumption3),[4](https://arxiv.org/html/2606.24999#Thmassumption4),[1](https://arxiv.org/html/2606.24999#Thmassumption1), there is a constantCCindependent ofϵ\\epsilonandNNsuch that

supφ∈ℋ\|L𝒢​\(φ\)−L​\(φ\)\|≤C​ϵ−4​N−1\.\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|L^\{\\mathscr\{G\}\}\(\\varphi\)\-L\(\\varphi\)\\right\|\\leq C\\epsilon^\{\-4\}N^\{\-1\}\.\(C\.22\)

ProofWe aim to bound the difference between the continuous\-time population lossL​\(φ\)L\(\\varphi\)and its discrete\-time approximationL𝒢​\(φ\)L^\{\\mathcal\{G\}\}\(\\varphi\)\. We prove the Hessian loss for illustration, and the value and gradient components are treated in the same way and are of lower order inϵ\\epsilon\. Let the estimator be denoted by

ξϵ​\(Z\):=Z​Z⊤−Idϵ2,f~​\(t,x\):=f​\(t,x,𝐔n−1​\(t,x\)\)\.\\xi^\{\\epsilon\}\(Z\):=\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\},\\quad\\tilde\{f\}\(t,x\):=f\(t,x,\\mathbf\{U\}\_\{n\-1\}\(t,x\)\)\.\(C\.23\)The continuous and discrete losses are defined as:

Lh​\(φh\)\\displaystyle L\_\{h\}\(\\varphi\_\{h\}\):=∫0T𝔼​\[eβ​t​\|φh​\(t,Xt\)−ξϵ​\(Z\)​Rt,Xt\+ϵ​Z​\(𝐔n−1\)\|2\]​dt,\\displaystyle:=\\int\_\{0\}^\{T\}\\mathbb\{E\}\\left\[e^\{\\beta t\}\\left\|\\varphi\_\{h\}\(t,X\_\{t\}\)\-\\xi^\{\\epsilon\}\(Z\)R^\{t,X\_\{t\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\|^\{2\}\\right\]\\mathrm\{d\}t,\(C\.24\)Lh𝒢​\(φh\)\\displaystyle L\_\{h\}^\{\\mathcal\{G\}\}\(\\varphi\_\{h\}\):=∑i=0N−1𝔼​\[eβ​ti​\|φh​\(ti,Xti\)−ξϵ​\(Z\)​R~ti,Xti\+ϵ​Z​\(𝐔n−1\)\|2\]​Δ​ti,\\displaystyle:=\\sum\_\{i=0\}^\{N\-1\}\\mathbb\{E\}\\left\[e^\{\\beta t\_\{i\}\}\\left\|\\varphi\_\{h\}\(t\_\{i\},X\_\{t\_\{i\}\}\)\-\\xi^\{\\epsilon\}\(Z\)\\tilde\{R\}^\{t\_\{i\},X\_\{t\_\{i\}\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\|^\{2\}\\right\]\\Delta t\_\{i\},\(C\.25\)whereR~t,x​\(𝐔n−1\)\\tilde\{R\}^\{t,x\}\(\\mathbf\{U\}\_\{n\-1\}\)uses the discrete reward approximation

R~t,x​\(𝐔n−1\)=g​\(XTt,x\)\+∑if​\(ti,Xtit,x,𝐔n−1​\(ti,Xtit,x\)\)​Δ​ti,\\tilde\{R\}^\{t,x\}\(\\mathbf\{U\}\_\{n\-1\}\)=g\(X\_\{T\}^\{t,x\}\)\+\\sum\_\{i\}f\(t\_\{i\},X\_\{t\_\{i\}\}^\{t,x\},\\mathbf\{U\}\_\{n\-1\}\(t\_\{i\},X\_\{t\_\{i\}\}^\{t,x\}\)\)\\Delta t\_\{i\},\(C\.26\)instead of the exactRt,x​\(𝐔n−1\)R^\{t,x\}\(\\mathbf\{U\}\_\{n\-1\}\)

Rt,x​\(𝐔n−1\)=g​\(XTt,x\)\+∫tTf​\(s,Xst,x,𝐔n−1​\(s,Xst,x\)\)​ds\.R^\{t,x\}\(\\mathbf\{U\}\_\{n\-1\}\)=g\(X\_\{T\}^\{t,x\}\)\+\\int\_\{t\}^\{T\}f\(s,X\_\{s\}^\{t,x\},\\mathbf\{U\}\_\{n\-1\}\(s,X\_\{s\}^\{t,x\}\)\)\\mathrm\{d\}s\.\(C\.27\)Denote

Δ​\(t,y\):=R~t,y​\(𝐔n−1\)−Rt,y​\(𝐔n−1\)\.\\Delta\(t,y\):=\\tilde\{R\}^\{t,y\}\(\\mathbf\{U\}\_\{n\-1\}\)\-R^\{t,y\}\(\\mathbf\{U\}\_\{n\-1\}\)\.Introduce the intermediate grid lossL¯h𝒢\\bar\{L\}\_\{h\}^\{\\mathscr\{G\}\}obtained by replacing the time integral inLhL\_\{h\}with the grid sum while keeping the exact rewardRR\.

L¯h​\(φh\)𝒢:=∑i=0N−1𝔼​\[eβ​ti​\|φh​\(ti,Xti\)−ξϵ​\(Z\)​Rti,Xti\+ϵ​Z​\(𝐔n−1\)\|2\]​Δ​ti\.\\bar\{L\}\_\{h\}\(\\varphi\_\{h\}\)^\{\\mathscr\{G\}\}:=\\sum\_\{i=0\}^\{N\-1\}\\mathbb\{E\}\\left\[e^\{\\beta t\_\{i\}\}\\left\|\\varphi\_\{h\}\(t\_\{i\},X\_\{t\_\{i\}\}\)\-\\xi^\{\\epsilon\}\(Z\)R^\{t\_\{i\},X\_\{t\_\{i\}\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\|^\{2\}\\right\]\\Delta t\_\{i\}\.\(C\.28\)
We have the following inner and outer error decompositions

\|Lh​\(φh\)−Lh𝒢​\(φh\)\|≤\|Lh𝒢​\(φh\)−L¯h𝒢​\(φh\)\|⏟Inner Discretization Error\+\|L¯h𝒢​\(φh\)−Lh​\(φh\)\|⏟Outer Discretization Error\.\\displaystyle\\left\|L\_\{h\}\(\\varphi\_\{h\}\)\-L\_\{h\}^\{\\mathcal\{G\}\}\(\\varphi\_\{h\}\)\\right\|\\leq\\underbrace\{\\left\|L\_\{h\}^\{\\mathscr\{G\}\}\(\\varphi\_\{h\}\)\-\\bar\{L\}\_\{h\}^\{\\mathscr\{G\}\}\(\\varphi\_\{h\}\)\\right\|\}\_\{\\textrm\{Inner Discretization Error\}\}\+\\underbrace\{\\left\|\\bar\{L\}\_\{h\}^\{\\mathscr\{G\}\}\(\\varphi\_\{h\}\)\-L\_\{h\}\(\\varphi\_\{h\}\)\\right\|\}\_\{\\textrm\{Outer Discretization Error\}\}\.\(C\.29\)Therefore, there are two types of error – the inner and the outer discretization errors\. For the inner discretization error, it suffices to bound

\|𝔼​\[‖φh​\(ti,Xti\)−ξϵ​\(Z\)​Rti,Xti\+ϵ​Z‖2\]−𝔼​\[‖φh​\(ti,Xti\)−ξϵ​\(Z\)​R~ti,Xti\+ϵ​Z‖2\]\|\\displaystyle\\left\|\\mathbb\{E\}\\left\[\\left\\\|\\varphi\_\{h\}\(t\_\{i\},X\_\{t\_\{i\}\}\)\-\\xi^\{\\epsilon\}\(Z\)R^\{t\_\{i\},X\_\{t\_\{i\}\}\+\\epsilon Z\}\\right\\\|^\{2\}\\right\]\-\\mathbb\{E\}\\left\[\\left\\\|\\varphi\_\{h\}\(t\_\{i\},X\_\{t\_\{i\}\}\)\-\\xi^\{\\epsilon\}\(Z\)\\tilde\{R\}^\{t\_\{i\},X\_\{t\_\{i\}\}\+\\epsilon Z\}\\right\\\|^\{2\}\\right\]\\right\|\(C\.30\)≤𝔼​\[\|2​φh​\(ti,Xti\)−ξϵ​\(Z\)​\(R~ti,Xti\+ϵ​Z\+Rti,Xti\+ϵ​Z\)\|​\|ξϵ​\(Z\)\|​\|Δ​\(ti,Xti\+ϵ​Z\)\|\]\.\\displaystyle\\leq\\mathbb\{E\}\\left\[\\left\|2\\varphi\_\{h\}\(t\_\{i\},X\_\{t\_\{i\}\}\)\-\\xi^\{\\epsilon\}\(Z\)\(\\tilde\{R\}^\{t\_\{i\},X\_\{t\_\{i\}\}\+\\epsilon Z\}\+R^\{t\_\{i\},X\_\{t\_\{i\}\}\+\\epsilon Z\}\)\\right\|\\left\|\\xi^\{\\epsilon\}\(Z\)\\right\|\\left\|\\Delta\(t\_\{i\},X\_\{t\_\{i\}\}\+\\epsilon Z\)\\right\|\\right\]\.Due to the polynomial growth condition and the moment bounds ofXtX\_\{t\}, it suffices to bound‖Δ​\(ti,Xti\+ϵ​Z\)‖L2\\\|\\Delta\(t\_\{i\},X\_\{t\_\{i\}\}\+\\epsilon Z\)\\\|\_\{L\_\{2\}\}\. To this end, recall thatRt,y=g​\(XTt,y\)\+∫tTf~​\(s,Xst,y\)​dsR^\{t,y\}=g\(X\_\{T\}^\{t,y\}\)\+\\int\_\{t\}^\{T\}\\tilde\{f\}\(s,X\_\{s\}^\{t,y\}\)\\mathrm\{d\}s\. The terminal part cancels and

Δ​\(ti,y\)=∑k=iN−1∫tktk\+1\[f~​\(tk,Xtkti,y\)−f~​\(s,Xsti,y\)\]​ds\.\\Delta\(t\_\{i\},y\)=\\sum\_\{k=i\}^\{N\-1\}\\int\_\{t\_\{k\}\}^\{t\_\{k\+1\}\}\\left\[\\tilde\{f\}\(t\_\{k\},X\_\{t\_\{k\}\}^\{t\_\{i\},y\}\)\-\\tilde\{f\}\(s,X\_\{s\}^\{t\_\{i\},y\}\)\\right\]\\mathrm\{d\}s\.By Assumption[4](https://arxiv.org/html/2606.24999#Thmassumption4), Itô’s formula gives

d​f~​\(s,Xsti,y\)=As​d​s\+Bs​d​Ws,𝔼​\[\|As\|2\+\|Bs\|2\]≤C​\(1\+\|y\|q\)\\mathrm\{d\}\\tilde\{f\}\(s,X\_\{s\}^\{t\_\{i\},y\}\)=A\_\{s\}\\,\\mathrm\{d\}s\+B\_\{s\}\\,\\mathrm\{d\}W\_\{s\},\\qquad\\mathbb\{E\}\\left\[\|A\_\{s\}\|^\{2\}\+\|B\_\{s\}\|^\{2\}\\right\]\\leq C\(1\+\|y\|^\{q\}\)for someq≥0q\\geq 0, uniformly inss\. Stochastic Fubini therefore yields

