Analysis of Adam Algorithms for Stochastic Dynamic Systems

# Analysis of Adam Algorithms for Stochastic Dynamic Systems
Source: [https://arxiv.org/html/2606.28879](https://arxiv.org/html/2606.28879)
Yifei Jinand Lei Guo\\IEEEmembershipFellow, IEEEThis paper was supported by the National Key Research and Development Program under Grant No\. 2024YFC3307200 and the National Natural Science Foundation of China under Grant No\. 12288201\.Xin Zheng and Lei Guo are with the State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, and also with the School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China\. \(e\-mails: zhengxin2021@amss\.ac\.cn, lguo@amss\.ac\.cn\)\.Yifei Jin is with the School of Advanced Interdisciplinary Sciences, University of Chinese Academy of Sciences, Beijing 101408, China, and also with the State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China\. \(e\-mail: jinyifei@amss\.ac\.cn\)\.

###### Abstract

The adaptive moment estimation algorithm, known as Adam, is widely used in modern machine learning, owing to its low per\-iteration complexity and strong empirical performance\. Despite its prevalent use, the theoretical foundation of Adam remains largely unexplored for time\-varying and nonstationary systems\. In fact, the existing theoretical analyses of Adam\-type algorithms are primarily concerned with time\-invariant model parameters and explicitly or implicitly rely on independent and identically distributed \(i\.i\.d\.\) data assumptions, under which the learning task can be formulated as minimizing a fixed expected objective with a static minimizer\. However, such assumptions are often violated in time\-varying and nonstationary systems, thereby calling for a theoretical investigation beyond the conventional yet idealized i\.i\.d\. setting\. The main objective of this paper is to solve this challenging problem by establishing a general theory of Adam for time\-varying and nonstationary stochastic systems\. We will introduce some new techniques for analyzing the products of nonstationary and dependent random matrices induced by Adam’s coupled first\- and second\-moment recursions, and will construct a new stochastic Lyapunov function that blends these two moment dynamics\. Under a stochastic excitation condition that allows nonstationary and dependent data, we will derive both parameter tracking and output prediction error bounds explicitly, quantifying the effects of stepsize, first\- and second\-momentum parameters, gradient noise and parameter drift\. These bounds not only provide guarantees for Adam performance, but also provide guidelines for hyperparameter selection\. Experiments on both synthetic and real\-world data validate our theory and design guidelines\.

\{IEEEkeywords\}

Stochastic systems, Adam, adaptive filtering, performance bound, stochastic excitation, products of random matrices, stochastic Lyapunov function\.

## 1Introduction

The adaptive moment estimation algorithm, known as Adam\[[1](https://arxiv.org/html/2606.28879#bib.bib1)\], is one of the most widely used and well\-known optimization methods in modern machine learning, owing to its adaptive per\-coordinate stepsizes via first\- and second\-moment estimates, which yield stable and robust updates in large\-scale stochastic optimization\. It has found broad applications in artificial intelligence, including generative adversarial networks \(GANs\)\[[2](https://arxiv.org/html/2606.28879#bib.bib2)\]and generative pre\-trained transformers \(GPTs\)\[[3](https://arxiv.org/html/2606.28879#bib.bib3)\], vision transformers \(ViTs\)\[[4](https://arxiv.org/html/2606.28879#bib.bib4)\], as well as reinforcement learning–based feedback control systems, e\.g\., soft actor\-critic \(SAC\)\[[5](https://arxiv.org/html/2606.28879#bib.bib5)\]\.

Despite its remarkable empirical success, the theoretical understanding of Adam remains limited\[[6](https://arxiv.org/html/2606.28879#bib.bib6),[7](https://arxiv.org/html/2606.28879#bib.bib7),[8](https://arxiv.org/html/2606.28879#bib.bib8)\], although substantial theoretical efforts have been devoted to analyzing the behavior of Adam \(e\.g\.,\[[9](https://arxiv.org/html/2606.28879#bib.bib9),[10](https://arxiv.org/html/2606.28879#bib.bib10),[11](https://arxiv.org/html/2606.28879#bib.bib11),[7](https://arxiv.org/html/2606.28879#bib.bib7),[12](https://arxiv.org/html/2606.28879#bib.bib12),[13](https://arxiv.org/html/2606.28879#bib.bib13),[14](https://arxiv.org/html/2606.28879#bib.bib14),[8](https://arxiv.org/html/2606.28879#bib.bib8)\]\)\. For example, Reddi et al\.\[[9](https://arxiv.org/html/2606.28879#bib.bib9)\]revealed potential divergence issues of Adam and proposed the stabilized variant AMSGrad to address these issues\. In contrast, Zhang et al\.\[[11](https://arxiv.org/html/2606.28879#bib.bib11)\]showed that, under appropriate conditions, the original Adam can still converge without any modification to its update rules\. More recent studies have relaxed smoothness or regularity assumptions and established refined convergence guarantees for Adam under broader settings, including non\-uniform smoothness conditions\[[7](https://arxiv.org/html/2606.28879#bib.bib7),[14](https://arxiv.org/html/2606.28879#bib.bib14)\], generalized affine\-variance noise assumptions\[[12](https://arxiv.org/html/2606.28879#bib.bib12)\], nonsmooth optimization\[[13](https://arxiv.org/html/2606.28879#bib.bib13)\], and unified convergence frameworks\[[8](https://arxiv.org/html/2606.28879#bib.bib8)\]\.

However, most existing analyses of Adam\-type algorithms are tailored to a static learning regime: they typically assume time\-invariant model parameters and i\.i\.d\. or stationary ergodic data\. Under these assumptions, the learning problem reduces to minimizing a fixed objective under mathematical expectation with a static minimizer\. In many applications of interest, however, Adam is deployed in time\-varying and nonstationary dynamical systems\[[15](https://arxiv.org/html/2606.28879#bib.bib15),[16](https://arxiv.org/html/2606.28879#bib.bib16)\], where the expected objective and its minimizer may drift over time\. This mismatch renders static guarantees inadequate and motivates theoretical frameworks that rigorously characterize tracking and prediction performance\. However, the corresponding theoretical foundations remain largely underdeveloped\.

To address this challenge, we draw on the well\-developed theory of adaptive identification under time\-varying and nonstationary scenarios, which provides powerful analytical tools for the performance analysis of Adam in such settings\. For example, Guo\[[17](https://arxiv.org/html/2606.28879#bib.bib17)\]pioneered the conditional excitation \(CE\) condition, which can be viewed as a stochastic relaxation of the deterministic persistent excitation \(PE\) requirement and is well suited to feedback\-driven stochastic control systems\. Under this condition, stability and convergence of Kalman filter–based parameter tracking were established for time\-varying stochastic regression models with nonstationary and correlated data\. Subsequently, a more general CE condition was proposed by Guo\[[18](https://arxiv.org/html/2606.28879#bib.bib18)\], leading to a unified theory for stability and performance analysis of stochastic regression models with time\-varying parameters and nonstationary data streams, covering several fundamental adaptive tracking algorithms such as the Kalman filter, least mean squares \(LMS\), and forgetting factor recursive least squares \(FFRLS\)\[[18](https://arxiv.org/html/2606.28879#bib.bib18)\]\. Moreover, inspired by the ideas in the analysis of recursive stochastic algorithms with vanishing gains\[[19](https://arxiv.org/html/2606.28879#bib.bib19)\], more accurate performance analysis was conducted in\[[20](https://arxiv.org/html/2606.28879#bib.bib20),[21](https://arxiv.org/html/2606.28879#bib.bib21),[22](https://arxiv.org/html/2606.28879#bib.bib22)\]for recursive tracking algorithms where the adaptation gains do not vanish\.It is worth noting that the Kalman filter and FFRLS employ matrix\-valued adaptation gains, reflecting their Newton\-type \(second\-order\) nature\. This typically incurs a heavy computational burden in high\-dimensional parameter settings, which partly explains the widespread use of gradient\-type methods, where the adaptation gain is usually scalar\.