Δ​\(ti,y\)=−∑k=iN−1∫tktk\+1\(tk\+1−u\)​Au​du−∑k=iN−1∫tktk\+1\(tk\+1−u\)​Bu​dWu\.\\displaystyle\\Delta\(t\_\{i\},y\)=\-\\sum\_\{k=i\}^\{N\-1\}\\int\_\{t\_\{k\}\}^\{t\_\{k\+1\}\}\(t\_\{k\+1\}\-u\)A\_\{u\}\\,\\mathrm\{d\}u\-\\sum\_\{k=i\}^\{N\-1\}\\int\_\{t\_\{k\}\}^\{t\_\{k\+1\}\}\(t\_\{k\+1\}\-u\)B\_\{u\}\\,\\mathrm\{d\}W\_\{u\}\.The drift term hasL2L^\{2\}norm at mostC​N​\(T/N\)2≤C/NCN\(T/N\)^\{2\}\\leq C/N\. For the martingale term, Itô’s isometry and orthogonality of martingale increments give

𝔼​\|∑k=iN−1∫tktk\+1\(tk\+1−u\)​Bu​dWu\|2≤C​N​h3≤CN2\.\\mathbb\{E\}\\left\|\\sum\_\{k=i\}^\{N\-1\}\\int\_\{t\_\{k\}\}^\{t\_\{k\+1\}\}\(t\_\{k\+1\}\-u\)B\_\{u\}\\,\\mathrm\{d\}W\_\{u\}\\right\|^\{2\}\\leq CNh^\{3\}\\leq\\frac\{C\}\{N^\{2\}\}\.Hence

‖Δ​\(ti,y\)‖L2≤C​\(1\+\|y\|q\)N\.\\\|\\Delta\(t\_\{i\},y\)\\\|\_\{L^\{2\}\}\\leq\\frac\{C\(1\+\|y\|^\{q\}\)\}\{N\}\.Conditioning onZZ, takingy=Xti\+ϵ​Zy=X\_\{t\_\{i\}\}\+\\epsilon Z, and using the moment bounds forXtiX\_\{t\_\{i\}\}together with Gaussian moments,

‖Δ​\(ti,Xti\+ϵ​Z\)‖L2≤CN\.\\left\\\|\\Delta\(t\_\{i\},X\_\{t\_\{i\}\}\+\\epsilon Z\)\\right\\\|\_\{L^\{2\}\}\\leq\\frac\{C\}\{N\}\.This along with \([C\.30](https://arxiv.org/html/2606.24999#A3.E30)\) yields

supφh∈ℋ\|Lh𝒢​\(φh\)−L¯h𝒢​\(φh\)\|≤CN​ϵ4\.\\sup\_\{\\varphi\_\{h\}\\in\\mathcal\{H\}\}\|L\_\{h\}^\{\\mathscr\{G\}\}\(\\varphi\_\{h\}\)\-\\bar\{L\}\_\{h\}^\{\\mathscr\{G\}\}\(\\varphi\_\{h\}\)\|\\leq\\frac\{C\}\{N\\epsilon^\{4\}\}\.
Now we bound the outer discretization error:

\|∫0Tl​\(t\)​dt−∑i=0N−1l​\(ti\)​Δ​ti\|,where​l​\(t\)=eβ​t​𝔼​\[‖φh​\(t,Xt\)−ξϵ​\(Z\)​Rt,Xt\+ϵ​Z‖2\]\.\\left\|\\int\_\{0\}^\{T\}l\(t\)\\mathrm\{d\}t\-\\sum\_\{i=0\}^\{N\-1\}l\(t\_\{i\}\)\\Delta t\_\{i\}\\right\|,\\quad\\text\{where \}l\(t\)=e^\{\\beta t\}\\mathbb\{E\}\\left\[\\left\\\|\\varphi\_\{h\}\(t,X\_\{t\}\)\-\\xi^\{\\epsilon\}\(Z\)R^\{t,X\_\{t\}\+\\epsilon Z\}\\right\\\|^\{2\}\\right\]\.\(C\.31\)It suffices to show that the time derivativedd​t​l​\(t\)\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}l\(t\)is bounded\. Indeed,

l​\(t\)\\displaystyle l\(t\)=eβ​t​𝔼​\[‖φh​\(t,Xt\)−ξϵ​\(Z\)​Rt,Xt\+ϵ​Z‖2\]\\displaystyle=e^\{\\beta t\}\\mathbb\{E\}\\left\[\\left\\\|\\varphi\_\{h\}\(t,X\_\{t\}\)\-\\xi^\{\\epsilon\}\(Z\)R^\{t,X\_\{t\}\+\\epsilon Z\}\\right\\\|^\{2\}\\right\]\(C\.32\)=eβ​t​\(𝔼​\[‖φh​\(t,Xt\)‖2\]\+𝔼​\[‖ξϵ​\(Z\)​Rt,Xt\+ϵ​Z‖2\]−2​𝔼​\[⟨φh​\(t,Xt\),ξϵ​\(Z\)⟩​Rt,Xt\+ϵ​Z\]\)\.\\displaystyle=e^\{\\beta t\}\\left\(\\mathbb\{E\}\\left\[\\left\\\|\\varphi\_\{h\}\(t,X\_\{t\}\)\\right\\\|^\{2\}\\right\]\+\\mathbb\{E\}\\left\[\\left\\\|\\xi^\{\\epsilon\}\(Z\)R^\{t,X\_\{t\}\+\\epsilon Z\}\\right\\\|^\{2\}\\right\]\-2\\mathbb\{E\}\\left\[\\langle\\varphi\_\{h\}\(t,X\_\{t\}\),\\xi^\{\\epsilon\}\(Z\)\\rangle R^\{t,X\_\{t\}\+\\epsilon Z\}\\right\]\\right\)\.Sinceφh\\varphi\_\{h\}is smooth and bounded, we need only to analyze the second and third terms\.

For the second term, using the conditional expectation, we have

I2​\(t\):=𝔼​\[‖ξϵ​\(Z\)​Rt,Xt\+ϵ​Z‖2\]=𝔼​\[‖ξϵ​\(Z\)‖2​𝔼​\[\(Rt,Xt\+ϵ​Z\)2∣Z,Xt\]\]\.I\_\{2\}\(t\):=\\mathbb\{E\}\\left\[\\left\\\|\\xi^\{\\epsilon\}\(Z\)R^\{t,X\_\{t\}\+\\epsilon Z\}\\right\\\|^\{2\}\\right\]=\\mathbb\{E\}\\left\[\\\|\\xi^\{\\epsilon\}\(Z\)\\\|^\{2\}\\mathbb\{E\}\\left\[\(R^\{t,X\_\{t\}\+\\epsilon Z\}\)^\{2\}\\mid Z,X\_\{t\}\\right\]\\right\]\.\(C\.33\)
Denoteu​\(t,x\):=𝔼​\[\(Rt,x\)2\]u\(t,x\):=\\mathbb\{E\}\[\(R^\{t,x\}\)^\{2\}\]\. Under Assumption[4](https://arxiv.org/html/2606.24999#Thmassumption4), the derivatives ofuusatisfy the polynomial growth condition\. Therefore, the time derivative of the expectation is:

dd​t​𝔼​\[‖ξϵ​\(Z\)​Rt,Xt\+ϵ​Z‖2\]\\displaystyle\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\mathbb\{E\}\\left\[\\left\\\|\\xi^\{\\epsilon\}\(Z\)R^\{t,X\_\{t\}\+\\epsilon Z\}\\right\\\|^\{2\}\\right\]=𝔼​\[‖ξϵ​\(Z\)‖2​dd​t​u​\(t,Xt\+ϵ​Z\)\]\\displaystyle=\\mathbb\{E\}\[\\\|\\xi^\{\\epsilon\}\(Z\)\\\|^\{2\}\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}u\(t,X\_\{t\}\+\\epsilon Z\)\]\(C\.34\)=𝔼​\[‖ξϵ​\(Z\)‖2​\(∂t\+ℒ\)​FZ​\(t,x\)\|x=Xt\],\\displaystyle=\\mathbb\{E\}\\left\[\\\|\\xi^\{\\epsilon\}\(Z\)\\\|^\{2\}\(\\partial\_\{t\}\+\\mathcal\{L\}\)F\_\{Z\}\(t,x\)\\big\|\_\{x=X\_\{t\}\}\\right\],whereFZ​\(t,x\)=u​\(t,x\+ϵ​Z\)F\_\{Z\}\(t,x\)=u\(t,x\+\\epsilon Z\)\. This confirms that\|dd​t​I2​\(t\)\|≤C​ϵ−4\|\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}I\_\{2\}\(t\)\|\\leq C\\epsilon^\{\-4\}\.

Similarly, for the cross term involving𝔼​\[⟨φ,ξϵ​\(Z\)​Rt,Xt\+ϵ​Z⟩\]\\mathbb\{E\}\[\\langle\\varphi,\\xi^\{\\epsilon\}\(Z\)R^\{t,X\_\{t\}\+\\epsilon Z\}\\rangle\], the derivative involves terms related to𝔼​\[Rt,x\]\\mathbb\{E\}\[R^\{t,x\}\]\. Using Assumption[4](https://arxiv.org/html/2606.24999#Thmassumption4), the derivative is bounded byC​ϵ−2C\\epsilon^\{\-2\}\.

Combining these bounds, the total discretization error is bounded by:

supφ∈ℋ\|Lh𝒢​\(φ\)−Lh​\(φ\)\|≤C​\(ϵ−4​N−1\+ϵ−2​N−1\)≤C​ϵ−4​N−1\.\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|L\_\{h\}^\{\\mathcal\{G\}\}\(\\varphi\)\-L\_\{h\}\(\\varphi\)\\right\|\\leq C\\left\(\\epsilon^\{\-4\}N^\{\-1\}\+\\epsilon^\{\-2\}N^\{\-1\}\\right\)\\leq C\\epsilon^\{\-4\}N^\{\-1\}\.\(C\.35\)This proves the desired result\.

The first term on the right hand side of \([C\.21](https://arxiv.org/html/2606.24999#A3.E21)\) is the classical statistical error, and we aim to bound it via the Rademacher complexity\. Typically, the Rademacher complexity framework works for bounded loss functions\. We first truncate the loss function and denoteGGas the implicit function andWWas the underlying random variable, such thatL𝒢​\(φ\)=𝔼​\[G​\(φ;W\)\]L^\{\\mathscr\{G\}\}\(\\varphi\)=\\mathbb\{E\}\[G\(\\varphi;W\)\]andL~\(B\)​\(φ\)=1B​∑i=1BG​\(φ;Wi\)\\tilde\{L\}^\{\(B\)\}\(\\varphi\)=\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}G\(\\varphi;W\_\{i\}\)\. Write the grid loss as

G​\(φ;W\)=∑j=0N−1Δ​tj​ℓj​\(φ;W\),G\(\\varphi;W\)=\\sum\_\{j=0\}^\{N\-1\}\\Delta t\_\{j\}\\,\\ell\_\{j\}\(\\varphi;W\),whereℓj\\ell\_\{j\}is the one\-point squared value\-gradient\-Hessian loss at timetjt\_\{j\}\. For each grid index, define the local truncation event

ΩK,j:=\{sup0≤t≤T\|Xt\|<K\}∩\{\|Z\|<K\}∩\{suptj≤s≤T\|Xstj,Xtj\+ϵ​Z\|<K\}\.\\Omega\_\{K,j\}:=\\left\\\{\\sup\_\{0\\leq t\\leq T\}\|X\_\{t\}\|<K\\right\\\}\\cap\\\{\|Z\|<K\\\}\\cap\\left\\\{\\sup\_\{t\_\{j\}\\leq s\\leq T\}\\left\|X\_\{s\}^\{t\_\{j\},X\_\{t\_\{j\}\}\+\\epsilon Z\}\\right\|<K\\right\\\}\.We clip the grid loss term by term:

G\(K\)​\(φ;W\):=∑j=0N−1Δ​tj​ℓj​\(φ;W\)​𝟏ΩK,j\.G^\{\(K\)\}\(\\varphi;W\):=\\sum\_\{j=0\}^\{N\-1\}\\Delta t\_\{j\}\\,\\ell\_\{j\}\(\\varphi;W\)\\mathbf\{1\}\_\{\\Omega\_\{K,j\}\}\.\(C\.36\)By Assumption[4](https://arxiv.org/html/2606.24999#Thmassumption4), there exists a finite exponentq≥1q\\geq 1, depending only on the polynomial growth orders ofg​\(x\)g\(x\),J1​\(t,x;φ\)J\_\{1\}\(t,x;\\varphi\),J2​\(t,x;φ\)J\_\{2\}\(t,x;\\varphi\), andf~​\(t,x;φ\)\\tilde\{f\}\(t,x;\\varphi\)uniformly overφ∈ℋ\\varphi\\in\\mathcal\{H\}, such that onΩK,j\\Omega\_\{K,j\}the one\-point value, gradient, and Hessian targets are bounded byC​ϵ−2​\(1\+Kq\)C\\epsilon^\{\-2\}\(1\+K^\{q\}\)\. As a consequence of this truncation,G\(K\)G^\{\(K\)\}has the following bound

\|G\(K\)​\(φ;W\)\|≤C​ϵ−4​\(1\+Kq\)\.\|G^\{\(K\)\}\(\\varphi;W\)\|\\leq C\\epsilon^\{\-4\}\(1\+K^\{q\}\)\.\(C\.37\)Next, we aim to derive the probability of the truncation sets\. To this end, we need the following lemmas\. To derive the tail probability of the trajectory ofXX, we will use the following Bernstein martingale inequality fromDzhaparidze and Van Zanten \([2001](https://arxiv.org/html/2606.24999#bib.bib18)\)\.

###### Lemma 20

LetMtM\_\{t\}be a continuous square integrable martingale\. Then

ℙ​\(sup0≤t≤T\|Mt\|≥z,⟨M⟩T≤L\)≤2​exp⁡\(−12​z2L\)\.\\mathbb\{P\}\\left\(\\sup\_\{0\\leq t\\leq T\}\|M\_\{t\}\|\\geq z,\\ \\langle M\\rangle\_\{T\}\\leq L\\right\)\\leq 2\\exp\\left\(\-\\frac\{1\}\{2\}\\frac\{z^\{2\}\}\{L\}\\right\)\.\(C\.38\)

Based on this lemma, we state the tail probability of the trajectory ofXX\.

###### Lemma 21\(Sub\-Gaussian Tail of Base Trajectory\)

Under Assumption[1](https://arxiv.org/html/2606.24999#Thmassumption1), letXXbe the solution to the SDE \([2\.2](https://arxiv.org/html/2606.24999#S2.E2)\) starting fromX0X\_\{0\}\. Then there exist constantsC1,C2\>0C\_\{1\},C\_\{2\}\>0depending only onT,Lb,σ0T,L\_\{b\},\\sigma\_\{0\}andX0X\_\{0\}, such that for anyK\>0K\>0:

ℙ​\(sup0≤t≤T\|Xt\|\>K\)≤C1​exp⁡\(−C2​K2\)\.\\mathbb\{P\}\\left\(\\sup\_\{0\\leq t\\leq T\}\|X\_\{t\}\|\>K\\right\)\\leq C\_\{1\}\\exp\(\-C\_\{2\}K^\{2\}\)\.\(C\.39\)

ProofThe SDE in integral form is given by:

Xt=X0\+∫0tb​\(s,Xs\)​ds\+Mt,where​Mt=2​∫0tσ​\(s,Xs\)​dWs\.X\_\{t\}=X\_\{0\}\+\\int\_\{0\}^\{t\}b\(s,X\_\{s\}\)\\,\\mathrm\{d\}s\+M\_\{t\},\\quad\\text\{where \}M\_\{t\}=\\sqrt\{2\}\\int\_\{0\}^\{t\}\\sigma\(s,X\_\{s\}\)\\,\\mathrm\{d\}W\_\{s\}\.\(C\.40\)Note thatMtM\_\{t\}is a continuous square\-integrable martingale\. Taking the norm and using the linear growth condition we get

\|Xt\|\\displaystyle\|X\_\{t\}\|≤\|X0\|\+∫0tLb​\(1\+\|Xs\|\)​ds\+\|Mt\|\\displaystyle\\leq\|X\_\{0\}\|\+\\int\_\{0\}^\{t\}L\_\{b\}\(1\+\|X\_\{s\}\|\)\\,\\mathrm\{d\}s\+\|M\_\{t\}\|=\(\|X0\|\+Lb​T\+\|Mt\|\)\+Lb​∫0t\|Xs\|​ds\.\\displaystyle=\\left\(\|X\_\{0\}\|\+L\_\{b\}T\+\|M\_\{t\}\|\\right\)\+L\_\{b\}\\int\_\{0\}^\{t\}\|X\_\{s\}\|\\,\\mathrm\{d\}s\.\(C\.41\)LetMT∗:=sup0≤t≤T\|Mt\|M^\{\*\}\_\{T\}:=\\sup\_\{0\\leq t\\leq T\}\|M\_\{t\}\|\. Applying Grönwall’s inequality yields a path\-wise bound:

sup0≤t≤T\|Xt\|≤\(\|X0\|\+Lb​T\+MT∗\)​eLb​T\.\\sup\_\{0\\leq t\\leq T\}\|X\_\{t\}\|\\leq\\left\(\|X\_\{0\}\|\+L\_\{b\}T\+M^\{\*\}\_\{T\}\\right\)e^\{L\_\{b\}T\}\.\(C\.42\)Therefore, the event\{sup0≤t≤T\|Xt\|\>K\}\\\{\\sup\_\{0\\leq t\\leq T\}\|X\_\{t\}\|\>K\\\}implies:

MT∗\>K​e−Lb​T−\(\|X0\|\+Lb​T\)\.M^\{\*\}\_\{T\}\>Ke^\{\-L\_\{b\}T\}\-\(\|X\_\{0\}\|\+L\_\{b\}T\)\.\(C\.43\)For sufficiently largeKK, the RHS is approximately proportional toKK\. Thus, it suffices to bound the tail ofMT∗M^\{\*\}\_\{T\}\. We invoke Bernstein’s inequality for continuous martingales \(Lemma[20](https://arxiv.org/html/2606.24999#Thmtheorem20)\)\. Under the bounded diffusion assumption, the quadratic variation is deterministically bounded:

⟨M⟩T=∫0T‖σ​\(s,Xs\)‖2​ds≤T​σmax2\.\\langle M\\rangle\_\{T\}=\\int\_\{0\}^\{T\}\\\|\\sigma\(s,X\_\{s\}\)\\\|^\{2\}\\,\\mathrm\{d\}s\\leq T\\sigma\_\{\\max\}^\{2\}\.\(C\.44\)SettingL=T​σmax2L=T\\sigma\_\{\\max\}^\{2\}, we obtain:

ℙ​\(MT∗≥z\)≤2​exp⁡\(−z22​T​σmax2\)\.\\mathbb\{P\}\(M^\{\*\}\_\{T\}\\geq z\)\\leq 2\\exp\\left\(\-\\frac\{z^\{2\}\}\{2T\\sigma\_\{\\max\}^\{2\}\}\\right\)\.\(C\.45\)Combining this with the Grönwall bound proves the lemma\. Recall theψ2\\psi\_\{2\}\-Orlicz norm \(sub\-Gaussian norm\) is defined as‖X‖ψ2=inf\{C\>0:𝔼​\[exp⁡\(\|X\|2/C2\)\]≤2\}\\\|X\\\|\_\{\\psi\_\{2\}\}=\\inf\\\{C\>0:\\mathbb\{E\}\[\\exp\(\|X\|^\{2\}/C^\{2\}\)\]\\leq 2\\\}\.

As a direct consequence of Lemma[21](https://arxiv.org/html/2606.24999#Thmtheorem21), the supremum of the base trajectory satisfies:

‖sup0≤t≤T\|Xt\|‖ψ2≤Cb​a​s​e,\\left\\\|\\sup\_\{0\\leq t\\leq T\}\|X\_\{t\}\|\\right\\\|\_\{\\psi\_\{2\}\}\\leq C\_\{base\},\(C\.46\)whereCb​a​s​eC\_\{base\}is a constant depending only on the SDE coefficients andTT\.

We now extend this analysis to the branching paths used in the ZOD estimation\. Let𝒢=\{t1,…,tN\}\\mathcal\{G\}=\\\{t\_\{1\},\\dots,t\_\{N\}\\\}be the time discretization grid\. For eachjj, letXstj,ξjX\_\{s\}^\{t\_\{j\},\\xi\_\{j\}\}denote the conditional path starting at timetjt\_\{j\}fromξj=Xtj\+ϵ​Z\\xi\_\{j\}=X\_\{t\_\{j\}\}\+\\epsilon Z\. The path satisfies:

Xstj,ξj=ξj\+∫tjsb​\(u,Xutj,ξj\)​du\+2​∫tjsσ​\(u,Xutj,ξj\)​dWu,s∈\[tj,T\]\.X\_\{s\}^\{t\_\{j\},\\xi\_\{j\}\}=\\xi\_\{j\}\+\\int\_\{t\_\{j\}\}^\{s\}b\(u,X\_\{u\}^\{t\_\{j\},\\xi\_\{j\}\}\)\\,\\mathrm\{d\}u\+\\sqrt\{2\}\\int\_\{t\_\{j\}\}^\{s\}\\sigma\(u,X\_\{u\}^\{t\_\{j\},\\xi\_\{j\}\}\)\\,\\mathrm\{d\}W\_\{u\},\\quad s\\in\[t\_\{j\},T\]\.\(C\.47\)
The following lemma bounds the branching trajectories\.