Among the gradient\-type algorithms, the well\-known LMS is probably the simplest one\. However,\[[23](https://arxiv.org/html/2606.28879#bib.bib23)\]showed that incorporating a heavy\-ball momentum term can accelerate the convergence of gradient\-type algorithms\. In particular, applying this idea to LMS leads to the momentum LMS \(MLMS\) algorithm\.\[[24](https://arxiv.org/html/2606.28879#bib.bib24)\]studied MLMS algorithms for time\-varying stochastic linear systems under the CE condition introduced in\[[17](https://arxiv.org/html/2606.28879#bib.bib17)\], establishing tracking and regret bounds under general nonstationary streaming data settings\. Unlike most existing LMS analyses, their approach augments the state vector to incorporate the momentum term, leading to a higher\-dimensional state\-space representation\. This reformulation enables a more tractable analysis of the resulting stochastic matrix products and offers useful insights for the study of the more popular Adam algorithms in this paper\.

The theoretical study of the Adam is technically more challenging, because Adam couples adaptive recursions for the first\- and second\-moment estimates and applies a coordinate\-wise, time\-varying rescaling, resulting in a data\-dependent, nonlinear update whose components evolve at different rates and are intricately coupled\. As a result, most existing analytical techniques developed for static or linear stochastic regimes are not directly applicable for establishing rigorous guarantees for time\-varying parameter with nonstationary dataset\. To overcome these technical difficulties and lay a theoretical foundation of Adam for stochastic dynamical systems, are the main objectives of this paper\. Our main contributions are summarized as follows\.

1. 1\.*First,*we establish explicit parameter tracking guarantees for Adam over a broad class of nonlinear stochastic systems with time\-varying parameters and nonstationary data\. In contrast to most existing Adam analyses for static objectives that do not provide parameter tracking guarantees, our results yield tracking\-type performance characterizations under a general CE condition\. Our results are established by introducing new methods in the analysis of the product of nonstationary and dependent random matrices with augmented dimension\.
2. 2\.*Second,*we derive explicit output prediction error bounds of Adam for time\-varying systems with nonstationary dataset\. In contrast to the existing prediction\-type analyses developed for time\-invariant parameter settings under i\.i\.d\. data, our results*do not*require any excitation condition on the system data\. Our results are established by introducing a new stochastic Lyapunov function constructed based on a mixture of the first\- and second\- moment in Adam\.
3. 3\.*Third,*both the parameter tracking and output prediction bounds are established for the first time, which quantify the effects of the hyperparameters, gradient noise, and drift magnitude, thereby providing theoretical guidelines for hyperparameter selection\.

The remainder of this paper is organized as follows\. Section[2](https://arxiv.org/html/2606.28879#S2)will present the problem formulation and the Adam algorithm\. Section[3](https://arxiv.org/html/2606.28879#S3)will establish the main results on parameter tracking and output prediction performance, followed by their proofs in Section[4](https://arxiv.org/html/2606.28879#S4)\. Section[5](https://arxiv.org/html/2606.28879#S5)will validate the theoretical findings via simulations and experiments on both synthetic and real\-world datasets, and the results support the proposed guidelines for hyperparameter selection\. Finally, Section[6](https://arxiv.org/html/2606.28879#S6)will conclude the paper with some remarks\.