###### Lemma 22\(Uniform Tail for Branching Paths\)

Under Assumption[1](https://arxiv.org/html/2606.24999#Thmassumption1), there exists a constantC\>0C\>0independent ofj,Nj,Nsuch that for anyK\>0K\>0:

sup0≤j<Nℙ​\(suptj≤s≤T\|Xstj,Xtj\+ϵ​Z\|\>K\)≤2​exp⁡\(−C​K2\)\.\\sup\_\{0\\leq j<N\}\\mathbb\{P\}\\left\(\\sup\_\{t\_\{j\}\\leq s\\leq T\}\\left\|X^\{t\_\{j\},X\_\{t\_\{j\}\}\+\\epsilon Z\}\_\{s\}\\right\|\>K\\right\)\\leq 2\\exp\(\-CK^\{2\}\)\.\(C\.48\)The same bounds hold withXtj−ϵ​ZX\_\{t\_\{j\}\}\-\\epsilon ZandXtjX\_\{t\_\{j\}\}in place ofXtj\+ϵ​ZX\_\{t\_\{j\}\}\+\\epsilon Z\.

ProofFix an indexj∈\{1,…,N\}j\\in\\\{1,\\dots,N\\\}\. LetXs\(j\):=Xstj,Xtj\+ϵ​ZX\_\{s\}^\{\(j\)\}:=X\_\{s\}^\{t\_\{j\},X\_\{t\_\{j\}\}\+\\epsilon Z\}\. By applying Grönwall’s inequality under the linear growth condition onbb, we obtain the pathwise bound:

sups∈\[tj,T\]\|Xs\(j\)\|≤eLb​T​\(\|Xtj\|\+ϵ​\|Z\|\+Lb​T\+sups∈\[tj,T\]\|Ms\(j\)\|\),\\sup\_\{s\\in\[t\_\{j\},T\]\}\|X\_\{s\}^\{\(j\)\}\|\\leq e^\{L\_\{b\}T\}\\left\(\|X\_\{t\_\{j\}\}\|\+\\epsilon\|Z\|\+L\_\{b\}T\+\\sup\_\{s\\in\[t\_\{j\},T\]\}\|M\_\{s\}^\{\(j\)\}\|\\right\),\(C\.49\)whereMs\(j\)=2​∫tjsσ​\(u,Xu\(j\)\)​dWuM\_\{s\}^\{\(j\)\}=\\sqrt\{2\}\\int\_\{t\_\{j\}\}^\{s\}\\sigma\(u,X\_\{u\}^\{\(j\)\}\)\\mathrm\{d\}W\_\{u\}\.

We analyze the Orliczψ2\\psi\_\{2\}\-norm of each term on the right\-hand side of the above:

1. 1\.Base Path:By \([C\.46](https://arxiv.org/html/2606.24999#A3.E46)\), we have‖sup\|Xt\|‖ψ2≤Cb​a​s​e\\\|\\sup\|X\_\{t\}\|\\\|\_\{\\psi\_\{2\}\}\\leq C\_\{base\}\.
2. 2\.Perturbation:SinceZ∼𝒩​\(0,Id\)Z\\sim\\mathcal\{N\}\(0,I\_\{d\}\), it is sub\-Gaussian with‖Z‖ψ2≤CG​a​u​s​s\\\|Z\\\|\_\{\\psi\_\{2\}\}\\leq C\_\{Gauss\}\.
3. 3\.Martingale:Since the diffusionσ\\sigmais uniformly bounded, the quadratic variation is bounded byσmax2​T\\sigma\_\{\\max\}^\{2\}T\. By Bernstein’s inequality for continuous martingales,sups\|Ms\(j\)\|\\sup\_\{s\}\|M\_\{s\}^\{\(j\)\}\|is sub\-Gaussian with norm bounded by a constantCm​a​r​tC\_\{mart\}independent of the path history\.

By the triangle inequality for theψ2\\psi\_\{2\}\-norm, the supremum of thejj\-th branching path is sub\-Gaussian:

‖suptj≤s≤T\|Xs\(j\)\|‖ψ2≤Cp​a​t​h,\\left\\\|\\sup\_\{t\_\{j\}\\leq s\\leq T\}\|X\_\{s\}^\{\(j\)\}\|\\right\\\|\_\{\\psi\_\{2\}\}\\leq C\_\{path\},\(C\.50\)whereCp​a​t​hC\_\{path\}is independent ofjjandNN\.

The tail bound follows directly from \([C\.50](https://arxiv.org/html/2606.24999#A3.E50)\) and the definition of theψ2\\psi\_\{2\}\-norm\. The proof for the branch started atXtj−ϵ​ZX\_\{t\_\{j\}\}\-\\epsilon Zis identical\.

Combining Lemma[21](https://arxiv.org/html/2606.24999#Thmtheorem21), the Gaussian tail ofZZ, and Lemma[22](https://arxiv.org/html/2606.24999#Thmtheorem22)gives the probability of the local truncation eventsΩK,jc\\Omega\_\{K,j\}^\{c\}used in the proof of Theorem[9](https://arxiv.org/html/2606.24999#Thmtheorem9)\.

###### Lemma 23

Under Assumption[1](https://arxiv.org/html/2606.24999#Thmassumption1), there exist constantsC1,C2\>0C\_\{1\},C\_\{2\}\>0, independent ofjjandNN, such that

sup0≤j<Nℙ​\(ΩK,jc\)≤C1​exp⁡\(−C2​K2\)\.\\sup\_\{0\\leq j<N\}\\mathbb\{P\}\(\\Omega\_\{K,j\}^\{c\}\)\\leq C\_\{1\}\\exp\(\-C\_\{2\}K^\{2\}\)\.\(C\.51\)

We are ready to obtain the upper bound of the first term on the right hand side of \([C\.21](https://arxiv.org/html/2606.24999#A3.E21)\)\. It can be decomposed as

supφ∈ℋ\|L~\(B\)​\(φ\)−L𝒢​\(φ\)\|\\displaystyle\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|\\tilde\{L\}^\{\(B\)\}\(\\varphi\)\-L^\{\\mathscr\{G\}\}\(\\varphi\)\\right\|=supφ∈ℋ\|1B​∑i=1BG​\(φ;Wi\)−𝔼​\[G​\(φ;W\)\]\|\\displaystyle=\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}G\(\\varphi;W\_\{i\}\)\-\\mathbb\{E\}\[G\(\\varphi;W\)\]\\right\|\(C\.52\)≤supφ∈ℋ\|1B​∑i=1BG\(K\)​\(φ;Wi\)−𝔼​\[G\(K\)​\(φ;W\)\]\|\\displaystyle\\leq\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}G^\{\(K\)\}\(\\varphi;W\_\{i\}\)\-\\mathbb\{E\}\[G^\{\(K\)\}\(\\varphi;W\)\]\\right\|\+supφ∈ℋ\|1B​∑i=1B\(G\(K\)​\(φ;Wi\)−G​\(φ;Wi\)\)\|\\displaystyle\\quad\+\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}\\left\(G^\{\(K\)\}\(\\varphi;W\_\{i\}\)\-G\(\\varphi;W\_\{i\}\)\\right\)\\right\|\+supφ∈ℋ𝔼​\[\|G\(K\)​\(φ;W\)−G​\(φ;W\)\|\]\.\\displaystyle\\quad\+\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\mathbb\{E\}\\left\[\\left\|G^\{\(K\)\}\(\\varphi;W\)\-G\(\\varphi;W\)\\right\|\\right\]\.Let

RK​\(W\):=supφ∈ℋ\|G​\(φ;W\)−G\(K\)​\(φ;W\)\|\.R\_\{K\}\(W\):=\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\|G\(\\varphi;W\)\-G^\{\(K\)\}\(\\varphi;W\)\|\.Since the networks inℋ\\mathcal\{H\}are uniformly bounded, the supremum overφ\\varphiin the local squared loss is bounded pointwise by the reward targets:

supφ∈ℋℓj​\(φ;W\)≤C​\(1\+\|Yv,j\|2\+‖Yg,jϵ‖2\+‖Yh,jϵ‖2\),\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\ell\_\{j\}\(\\varphi;W\)\\leq C\\left\(1\+\|Y\_\{v,j\}\|^\{2\}\+\\\|Y\_\{g,j\}^\{\\epsilon\}\\\|^\{2\}\+\\\|Y\_\{h,j\}^\{\\epsilon\}\\\|^\{2\}\\right\),where

Yv,j:=R~tj,Xtj​\(𝐔n−1\),Yg,jϵ:=Zϵ​R~tj,Xtj\+ϵ​Z​\(𝐔n−1\),Y\_\{v,j\}:=\\tilde\{R\}^\{t\_\{j\},X\_\{t\_\{j\}\}\}\(\\mathbf\{U\}\_\{n\-1\}\),\\quad Y\_\{g,j\}^\{\\epsilon\}:=\\frac\{Z\}\{\\epsilon\}\\tilde\{R\}^\{t\_\{j\},X\_\{t\_\{j\}\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\),and

Yh,jϵ:=Z​Z⊤−Idϵ2​R~tj,Xtj\+ϵ​Z​\(𝐔n−1\)\.Y\_\{h,j\}^\{\\epsilon\}:=\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{\\epsilon^\{2\}\}\\tilde\{R\}^\{t\_\{j\},X\_\{t\_\{j\}\}\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\.The assumption[4](https://arxiv.org/html/2606.24999#Thmassumption4)and the polynomial growth ofggimply that, for someq≥0q\\geq 0,

\|Rtj,y​\(𝐔n−1\)\|\+\|R~tj,y​\(𝐔n−1\)\|≤C​\(1\+suptj≤s≤T\|Xstj,y\|q\)\.\|R^\{t\_\{j\},y\}\(\\mathbf\{U\}\_\{n\-1\}\)\|\+\|\\tilde\{R\}^\{t\_\{j\},y\}\(\\mathbf\{U\}\_\{n\-1\}\)\|\\leq C\\left\(1\+\\sup\_\{t\_\{j\}\\leq s\\leq T\}\|X\_\{s\}^\{t\_\{j\},y\}\|^\{q\}\\right\)\.Together with Assumption[1](https://arxiv.org/html/2606.24999#Thmassumption1)and the Gaussian moments ofZZ, this gives

sup0≤j<N𝔼​\[\(supφ∈ℋℓj​\(φ;W\)\)2\]≤C​ϵ−8\.\\sup\_\{0\\leq j<N\}\\mathbb\{E\}\\\!\\left\[\\left\(\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\ell\_\{j\}\(\\varphi;W\)\\right\)^\{2\}\\right\]\\leq C\\epsilon^\{\-8\}\.\(C\.53\)Hence, by Cauchy–Schwarz applied to each grid summand, Lemma[23](https://arxiv.org/html/2606.24999#Thmtheorem23), and∑j=0N−1Δ​tj=T\\sum\_\{j=0\}^\{N\-1\}\\Delta t\_\{j\}=T,

𝔼​RK​\(W\)\\displaystyle\\mathbb\{E\}R\_\{K\}\(W\)≤∑j=0N−1Δ​tj​𝔼​\[supφ∈ℋℓj​\(φ;W\)​𝟏ΩK,jc\]\\displaystyle\\leq\\sum\_\{j=0\}^\{N\-1\}\\Delta t\_\{j\}\\mathbb\{E\}\\left\[\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\ell\_\{j\}\(\\varphi;W\)\\mathbf\{1\}\_\{\\Omega\_\{K,j\}^\{c\}\}\\right\]\(C\.54\)≤∑j=0N−1Δ​tj​\(𝔼​\[\(supφ∈ℋℓj​\(φ;W\)\)2\]\)1/2​ℙ​\(ΩK,jc\)1/2\\displaystyle\\leq\\sum\_\{j=0\}^\{N\-1\}\\Delta t\_\{j\}\\left\(\\mathbb\{E\}\\\!\\left\[\\left\(\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\ell\_\{j\}\(\\varphi;W\)\\right\)^\{2\}\\right\]\\right\)^\{1/2\}\\mathbb\{P\}\(\\Omega\_\{K,j\}^\{c\}\)^\{1/2\}≤C​ϵ−4​exp⁡\(−C​K2\)\.\\displaystyle\\leq C\\epsilon^\{\-4\}\\exp\(\-CK^\{2\}\)\.In particular,

supφ∈ℋ𝔼​\[\|G\(K\)​\(φ;W\)−G​\(φ;W\)\|\]≤𝔼​RK​\(W\)≤C​ϵ−4​exp⁡\(−C​K2\)\.\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\mathbb\{E\}\\left\[\|G^\{\(K\)\}\(\\varphi;W\)\-G\(\\varphi;W\)\|\\right\]\\leq\\mathbb\{E\}R\_\{K\}\(W\)\\leq C\\epsilon^\{\-4\}\\exp\(\-CK^\{2\}\)\.Therefore, by Markov’s inequality, with probability at least1−δ1\-\\delta,

1B​∑i=1BRK​\(Wi\)\+𝔼​RK​\(W\)≤C​δ−1​ϵ−4​exp⁡\(−C​K2\)\.\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}R\_\{K\}\(W\_\{i\}\)\+\\mathbb\{E\}R\_\{K\}\(W\)\\leq C\\delta^\{\-1\}\\epsilon^\{\-4\}\\exp\(\-CK^\{2\}\)\.\(C\.55\)
For the first term on the right hand side of \([C\.52](https://arxiv.org/html/2606.24999#A3.E52)\), we use the Rademacher complexity\. DenoteS=\(W1,…,WB\)S=\(W\_\{1\},\\dots,W\_\{B\}\)and define

Φ​\(S\):=supφ\|1B​∑i=1BG\(K\)​\(φ;Wi\)−𝔼​\[G\(K\)​\(φ;W\)\]\|\.\\Phi\(S\):=\\sup\_\{\\varphi\}\\left\|\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}G^\{\(K\)\}\(\\varphi;W\_\{i\}\)\-\\mathbb\{E\}\\left\[G^\{\(K\)\}\(\\varphi;W\)\\right\]\\right\|\.\(C\.56\)LetS′=\(W1,…,Wj′,…,WB\)S^\{\\prime\}=\(W\_\{1\},\\dots,W\_\{j\}^\{\\prime\},\\dots,W\_\{B\}\)differ fromSSonly atjj\-th component\. It follows from Jensen’s inequality and the upper bound ofG\(K\)G^\{\(K\)\}that

\|Φ​\(S\)−Φ​\(S′\)\|≤supφ∈ℋ1B​\|G\(K\)​\(φ;Wj\)−G\(K\)​\(φ;Wj′\)\|≤C​ϵ−4​\(1\+Kq\)B\.\\displaystyle\\left\|\\Phi\(S\)\-\\Phi\(S^\{\\prime\}\)\\right\|\\leq\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\frac\{1\}\{B\}\\left\|G^\{\(K\)\}\(\\varphi;W\_\{j\}\)\-G^\{\(K\)\}\(\\varphi;W\_\{j\}^\{\\prime\}\)\\right\|\\leq\\frac\{C\\epsilon^\{\-4\}\(1\+K^\{q\}\)\}\{B\}\.\(C\.57\)By McDiarmid’s inequality,

ℙ​\(Φ​\(S\)−𝔼​\[Φ​\(S\)\]\>C​ϵ−4​\(1\+Kq\)​log⁡\(1/δ\)B\)<δ\.\\mathbb\{P\}\\left\(\\Phi\(S\)\-\\mathbb\{E\}\[\\Phi\(S\)\]\>C\\epsilon^\{\-4\}\(1\+K^\{q\}\)\\sqrt\{\\frac\{\\log\(1/\\delta\)\}\{B\}\}\\right\)<\\delta\.\(C\.58\)Now we use the Rademacher complexity to bound𝔼​\[Φ\]\\mathbb\{E\}\[\\Phi\]\. LetW′W^\{\\prime\}be the independent copy ofWW, and\{σi\}i=1B\\\{\\sigma\_\{i\}\\\}\_\{i=1\}^\{B\}are i\.i\.d\. random variables withℙ​\(σi=1\)=ℙ​\(σi=−1\)=12\\mathbb\{P\}\(\\sigma\_\{i\}=1\)=\\mathbb\{P\}\(\\sigma\_\{i\}=\-1\)=\\frac\{1\}\{2\}\. Then

𝔼​\[Φ​\(S\)\]\\displaystyle\\mathbb\{E\}\[\\Phi\(S\)\]=𝔼​\[supφ\|1B​∑i=1BG\(K\)​\(φ;Wi\)−𝔼​\[G\(K\)​\(φ;W\)\]\|\]\\displaystyle=\\mathbb\{E\}\\left\[\\sup\_\{\\varphi\}\\left\|\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}G^\{\(K\)\}\(\\varphi;W\_\{i\}\)\-\\mathbb\{E\}\\left\[G^\{\(K\)\}\(\\varphi;W\)\\right\]\\right\|\\right\]\(C\.59\)=𝔼​\[supφ∈ℋ\|𝔼W′​\[1B​∑i=1B\(G\(K\)​\(φ;Wj′\)−G\(K\)​\(φ;Wj\)\)\]\|\]\\displaystyle=\\mathbb\{E\}\\left\[\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|\\mathbb\{E\}\_\{W^\{\\prime\}\}\\left\[\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}\\left\(G^\{\(K\)\}\(\\varphi;W\_\{j\}^\{\\prime\}\)\-G^\{\(K\)\}\(\\varphi;W\_\{j\}\)\\right\)\\right\]\\right\|\\right\]=𝔼​\[supφ∈ℋ\|𝔼W′,σ​\[1B​∑i=1Bσi​\(G\(K\)​\(φ;Wj′\)−G\(K\)​\(φ;Wj\)\)\]\|\]\\displaystyle=\\mathbb\{E\}\\left\[\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|\\mathbb\{E\}\_\{W^\{\\prime\},\\sigma\}\\left\[\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}\\sigma\_\{i\}\\left\(G^\{\(K\)\}\(\\varphi;W\_\{j\}^\{\\prime\}\)\-G^\{\(K\)\}\(\\varphi;W\_\{j\}\)\\right\)\\right\]\\right\|\\right\]≤𝔼W,W′,σ​\[supφ∈ℋ\|1B​∑i=1Bσi​G\(K\)​\(φ;Wi′\)\|\]\+𝔼W,W′,σ​\[supφ∈ℋ\|1B​∑i=1Bσi​G\(K\)​\(φ;Wi\)\|\]\\displaystyle\\leq\\mathbb\{E\}\_\{W,W^\{\\prime\},\\sigma\}\\left\[\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}\\sigma\_\{i\}G^\{\(K\)\}\(\\varphi;W\_\{i\}^\{\\prime\}\)\\right\|\\right\]\+\\mathbb\{E\}\_\{W,W^\{\\prime\},\\sigma\}\\left\[\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}\\sigma\_\{i\}G^\{\(K\)\}\(\\varphi;W\_\{i\}\)\\right\|\\right\]≤2ℛB\(G\(K\)\(ℋ\)∪−G\(K\)\(ℋ\)\)\\displaystyle\\leq 2\\mathcal\{R\}\_\{B\}\(G^\{\(K\)\}\(\\mathcal\{H\}\)\\cup\-G^\{\(K\)\}\(\\mathcal\{H\}\)\)whereG\(K\)​\(ℋ\)=\{G\(K\)​\(φ;⋅\)\|φ∈ℋ\}G^\{\(K\)\}\(\\mathcal\{H\}\)=\\left\\\{G^\{\(K\)\}\(\\varphi;\\cdot\)\|\\varphi\\in\\mathcal\{H\}\\right\\\}\. For the Rademacher complexity in terms of the parameterized function class, we have the following lemma byJiaoet al\.\([2024](https://arxiv.org/html/2606.24999#bib.bib16)\)\.

###### Lemma 24

Let𝒢\\mathcal\{G\}be a parameterized function class𝒢=\{f​\(⋅;θ\):ℝd→ℝ\|θ∈Θ\}\\mathcal\{G\}=\\left\\\{f\(\\cdot;\\theta\):\\mathbb\{R\}^\{d\}\\rightarrow\\mathbb\{R\}\|\\theta\\in\\Theta\\right\\\}, whereΘ=\{θ\|θ∈ℝp,‖θ‖∞<R\}\\Theta=\\left\\\{\\theta\|\\theta\\in\\mathbb\{R\}^\{p\},\\\|\\theta\\\|\_\{\\infty\}<R\\right\\\}\. If there are constantsC1,C2C\_\{1\},C\_\{2\}, such that

supx∈ℝd\|f​\(x;θ1\)−f​\(x;θ2\)\|\\displaystyle\\sup\_\{x\\in\\mathbb\{R\}^\{d\}\}\|f\(x;\\theta\_\{1\}\)\-f\(x;\\theta\_\{2\}\)\|≤C1​‖θ1−θ2‖∞,\\displaystyle\\leq C\_\{1\}\\\|\\theta\_\{1\}\-\\theta\_\{2\}\\\|\_\{\\infty\},\(C\.60\)supθ∈Θsupx∈ℝd\|f​\(x;θ\)\|\\displaystyle\\sup\_\{\\theta\\in\\Theta\}\\sup\_\{x\\in\\mathbb\{R\}^\{d\}\}\|f\(x;\\theta\)\|≤C2,\\displaystyle\\leq C\_\{2\},then

ℛm​\(𝒢\)≤4m\+6​p​C2m​log⁡\(2​R​C1​p​m\)\.\\mathcal\{R\}\_\{m\}\(\\mathcal\{G\}\)\\leq\\frac\{4\}\{\\sqrt\{m\}\}\+\\frac\{6\\sqrt\{p\}C\_\{2\}\}\{\\sqrt\{m\}\}\\sqrt\{\\log\(2RC\_\{1\}p\\sqrt\{m\}\)\}\.\(C\.61\)

Based on this lemma, it follows from Assumption[3](https://arxiv.org/html/2606.24999#Thmassumption3)that

ℛB\(G\(K\)\(ℋ\)∪−G\(K\)\(ℋ\)\)\\displaystyle\\mathcal\{R\}\_\{B\}\(G^\{\(K\)\}\(\\mathcal\{H\}\)\\cup\-G^\{\(K\)\}\(\\mathcal\{H\}\)\)\(C\.62\)≤4B\+6​p​C​\(1\+ϵ−4​\(1\+Kq\)\)B​log⁡\(2​R​C​ϵ−2​\(1\+Kq\)​p​B\),\\displaystyle\\qquad\\leq\\frac\{4\}\{\\sqrt\{B\}\}\+\\frac\{6\\sqrt\{p\}C\(1\+\\epsilon^\{\-4\}\(1\+K^\{q\}\)\)\}\{\\sqrt\{B\}\}\\sqrt\{\\log\(2RC\\epsilon^\{\-2\}\(1\+K^\{q\}\)p\\sqrt\{B\}\)\},whereCCis a constant depending only on the coefficients of \([2\.1](https://arxiv.org/html/2606.24999#S2.E1)\)\. The desired result follows now from \([C\.58](https://arxiv.org/html/2606.24999#A3.E58)\), \([C\.54](https://arxiv.org/html/2606.24999#A3.E54)\), \([C\.59](https://arxiv.org/html/2606.24999#A3.E59)\) and \([C\.62](https://arxiv.org/html/2606.24999#A3.E62)\) by choosingK=C​\(log⁡B\+log⁡\(1/δ\)\)1/2K=C\(\\log B\+\\log\(1/\\delta\)\)^\{1/2\}withCClarge enough, which gives the factorLB,δqL\_\{B,\\delta\}^\{q\}after increasingqqonce more if necessary\.