Notation\.For a matrixAA,λmin​\(A\)\\lambda\_\{\\min\}\(A\)andλmax​\(A\)\\lambda\_\{\\max\}\(A\)denote its minimum and maximum eigenvalues, respectively\. For symmetric matricesAAandBB,A⪰BA\\succeq B\(resp\.,A≻BA\\succ B\) meansA−BA\-Bis positive semidefinite \(resp\., positive definite\), andA⪯BA\\preceq B\(resp\.,A≺BA\\prec B\) meansB⪰AB\\succeq A\(resp\.,B≻AB\\succ A\)\. The transpose ofAAis denoted byATA^\{\\mathrm\{T\}\}, and‖A‖\\\|A\\\|denotes the induced Euclidean\-norm‖A‖=λmax​\(AT​A\)\\\|A\\\|=\\sqrt\{\\lambda\_\{\\max\}\(A^\{\\mathrm\{T\}\}A\)\}\. The mathematical expectation operator is denoted by𝔼​\[⋅\]\\mathbb\{E\}\[\\cdot\], and𝔼\[⋅∣ℱk\]\\mathbb\{E\}\[\\cdot\\mid\\mathcal\{F\}\_\{k\}\]denotes the conditional expectation with respect toℱk\\mathcal\{F\}\_\{k\}, where\{ℱk\}k≥0\\\{\\mathcal\{F\}\_\{k\}\\\}\_\{k\\geq 0\}is a nondecreasing sequence ofσ\\sigma\-algebras\. A stochastic sequence\{xk,ℱk\}\\\{x\_\{k\},\\mathcal\{F\}\_\{k\}\\\}is adapted ifxkx\_\{k\}isℱk\\mathcal\{F\}\_\{k\}\-measurable for allk≥0k\\geq 0\. For real sequences\{ak\}\\\{a\_\{k\}\\\}and\{bk\}\\\{b\_\{k\}\\\}withbk\>0b\_\{k\}\>0,ak=o​\(bk\)a\_\{k\}=o\(b\_\{k\}\)meansak/bk→0a\_\{k\}/b\_\{k\}\\to 0, andak=O​\(bk\)a\_\{k\}=O\(b\_\{k\}\)means\|ak\|≤M​bk\|a\_\{k\}\|\\leq Mb\_\{k\}for some constantM\>0M\>0\. Finally,𝟎\\mathbf\{0\}andIIdenote the zero and identity matrices, respectively\.

## 2Problem formulation and algorithm

### 2\.1Problem formulation

We consider the time\-varying stochastic nonlinear model

yk\+1=Gk​\(ϕk,θk,εk\+1\),k≥0,y\_\{k\+1\}=G\_\{k\}\(\\phi\_\{k\},\\theta\_\{k\},\\varepsilon\_\{k\+1\}\),~~k\\geq 0,\(1\)whereyk\+1∈ℝy\_\{k\+1\}\\in\\mathbb\{R\}is the system observation,ϕk∈ℝd1\\phi\_\{k\}\\in\\mathbb\{R\}^\{d\_\{1\}\}\(d1≥1d\_\{1\}\\geq 1\) is the stochastic regressor,θk∈ℝd2\\theta\_\{k\}\\in\\mathbb\{R\}^\{d\_\{2\}\}\(d2≥1d\_\{2\}\\geq 1\) is the unknown time\-varying parameter to be estimated, andεk\+1\\varepsilon\_\{k\+1\}denotes the random noise\. The mappingGk:ℝd1×ℝd2×ℝ→ℝG\_\{k\}:\\mathbb\{R\}^\{d\_\{1\}\}\\times\\mathbb\{R\}^\{d\_\{2\}\}\\times\\mathbb\{R\}\\to\\mathbb\{R\}is known\. Furthermore, the parameter variation process is characterized by the sequence\{Δk\}k≥1\\\{\\Delta\_\{k\}\\\}\_\{k\\geq 1\}, where

Δk≜θk−θk−1,k≥1\.\\Delta\_\{k\}\\triangleq\\theta\_\{k\}\-\\theta\_\{k\-1\},~~k\\geq 1\.\(2\)
Given the data stream generated by \([1](https://arxiv.org/html/2606.28879#S2.E1)\), at each timekkwe form a one\-step\-ahead predictory^k\+1\\hat\{y\}\_\{k\+1\}based onℱk′\\mathcal\{F\}\_\{k\}^\{\\prime\}, whereℱk′\\mathcal\{F\}\_\{k\}^\{\{\}^\{\\prime\}\}is theσ\\sigma\-algebra generated by the available information up to timekk, i\.e\.,ℱk′=σ​\{yi,ϕi,i≤k\}\\mathcal\{F\}\_\{k\}^\{\\prime\}=\\sigma\\\{y\_\{i\},\\phi\_\{i\},i\\leq k\\\}\. After observingyk\+1y\_\{k\+1\}, the prediction error is quantified by the instantaneous loss

ℒ​\(yk\+1,y^k\+1\),\\mathcal\{L\}\\\!\\left\(y\_\{k\+1\},\\hat\{y\}\_\{k\+1\}\\right\),\(3\)whereℒ:ℝ×ℝ→ℝ\+\\mathcal\{L\}:\\mathbb\{R\}\\times\\mathbb\{R\}\\to\\mathbb\{R\}^\{\+\}is the given loss function that measures the discrepancy between the observationyk\+1y\_\{k\+1\}and the predictiony^k\+1\\hat\{y\}\_\{k\+1\}\. Without loss of generality, we assume thaty^k\+1\\hat\{y\}\_\{k\+1\}can be described by the following parametric representation

y^k\+1​\(ϑ\)=fk​\(ϕk,ϑ\),k≥0,\\hat\{y\}\_\{k\+1\}\(\\vartheta\)=f\_\{k\}\(\\phi\_\{k\},\\vartheta\),~~k\\geq 0,\(4\)where the mappingfk​\(⋅,⋅\)f\_\{k\}\(\\cdot,\\cdot\)is known, andϑ∈ℝd2\\vartheta\\in\\mathbb\{R\}^\{d\_\{2\}\}\.

Then anℱk′\\mathcal\{F\}\_\{k\}^\{\\prime\}\-measurable one\-step\-ahead oracle predictory^k\+1∗\\hat\{y\}\_\{k\+1\}^\{\\ast\}can be defined asy^k\+1∗=fk​\(ϕk,θ^k∗\),\\hat\{y\}\_\{k\+1\}^\{\\ast\}=f\_\{k\}\(\\phi\_\{k\},\\hat\{\\theta\}\_\{k\}^\{\\ast\}\),whereθ^k∗∈arg​minϑ∈ℝd2⁡𝔼​\[ℒ​\(yk\+1,y^k\+1​\(ϑ\)\)∣ℱk′\]\.\\hat\{\\theta\}\_\{k\}^\{\\ast\}\\in\\operatorname\*\{arg\\,min\}\\limits\_\{\\vartheta\\in\\mathbb\{R\}^\{d\_\{2\}\}\}\\mathbb\{E\}\\\!\\left\[\\mathcal\{L\}\\\!\\left\(y\_\{k\+1\},\\hat\{y\}\_\{k\+1\}\(\\vartheta\)\\right\)\\mid\\mathcal\{F\}\_\{k\}^\{\\prime\}\\right\]\.

Our objective is to track the time\-varying parameterθk\\theta\_\{k\}and to minimize the averaged prediction error under the lossℒ​\(⋅,⋅\)\\mathcal\{L\}\(\\cdot,\\cdot\)\. To this end, we first introduce the commonly used Adam algorithm in the next subsection\.