We now prove Theorem[9](https://arxiv.org/html/2606.24999#Thmtheorem9)\-\(b\) with the strong simulator\. The only parts that differ from the preceding one\-point proof are the ZOD target, the discretization estimate, and the envelope used in the empirical\-process bound\. The first entry in the minimum definingΓϵ,B,δ\\Gamma\_\{\\epsilon,B,\\delta\}follows from the same truncation and Rademacher\-complexity argument as in the weak\-simulator case\. We therefore focus below on the second entry of the minimum under the strong\-simulator coupling\. In the remainder of this proof, all the rewards

Rt,x\+ϵ​Z​\(𝐔n−1\),Rt,x−ϵ​Z​\(𝐔n−1\),Rt,x​\(𝐔n−1\)R^\{t,x\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\),\\qquad R^\{t,x\-\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\),\\qquad R^\{t,x\}\(\\mathbf\{U\}\_\{n\-1\}\)are evaluated under the same Brownian path within one strong\-simulator query\. The exact multi\-point derivative targets are the two central differences

Z2​ϵ​\(Rt,x\+ϵ​Z​\(𝐔n−1\)−Rt,x−ϵ​Z​\(𝐔n−1\)\),\\frac\{Z\}\{2\\epsilon\}\\left\(R^\{t,x\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\-R^\{t,x\-\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\),and

Z​Z⊤−Id2​ϵ2​\(Rt,x\+ϵ​Z​\(𝐔n−1\)\+Rt,x−ϵ​Z​\(𝐔n−1\)−2​Rt,x​\(𝐔n−1\)\)\.\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{2\\epsilon^\{2\}\}\\left\(R^\{t,x\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\+R^\{t,x\-\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\-2R^\{t,x\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\)\.By Gaussian symmetry and linearity, their conditional means areGϵG^\{\\epsilon\}andHϵH^\{\\epsilon\}\. LetLmL\_\{m\}be the corresponding population loss and letL~m\(B\)\\tilde\{L\}\_\{m\}^\{\(B\)\}be its quadrature empirical version\. By the same conditional\-expectation decomposition as in Lemma[7](https://arxiv.org/html/2606.24999#Thmtheorem7),

Lm​\(φ,𝐔n−1\)=‖φ−𝒮ϵ​\(𝐔n−1\)‖β,mix2\+Cm,L\_\{m\}\(\\varphi,\\mathbf\{U\}\_\{n\-1\}\)=\\\|\\varphi\-\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\\\|\_\{\\beta,\\mathrm\{mix\}\}^\{2\}\+C\_\{m\},\(C\.63\)whereCmC\_\{m\}is independent ofφ\\varphi\. Hence the ERM argument in Lemma[18](https://arxiv.org/html/2606.24999#Thmtheorem18)gives

En2≤\\displaystyle E\_\{n\}^\{2\}\\leq4​supφ∈ℋ\|Lm​\(φ\)−L~m\(B\)​\(φ\)\|\+2​infφ∈ℋ‖φ−𝒮ϵ​\(𝐔n−1\)‖β,mix2\\displaystyle 4\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|L\_\{m\}\(\\varphi\)\-\\tilde\{L\}\_\{m\}^\{\(B\)\}\(\\varphi\)\\right\|\+2\\inf\_\{\\varphi\\in\\mathcal\{H\}\}\\left\\\|\\varphi\-\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}\(C\.64\)\+2​‖𝒮ϵ​\(𝐔n−1\)−𝒮​\(𝐔n−1\)‖β,mix2\.\\displaystyle\\quad\+2\\left\\\|\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}\.By Proposition[5](https://arxiv.org/html/2606.24999#Thmtheorem5), the multi\-point estimators have the sameO​\(ϵ2\)O\(\\epsilon^\{2\}\)bias as the one\-point estimators under the stated smoothness assumptions\. Therefore,

‖𝒮ϵ​\(𝐔n−1\)−𝒮​\(𝐔n−1\)‖β,mix2≤C​ϵ4\.\\left\\\|\\mathcal\{S\}^\{\\epsilon\}\(\\mathbf\{U\}\_\{n\-1\}\)\-\\mathcal\{S\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\\\|\_\{\\beta,\\text\{mix\}\}^\{2\}\\leq C\\epsilon^\{4\}\.\(C\.65\)
It remains to bound the statistical error term in \([C\.64](https://arxiv.org/html/2606.24999#A3.E64)\)\. LetLm𝒢L\_\{m\}^\{\\mathscr\{G\}\}be the time\-discretized version ofLmL\_\{m\}\. As before,

supφ∈ℋ\|Lm​\(φ\)−L~m\(B\)​\(φ\)\|≤supφ∈ℋ\|L~m\(B\)​\(φ\)−Lm𝒢​\(φ\)\|\+supφ∈ℋ\|Lm𝒢​\(φ\)−Lm​\(φ\)\|\.\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|L\_\{m\}\(\\varphi\)\-\\tilde\{L\}\_\{m\}^\{\(B\)\}\(\\varphi\)\\right\|\\leq\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|\\tilde\{L\}\_\{m\}^\{\(B\)\}\(\\varphi\)\-L\_\{m\}^\{\\mathscr\{G\}\}\(\\varphi\)\\right\|\+\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|L\_\{m\}^\{\\mathscr\{G\}\}\(\\varphi\)\-L\_\{m\}\(\\varphi\)\\right\|\.\(C\.66\)We first control the discretization term\. To separate the two sources of time discretization, letL¯m𝒢\\bar\{L\}\_\{m\}^\{\\mathscr\{G\}\}denote the time\-grid population loss obtained fromLmL\_\{m\}by replacing the time integral with the grid sum while keeping the exact rewardsRt,x​\(𝐔n−1\)R^\{t,x\}\(\\mathbf\{U\}\_\{n\-1\}\)unchanged\. Then

supφ∈ℋ\|Lm𝒢​\(φ\)−Lm​\(φ\)\|≤supφ∈ℋ\|Lm𝒢​\(φ\)−L¯m𝒢​\(φ\)\|\+supφ∈ℋ\|L¯m𝒢​\(φ\)−Lm​\(φ\)\|\.\\displaystyle\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|L\_\{m\}^\{\\mathscr\{G\}\}\(\\varphi\)\-L\_\{m\}\(\\varphi\)\\right\|\\leq\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|L\_\{m\}^\{\\mathscr\{G\}\}\(\\varphi\)\-\\bar\{L\}\_\{m\}^\{\\mathscr\{G\}\}\(\\varphi\)\\right\|\+\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|\\bar\{L\}\_\{m\}^\{\\mathscr\{G\}\}\(\\varphi\)\-L\_\{m\}\(\\varphi\)\\right\|\.\(C\.67\)Write

Δ​\(t,x\):=R~t,x​\(𝐔n−1\)−Rt,x​\(𝐔n−1\)\.\\Delta\(t,x\):=\\tilde\{R\}^\{t,x\}\(\\mathbf\{U\}\_\{n\-1\}\)\-R^\{t,x\}\(\\mathbf\{U\}\_\{n\-1\}\)\.Because the three rewards in the multi\-point estimator are generated by the same Brownian path and the same quadrature grid, the discretization error enters through the finite differences

Z2​ϵ​\{Δ​\(t,x\+ϵ​Z\)−Δ​\(t,x−ϵ​Z\)\}\\frac\{Z\}\{2\\epsilon\}\\\{\\Delta\(t,x\+\\epsilon Z\)\-\\Delta\(t,x\-\\epsilon Z\)\\\}and

Z​Z⊤−Id2​ϵ2​\{Δ​\(t,x\+ϵ​Z\)\+Δ​\(t,x−ϵ​Z\)−2​Δ​\(t,x\)\}\.\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{2\\epsilon^\{2\}\}\\\{\\Delta\(t,x\+\\epsilon Z\)\+\\Delta\(t,x\-\\epsilon Z\)\-2\\Delta\(t,x\)\\\}\.Under the regularity assumptionsb,σ∈Cb0,5b,\\sigma\\in C\_\{b\}^\{0,5\},f​\(⋅,⋅,𝐔n−1​\(⋅,⋅\)\)∈Cp1,5f\(\\cdot,\\cdot,\\mathbf\{U\}\_\{n\-1\}\(\\cdot,\\cdot\)\)\\in C\_\{p\}^\{1,5\}, andg∈Cp4g\\in C\_\{p\}^\{4\}, using the same method as in the proof of Lemma[19](https://arxiv.org/html/2606.24999#Thmtheorem19), there are constantsC\>0,q\>0C\>0,q\>0such that

‖∂xrΔ​\(t,x\)‖L2≤CN​\(1\+\|x\|q\)\.\\left\\\|\\partial\_\{x\}^\{r\}\\Delta\(t,x\)\\right\\\|\_\{L^\{2\}\}\\leq\\frac\{C\}\{N\}\(1\+\|x\|^\{q\}\)\.\(C\.68\)Applying Taylor’s expansion yields

‖Z2​ϵ​\{Δ​\(t,Xt\+ϵ​Z\)−Δ​\(t,Xt−ϵ​Z\)\}‖L2\\displaystyle\\left\\\|\\frac\{Z\}\{2\\epsilon\}\\\{\\Delta\(t,X\_\{t\}\+\\epsilon Z\)\-\\Delta\(t,X\_\{t\}\-\\epsilon Z\)\\\}\\right\\\|\_\{L^\{2\}\}≤CN,\\displaystyle\\leq\\frac\{C\}\{N\},\(C\.69\)‖Z​Z⊤−Id2​ϵ2​\{Δ​\(t,Xt\+ϵ​Z\)\+Δ​\(t,Xt−ϵ​Z\)−2​Δ​\(t,Xt\)\}‖L2\\displaystyle\\left\\\|\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{2\\epsilon^\{2\}\}\\\{\\Delta\(t,X\_\{t\}\+\\epsilon Z\)\+\\Delta\(t,X\_\{t\}\-\\epsilon Z\)\-2\\Delta\(t,X\_\{t\}\)\\\}\\right\\\|\_\{L^\{2\}\}≤CN\.\\displaystyle\\leq\\frac\{C\}\{N\}\.\(C\.70\)Using the same method as in the proof of Lemma[19](https://arxiv.org/html/2606.24999#Thmtheorem19)gives

supφ∈ℋ\|Lm𝒢​\(φ\)−L¯m𝒢​\(φ\)\|≤CN\.\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|L\_\{m\}^\{\\mathscr\{G\}\}\(\\varphi\)\-\\bar\{L\}\_\{m\}^\{\\mathscr\{G\}\}\(\\varphi\)\\right\|\\leq\\frac\{C\}\{N\}\.\(C\.71\)The outer discretization error is treated in the same way, leading to

supφ∈ℋ\|Lm𝒢​\(φ\)−Lm​\(φ\)\|≤CN\.\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|L\_\{m\}^\{\\mathscr\{G\}\}\(\\varphi\)\-L\_\{m\}\(\\varphi\)\\right\|\\leq\\frac\{C\}\{N\}\.\(C\.72\)
For the first term on the right hand side of \([C\.66](https://arxiv.org/html/2606.24999#A3.E66)\), denote and note that