### 2\.2The Adam algorithm

To facilitate the presentation of the algorithm, we need several definitions\. For anyk≥0k\\geq 0and anyϑ∈ℝd2\\vartheta\\in\\mathbb\{R\}^\{d\_\{2\}\}, define the Clarke subdifferential selections by

ℓk\+1​\(ϑ\)\\displaystyle\\ell\_\{k\+1\}\(\\vartheta\)∈∂y^ℒ​\(yk\+1,fk​\(ϕk,ϑ\)\),\\displaystyle\\in\\partial\_\{\\hat\{y\}\}\\mathcal\{L\}\\\!\\left\(y\_\{k\+1\},f\_\{k\}\(\\phi\_\{k\},\\vartheta\)\\right\),\(5\)gk​\(ϑ\)\\displaystyle g\_\{k\}\(\\vartheta\)∈∂ϑfk​\(ϕk,ϑ\)⊂ℝd2,\\displaystyle\\in\\partial\_\{\\vartheta\}f\_\{k\}\(\\phi\_\{k\},\\vartheta\)\\subset\\mathbb\{R\}^\{d\_\{2\}\},where∂\\partialdenotes the Clarke subdifferential\[[25](https://arxiv.org/html/2606.28879#bib.bib25)\]\. Evaluating the above quantities at the current estimateθ^k\\hat\{\\theta\}\_\{k\}generated based on the observations, we define

ℓk\+1≜ℓk\+1​\(θ^k\),gk≜gk​\(θ^k\)\.\\ell\_\{k\+1\}\\triangleq\\ell\_\{k\+1\}\(\\hat\{\\theta\}\_\{k\}\),~~g\_\{k\}\\triangleq g\_\{k\}\(\\hat\{\\theta\}\_\{k\}\)\.\(6\)We also use the Frobenius norm∥⋅∥Q\\\|\\cdot\\\|\_\{Q\}induced by a positive definite matrixQQ, defined by‖x‖Q2≜xT​Q​x\\\|x\\\|\_\{Q\}^\{2\}\\triangleq x^\{\\mathrm\{T\}\}Qxforx∈ℝd2x\\in\\mathbb\{R\}^\{d\_\{2\}\}\. To keep the parameter estimates in a prescribed admissible set, we employ the projection operatorΠQ​\(⋅\)\\Pi\_\{Q\}\(\\cdot\)below \(see, e\.g\.,\[[26](https://arxiv.org/html/2606.28879#bib.bib26)\]\)\.

###### Definition 1

Given a positive definite matrixQQ, define the projection mappingΠQ:ℝd2→D\\Pi\_\{Q\}:\\mathbb\{R\}^\{d\_\{2\}\}\\to Dby

ΠQ​\(x1\)≜arg⁡minx2∈D⁡‖x1−x2‖Q,∀x1∈ℝd2,\\Pi\_\{Q\}\(x\_\{1\}\)\\triangleq\\arg\\min\_\{x\_\{2\}\\in D\}\\,\\\|x\_\{1\}\-x\_\{2\}\\\|\_\{Q\},~~\\forall~x\_\{1\}\\in\\mathbb\{R\}^\{d\_\{2\}\},\(7\)whereD⊂ℝd2D\\subset\\mathbb\{R\}^\{d\_\{2\}\}is a given compact convex set such thatθk∈D\\theta\_\{k\}\\in Dfor allk≥0k\\geq 0\(see Assumption[1](https://arxiv.org/html/2606.28879#Thmassumption1)\)\.

We are now in a position to present the Adam algorithm\. In this algorithm,⊙\\odotand⊘\\oslashdenote elementwise multiplication and division, respectively, and the square root⋅\\sqrt\{\\cdot\}is taken elementwise for vectors\. Moreover,𝟏∈ℝd2\\mathbf\{1\}\\in\\mathbb\{R\}^\{d\_\{2\}\}denotes the all\-ones vector, anddiag⁡\(⋅\)\\operatorname\{diag\}\(\\cdot\)maps a vector to a diagonal matrix, which will be used in the subsequent analysis\.

Algorithm 1The Adam Algorithm1:stepsize

α\>0\\alpha\>0, hyperparameters

β1,β2∈\[0,1\)\\beta\_\{1\},\\beta\_\{2\}\\in\[0,1\), offset

c0\>0c\_\{0\}\>0, admissible set

DD
2:Initialize

θ^0∈D\\hat\{\\theta\}\_\{0\}\\in D,

m0=0m\_\{0\}=0,

v0=0v\_\{0\}=0
3:for

k=0,1,2,…k=0,1,2,\\ldotsdo

4:

mk\+1←β1​mk\+\(1−β1\)​ℓk\+1​gkm\_\{k\+1\}\\leftarrow\\beta\_\{1\}m\_\{k\}\+\(1\-\\beta\_\{1\}\)\\,\\ell\_\{k\+1\}g\_\{k\}
5:

vk\+1←β2​vk\+\(1−β2\)​ℓk\+12​\(gk⊙gk\)v\_\{k\+1\}\\leftarrow\\beta\_\{2\}v\_\{k\}\+\(1\-\\beta\_\{2\}\)\\,\\ell\_\{k\+1\}^\{2\}\\,\(g\_\{k\}\\odot g\_\{k\}\)
6:

sk\+1←vk\+1\+c0​1s\_\{k\+1\}\\leftarrow\\sqrt\{v\_\{k\+1\}\}\+c\_\{0\}\\,\\mathbf\{1\}
7:

Qk\+1≜diag⁡\(sk\+1\)\(≻0\)Q\_\{k\+1\}\\triangleq\\operatorname\{diag\}\(s\_\{k\+1\}\)\\ \\ \(\\succ 0\)
8:

θ^k\+1←ΠQk\+1​\(θ^k−α​\(mk\+1⊘sk\+1\)\)\\hat\{\\theta\}\_\{k\+1\}\\leftarrow\\Pi\_\{Q\_\{k\+1\}\}\\\!\\Bigl\(\\hat\{\\theta\}\_\{k\}\-\\alpha\\,\(m\_\{k\+1\}\\oslash s\_\{k\+1\}\)\\Bigr\)
9:endfor

Withvkv\_\{k\}defined as in Algorithm[1](https://arxiv.org/html/2606.28879#alg1), we define

Vk≜diag⁡\(vk\),V¯k≜\(Vk12\+c0​I\)2≻𝟎\.V\_\{k\}\\triangleq\\operatorname\{diag\}\(v\_\{k\}\),~~\\bar\{V\}\_\{k\}\\triangleq\(V\_\{k\}^\{\\frac\{1\}\{2\}\}\+c\_\{0\}I\)^\{2\}\\succ\\mathbf\{0\}\.\(8\)ThenQkQ\_\{k\}can be rewritten asdiag⁡\(sk\)=V¯k12\\operatorname\{diag\}\(s\_\{k\}\)=\\bar\{V\}\_\{k\}^\{\\frac\{1\}\{2\}\}andmk⊘sk=V¯k−12​mkm\_\{k\}\\oslash s\_\{k\}=\\bar\{V\}\_\{k\}^\{\-\\frac\{1\}\{2\}\}m\_\{k\}\. Hence, the update admits the matrix form

θ^k\+1=ΠV¯k\+112​\(θ^k−α​V¯k\+1−12​mk\+1\)\.\\hat\{\\theta\}\_\{k\+1\}=\\Pi\_\{\\bar\{V\}\_\{k\+1\}^\{\\frac\{1\}\{2\}\}\}\\\!\\Bigl\(\\hat\{\\theta\}\_\{k\}\-\\alpha\\,\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{2\}\}m\_\{k\+1\}\\Bigr\)\.\(9\)All subsequent analysis will be based on \([8](https://arxiv.org/html/2606.28879#S2.E8)\) and \([9](https://arxiv.org/html/2606.28879#S2.E9)\)\.

## 3The main results

In this section, we derive bounds both on the parameter tracking error and the output prediction error of Algorithm[1](https://arxiv.org/html/2606.28879#alg1)\.