Ygϵ,m​\(t,x\)\\displaystyle Y\_\{g\}^\{\\epsilon,m\}\(t,x\):=Z2​ϵ​\(R~t,x\+ϵ​Z​\(𝐔n−1\)−R~t,x−ϵ​Z​\(𝐔n−1\)\)\\displaystyle=\\frac\{Z\}\{2\\epsilon\}\\left\(\\tilde\{R\}^\{t,x\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\-\\tilde\{R\}^\{t,x\-\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\)\(C\.73\)=Z​∫−11∇xR~t,x\+s​ϵ​Z​\(𝐔n−1\)⊤​Z​d​s2,\\displaystyle=Z\\int\_\{\-1\}^\{1\}\\nabla\_\{x\}\\tilde\{R\}^\{t,x\+s\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)^\{\\top\}Z\\,\\frac\{\\mathrm\{d\}s\}\{2\},Yhϵ,m​\(t,x\)\\displaystyle Y\_\{h\}^\{\\epsilon,m\}\(t,x\):=Z​Z⊤−Id2​ϵ2​\(R~t,x\+ϵ​Z​\(𝐔n−1\)\+R~t,x−ϵ​Z​\(𝐔n−1\)−2​R~t,x​\(𝐔n−1\)\)\\displaystyle=\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{2\\epsilon^\{2\}\}\\left\(\\tilde\{R\}^\{t,x\+\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\+\\tilde\{R\}^\{t,x\-\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)\-2\\tilde\{R\}^\{t,x\}\(\\mathbf\{U\}\_\{n\-1\}\)\\right\)=Z​Z⊤−Id2​∫−11\(1−\|s\|\)​Z⊤​∇x2R~t,x\+s​ϵ​Z​\(𝐔n−1\)​Z​ds\.\\displaystyle=\\frac\{ZZ^\{\\top\}\-I\_\{d\}\}\{2\}\\int\_\{\-1\}^\{1\}\(1\-\|s\|\)Z^\{\\top\}\\nabla\_\{x\}^\{2\}\\tilde\{R\}^\{t,x\+s\\epsilon Z\}\(\\mathbf\{U\}\_\{n\-1\}\)Z\\,\\mathrm\{d\}s\.Set

f~​\(s,x\):=f​\(s,x,𝐔n−1​\(s,x\)\),Ds\(r\):=∂xrXst,x,r=1,…,4\.\\tilde\{f\}\(s,x\):=f\(s,x,\\mathbf\{U\}\_\{n\-1\}\(s,x\)\),\\qquad D\_\{s\}^\{\(r\)\}:=\\partial\_\{x\}^\{r\}X\_\{s\}^\{t,x\},\\quad r=1,\\ldots,4\.ForD\(1\)D^\{\(1\)\}, differentiating the SDE gives a linear equation with bounded drift and multiplicative diffusion coefficients:

d​Ds\(1\)=∇xb​\(s,Xst,x\)​Ds\(1\)​d​s\+2​∑ℓ=1d∇xσℓ​\(s,Xst,x\)​Ds\(1\)​d​Wsℓ\.\\mathrm\{d\}D\_\{s\}^\{\(1\)\}=\\nabla\_\{x\}b\(s,X\_\{s\}^\{t,x\}\)D\_\{s\}^\{\(1\)\}\\,\\mathrm\{d\}s\+\\sqrt\{2\}\\sum\_\{\\ell=1\}^\{d\}\\nabla\_\{x\}\\sigma\_\{\\ell\}\(s,X\_\{s\}^\{t,x\}\)D\_\{s\}^\{\(1\)\}\\,\\mathrm\{d\}W\_\{s\}^\{\\ell\}\.Forr=2,3,4r=2,3,4,D\(r\)D^\{\(r\)\}satisfies

d​Ds\(r\)=As\(r\)​Ds\(r\)​d​s\+2​∑ℓ=1dBℓ,s\(r\)​Ds\(r\)​d​Wsℓ\+Fs\(r\)​d​s\+2​∑ℓ=1dHℓ,s\(r\)​d​Wsℓ,\\mathrm\{d\}D\_\{s\}^\{\(r\)\}=A\_\{s\}^\{\(r\)\}D\_\{s\}^\{\(r\)\}\\,\\mathrm\{d\}s\+\\sqrt\{2\}\\sum\_\{\\ell=1\}^\{d\}B\_\{\\ell,s\}^\{\(r\)\}D\_\{s\}^\{\(r\)\}\\,\\mathrm\{d\}W\_\{s\}^\{\\ell\}\+F\_\{s\}^\{\(r\)\}\\,\\mathrm\{d\}s\+\\sqrt\{2\}\\sum\_\{\\ell=1\}^\{d\}H\_\{\\ell,s\}^\{\(r\)\}\\,\\mathrm\{d\}W\_\{s\}^\{\\ell\},\(C\.74\)whereAs\(r\)A\_\{s\}^\{\(r\)\}andBℓ,s\(r\)B\_\{\\ell,s\}^\{\(r\)\}are bounded, and the forcing terms depend polynomially on lower\-order derivatives:

\|Fs\(2\)\|\+∑ℓ\|Hℓ,s\(2\)\|\\displaystyle\|F\_\{s\}^\{\(2\)\}\|\+\\sum\_\{\\ell\}\|H\_\{\\ell,s\}^\{\(2\)\}\|≤C​\|Ds\(1\)\|2,\\displaystyle\\leq C\|D\_\{s\}^\{\(1\)\}\|^\{2\},\|Fs\(3\)\|\+∑ℓ\|Hℓ,s\(3\)\|\\displaystyle\|F\_\{s\}^\{\(3\)\}\|\+\\sum\_\{\\ell\}\|H\_\{\\ell,s\}^\{\(3\)\}\|≤C​\(\|Ds\(1\)\|3\+\|Ds\(1\)\|​\|Ds\(2\)\|\),\\displaystyle\\leq C\\left\(\|D\_\{s\}^\{\(1\)\}\|^\{3\}\+\|D\_\{s\}^\{\(1\)\}\|\|D\_\{s\}^\{\(2\)\}\|\\right\),\|Fs\(4\)\|\+∑ℓ\|Hℓ,s\(4\)\|\\displaystyle\|F\_\{s\}^\{\(4\)\}\|\+\\sum\_\{\\ell\}\|H\_\{\\ell,s\}^\{\(4\)\}\|≤C​\(\|Ds\(1\)\|4\+\|Ds\(1\)\|2​\|Ds\(2\)\|\+\|Ds\(2\)\|2\+\|Ds\(1\)\|​\|Ds\(3\)\|\)\.\\displaystyle\\leq C\\left\(\|D\_\{s\}^\{\(1\)\}\|^\{4\}\+\|D\_\{s\}^\{\(1\)\}\|^\{2\}\|D\_\{s\}^\{\(2\)\}\|\+\|D\_\{s\}^\{\(2\)\}\|^\{2\}\+\|D\_\{s\}^\{\(1\)\}\|\|D\_\{s\}^\{\(3\)\}\|\\right\)\.Applying BDG and Grönwall first toD\(1\)D^\{\(1\)\}and then inductively to \([C\.74](https://arxiv.org/html/2606.24999#A3.E74)\) yields, for everyp≥2p\\geq 2,

supt,x‖1\+∑r=14supt≤s≤T\|Ds\(r\)\|‖Lp≤C​ec​p\.\\sup\_\{t,x\}\\left\\\|1\+\\sum\_\{r=1\}^\{4\}\\sup\_\{t\\leq s\\leq T\}\|D\_\{s\}^\{\(r\)\}\|\\right\\\|\_\{L^\{p\}\}\\leq Ce^\{cp\}\.\(C\.75\)Since

R~tj,x​\(𝐔n−1\)=g​\(XTtj,x\)\+∑i=jN−1f~​\(ti,Xtitj,x\)​Δ​ti,\\tilde\{R\}^\{t\_\{j\},x\}\(\\mathbf\{U\}\_\{n\-1\}\)=g\(X\_\{T\}^\{t\_\{j\},x\}\)\+\\sum\_\{i=j\}^\{N\-1\}\\tilde\{f\}\(t\_\{i\},X\_\{t\_\{i\}\}^\{t\_\{j\},x\}\)\\Delta t\_\{i\},Faà di Bruno’s formula, along with the fact thatg∈Cp4g\\in C\_\{p\}^\{4\},f~∈Cp1,5\\tilde\{f\}\\in C\_\{p\}^\{1,5\}and the bound \([C\.75](https://arxiv.org/html/2606.24999#A3.E75)\), implies that there are constantC,c\>0C,c\>0andq\>0q\>0such that for everyp≥2p\\geq 2,

supj‖\|∇xR~tj,x​\(𝐔n−1\)\|\+\|∇x2R~tj,x​\(𝐔n−1\)\|‖Lp≤C​ec​p​\(1\+\|x\|q\)\.\\sup\_\{j\}\\left\\\|\|\\nabla\_\{x\}\\tilde\{R\}^\{t\_\{j\},x\}\(\\mathbf\{U\}\_\{n\-1\}\)\|\+\|\\nabla\_\{x\}^\{2\}\\tilde\{R\}^\{t\_\{j\},x\}\(\\mathbf\{U\}\_\{n\-1\}\)\|\\right\\\|\_\{L^\{p\}\}\\leq Ce^\{cp\}\(1\+\|x\|^\{q\}\)\.\(C\.76\)Combining \([C\.73](https://arxiv.org/html/2606.24999#A3.E73)\) with

‖Z‖Lp≤C​p1/2,‖Z​Z⊤−Id‖Lp≤C​p,\\\|Z\\\|\_\{L^\{p\}\}\\leq Cp^\{1/2\},\\qquad\\\|ZZ^\{\\top\}\-I\_\{d\}\\\|\_\{L^\{p\}\}\\leq Cp,and the moment bound ofXtX\_\{t\}gives

sup0<ϵ≤1supj‖\|Ygϵ,m​\(tj,Xtj\)\|\+\|Yhϵ,m​\(tj,Xtj\)\|‖Lp≤C​ec​p,p≥2\.\\sup\_\{0<\\epsilon\\leq 1\}\\sup\_\{j\}\\left\\\|\|Y\_\{g\}^\{\\epsilon,m\}\(t\_\{j\},X\_\{t\_\{j\}\}\)\|\+\|Y\_\{h\}^\{\\epsilon,m\}\(t\_\{j\},X\_\{t\_\{j\}\}\)\|\\right\\\|\_\{L^\{p\}\}\\leq Ce^\{cp\},\\qquad p\\geq 2\.\(C\.77\)
Define the truncation event

ΩK,m,j:=\{suptj≤t≤T\|Xttj,Xtj\|<K\}∩\{\|Ygϵ,m​\(tj,Xtj\)\|≤K\}∩\{\|Yhϵ,m​\(tj,Xtj\)\|≤K\}\.\\Omega\_\{K,m,j\}:=\\left\\\{\\sup\_\{t\_\{j\}\\leq t\\leq T\}\\left\|X\_\{t\}^\{t\_\{j\},X\_\{t\_\{j\}\}\}\\right\|<K\\right\\\}\\cap\\\{\|Y\_\{g\}^\{\\epsilon,m\}\(t\_\{j\},X\_\{t\_\{j\}\}\)\|\\leq K\\\}\\cap\\\{\|Y\_\{h\}^\{\\epsilon,m\}\(t\_\{j\},X\_\{t\_\{j\}\}\)\|\\leq K\\\}\.WithqK=max⁡\{2,⌊α​log⁡\(1\+K\)⌋\}q\_\{K\}=\\max\\\{2,\\lfloor\\alpha\\log\(1\+K\)\\rfloor\\\}, Markov’s inequality and \([C\.77](https://arxiv.org/html/2606.24999#A3.E77)\) give

ℙ​\(\|Ygϵ,m​\(tj,Xtj\)\|\>K\)\\displaystyle\\mathbb\{P\}\(\|Y\_\{g\}^\{\\epsilon,m\}\(t\_\{j\},X\_\{t\_\{j\}\}\)\|\>K\)≤\(C​ec​qKK\)qK,\\displaystyle\\leq\\left\(\\frac\{Ce^\{cq\_\{K\}\}\}\{K\}\\right\)^\{q\_\{K\}\},ℙ​\(\|Yhϵ,m​\(tj,Xtj\)\|\>K\)\\displaystyle\\mathbb\{P\}\(\|Y\_\{h\}^\{\\epsilon,m\}\(t\_\{j\},X\_\{t\_\{j\}\}\)\|\>K\)≤\(C​ec​qKK\)qK\.\\displaystyle\\leq\\left\(\\frac\{Ce^\{cq\_\{K\}\}\}\{K\}\\right\)^\{q\_\{K\}\}\.Choosingα\>0\\alpha\>0small enough andKKlarge enough yields

sup1≤j≤Nℙ\(ΩK,m,jc\)≤C1exp\(−C2\(log\(1\+K\)\)2\)\.K≥2,\\sup\_\{1\\leq j\\leq N\}\\mathbb\{P\}\(\\Omega\_\{K,m,j\}^\{c\}\)\\leq C\_\{1\}\\exp\\\!\\left\(\-C\_\{2\}\(\\log\(1\+K\)\)^\{2\}\\right\)\.\\qquad K\\geq 2,\(C\.78\)Write the time\-discretized sample loss as

Gm​\(φ;W\)=∑j=0N−1Δ​tj​ℓm,j​\(φ;W\),G\_\{m\}\(\\varphi;W\)=\\sum\_\{j=0\}^\{N\-1\}\\Delta t\_\{j\}\\,\\ell\_\{m,j\}\(\\varphi;W\),whereℓm,j\\ell\_\{m,j\}is the squared value\-gradient\-Hessian loss at the grid timetjt\_\{j\}\. Clip the grid loss term by the term:

Gm\(K\)​\(φ;W\):=∑j=0N−1Δ​tj​ℓm,j​\(φ;W\)​𝟏ΩK,m,j\.G\_\{m\}^\{\(K\)\}\(\\varphi;W\):=\\sum\_\{j=0\}^\{N\-1\}\\Delta t\_\{j\}\\,\\ell\_\{m,j\}\(\\varphi;W\)\\mathbf\{1\}\_\{\\Omega\_\{K,m,j\}\}\.Hence, after summing over the time grid and using∑j=0N−1Δ​tj=T\\sum\_\{j=0\}^\{N\-1\}\\Delta t\_\{j\}=T,

\|Gm\(K\)​\(φ;W\)\|≤C​\(1\+Kq\),\|G\_\{m\}^\{\(K\)\}\(\\varphi;W\)\|\\leq C\(1\+K^\{q\}\),\(C\.79\)whereqqdepends on the polynomial growth condition ofggandff\. Recall the decomposition \([C\.52](https://arxiv.org/html/2606.24999#A3.E52)\), by McDiarmid’s inequality and the parameterized Rademacher bound,

supφ∈ℋ\|1B​∑i=1BGm\(K\)​\(φ;Wi\)−𝔼​\[Gm\(K\)​\(φ;W\)\]\|\\displaystyle\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}G\_\{m\}^\{\(K\)\}\(\\varphi;W\_\{i\}\)\-\\mathbb\{E\}\[G\_\{m\}^\{\(K\)\}\(\\varphi;W\)\]\\right\|\(C\.80\)≤C​\(1\+log⁡\(1/δ\)\)​𝒫​\(d\)​\(1\+Kq\)B​log⁡\(C​R​p​\(1\+Kq\)​B\)\.\\displaystyle\\qquad\\leq C\(1\+\\sqrt\{\\log\(1/\\delta\)\}\)\\frac\{\\mathcal\{P\}\(d\)\(1\+K^\{q\}\)\}\{\\sqrt\{B\}\}\\sqrt\{\\log\(CRp\(1\+K^\{q\}\)\\sqrt\{B\}\)\}\.with probability at least1−δ1\-\\delta\. Let

RK​\(W\):=supφ∈ℋ\|Gm​\(φ;W\)−Gm\(K\)​\(φ;W\)\|\.R\_\{K\}\(W\):=\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\|G\_\{m\}\(\\varphi;W\)\-G\_\{m\}^\{\(K\)\}\(\\varphi;W\)\|\.Since the networks inℋ\\mathcal\{H\}are uniformly bounded,

supφ∈ℋℓm,j​\(φ;W\)≤C​\(1\+\|Yv\|2\+‖Ygϵ,m‖2\+‖Yhϵ,m‖2\)\.\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\ell\_\{m,j\}\(\\varphi;W\)\\leq C\\left\(1\+\|Y\_\{v\}\|^\{2\}\+\\\|Y\_\{g\}^\{\\epsilon,m\}\\\|^\{2\}\+\\\|Y\_\{h\}^\{\\epsilon,m\}\\\|^\{2\}\\right\)\.Therefore,

𝔼​RK​\(W\)\\displaystyle\\mathbb\{E\}R\_\{K\}\(W\)≤∑j=0N−1Δ​tj​𝔼​\[supφ∈ℋℓm,j​\(φ;W\)​𝟏ΩK,m,jc\]\\displaystyle\\leq\\sum\_\{j=0\}^\{N\-1\}\\Delta t\_\{j\}\\mathbb\{E\}\\left\[\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\ell\_\{m,j\}\(\\varphi;W\)\\mathbf\{1\}\_\{\\Omega\_\{K,m,j\}^\{c\}\}\\right\]≤C​∑j=0N−1Δ​tj​‖1\+\|Yv\|2\+‖Ygϵ,m‖2\+‖Yhϵ,m‖2‖L2​ℙ​\(ΩK,m,jc\)1/2\.\\displaystyle\\leq C\\sum\_\{j=0\}^\{N\-1\}\\Delta t\_\{j\}\\left\\\|1\+\|Y\_\{v\}\|^\{2\}\+\\\|Y\_\{g\}^\{\\epsilon,m\}\\\|^\{2\}\+\\\|Y\_\{h\}^\{\\epsilon,m\}\\\|^\{2\}\\right\\\|\_\{L^\{2\}\}\\mathbb\{P\}\(\\Omega\_\{K,m,j\}^\{c\}\)^\{1/2\}\.It follows from the moment bound \([C\.77](https://arxiv.org/html/2606.24999#A3.E77)\), the moments ofXtjX\_\{t\_\{j\}\}, the truncation tail \([C\.78](https://arxiv.org/html/2606.24999#A3.E78)\), and the identity∑j=0N−1Δ​tj=T\\sum\_\{j=0\}^\{N\-1\}\\Delta t\_\{j\}=Tthat

𝔼​RK​\(W\)≤C​exp⁡\(−C​\(log⁡\(1\+K\)\)2\)\.\\mathbb\{E\}R\_\{K\}\(W\)\\leq C\\exp\\\!\\left\(\-C\(\\log\(1\+K\)\)^\{2\}\\right\)\.Markov’s inequality therefore gives, with probability at least1−δ1\-\\delta,

1B​∑i=1BRK​\(Wi\)\+𝔼​RK​\(W\)≤C​δ−1​exp⁡\(−C​\(log⁡\(1\+K\)\)2\)\.\\frac\{1\}\{B\}\\sum\_\{i=1\}^\{B\}R\_\{K\}\(W\_\{i\}\)\+\\mathbb\{E\}R\_\{K\}\(W\)\\leq C\\delta^\{\-1\}\\exp\\\!\\left\(\-C\(\\log\(1\+K\)\)^\{2\}\\right\)\.ChoosingK=exp⁡\(C​log⁡B\+log⁡\(1/δ\)\)K=\\exp\\\!\\left\(C\\sqrt\{\\log B\+\\log\(1/\\delta\)\}\\right\), we obtain

supφ∈ℋ\|L~m\(B\)​\(φ\)−Lm𝒢​\(φ\)\|\\displaystyle\\sup\_\{\\varphi\\in\\mathcal\{H\}\}\\left\|\\tilde\{L\}\_\{m\}^\{\(B\)\}\(\\varphi\)\-L\_\{m\}^\{\\mathscr\{G\}\}\(\\varphi\)\\right\|\(C\.81\)≤C​\(1\+log⁡\(1/δ\)\)​𝒫​\(d\)B1/2​exp⁡\(C​log⁡B\+log⁡\(1/δ\)\)\.\\displaystyle\\qquad\\leq C\\left\(1\+\\sqrt\{\\log\(1/\\delta\)\}\\right\)\\frac\{\\mathcal\{P\}\(d\)\}\{B^\{1/2\}\}\\exp\\\!\\left\(C\\sqrt\{\\log B\+\\log\(1/\\delta\)\}\\right\)\.Combining \([C\.64](https://arxiv.org/html/2606.24999#A3.E64)\), \([C\.65](https://arxiv.org/html/2606.24999#A3.E65)\), \([C\.66](https://arxiv.org/html/2606.24999#A3.E66)\), \([C\.72](https://arxiv.org/html/2606.24999#A3.E72)\), and \([C\.81](https://arxiv.org/html/2606.24999#A3.E81)\) proves the strong\-simulator bound\.

### C\.8Proof of Theorem[10](https://arxiv.org/html/2606.24999#Thmtheorem10)

The conclusion follows by takingδ=δ/n\\delta=\\delta/nin Theorem[9](https://arxiv.org/html/2606.24999#Thmtheorem9), and combining with the conclusion in Theorem[6](https://arxiv.org/html/2606.24999#Thmtheorem6)\.

## References

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