To facilitate the presentation of the results, letℱk\\mathcal\{F\}\_\{k\}denote theσ\\sigma\-algebra generated by\{yi,ϕi,θi,i≤k\}\\\{y\_\{i\},\\phi\_\{i\},\\theta\_\{i\},\\,i\\leq k\\\}\. For anyk≥0k\\geq 0andϑ∈ℝd2\\vartheta\\in\\mathbb\{R\}^\{d\_\{2\}\}, define

ψk​\(ϑ\)≜𝔼​\[ℓk\+1​\(ϑ\)∣ℱk\],wk\+1​\(ϑ\)≜ℓk\+1​\(ϑ\)−ψk​\(ϑ\)\.\\psi\_\{k\}\(\\vartheta\)\\triangleq\\mathbb\{E\}\\left\[\\ell\_\{k\+1\}\(\\vartheta\)\\mid\\mathcal\{F\}\_\{k\}\\right\],~~w\_\{k\+1\}\(\\vartheta\)\\triangleq\\ell\_\{k\+1\}\(\\vartheta\)\-\\psi\_\{k\}\(\\vartheta\)\.\(10\)Thenℓk\+1​\(ϑ\)=ψk​\(ϑ\)\+wk\+1​\(ϑ\)\\ell\_\{k\+1\}\(\\vartheta\)=\\psi\_\{k\}\(\\vartheta\)\+w\_\{k\+1\}\(\\vartheta\)\. Moreover, for anyϑ∈ℝd2\\vartheta\\in\\mathbb\{R\}^\{d\_\{2\}\}, wheneverℓk\+1​\(ϑ\)\\ell\_\{k\+1\}\(\\vartheta\)is integrable,\{wk\+1​\(ϑ\),ℱk\}\\\{w\_\{k\+1\}\(\\vartheta\),\\mathcal\{F\}\_\{k\}\\\}forms a martingale difference sequence, i\.e\.,

𝔼​\[wk\+1​\(ϑ\)∣ℱk\]=0,k≥0\.\\mathbb\{E\}\\\!\\left\[w\_\{k\+1\}\(\\vartheta\)\\mid\\mathcal\{F\}\_\{k\}\\right\]=0,~~k\\geq 0\.
To establish tracking and prediction performance guarantees for Algorithm[1](https://arxiv.org/html/2606.28879#alg1), a standard approach is to study the squared parameter tracking error‖θ~k‖2\\\|\\tilde\{\\theta\}\_\{k\}\\\|^\{2\}, whereθ~k≜θk−θ^k\\tilde\{\\theta\}\_\{k\}\\triangleq\\theta\_\{k\}\-\\hat\{\\theta\}\_\{k\}\. However, such a direct analysis is impossible for Adam due to the coupling between the first momentummkm\_\{k\}and the second momentumV¯k\\bar\{V\}\_\{k\}in the update ofθ^k\\hat\{\\theta\}\_\{k\}\. These interactions induce intrinsically coupled error dynamics, rendering the stability and tracking analysis highly nontrivial\.

To address this challenge, we introduce the following augmented state vector transformed from the parameter estimation error and the first momentum, scaled respectively by the fourth root and the negative fourth root of the second momentum:

Θ~k≜\[V¯k14​θ~kβ1​V¯k−14​mk\],k≥0,\\tilde\{\\Theta\}\_\{k\}\\triangleq\\begin\{bmatrix\}\\bar\{V\}\_\{k\}^\{\\frac\{1\}\{4\}\}\\tilde\{\\theta\}\_\{k\}\\\\\[3\.0pt\] \\beta\_\{1\}\\bar\{V\}\_\{k\}^\{\-\\frac\{1\}\{4\}\}m\_\{k\}\\end\{bmatrix\},~~k\\geq 0,\(11\)and construct a Lyapunov function‖Θ~k‖2\\\|\\tilde\{\\Theta\}\_\{k\}\\\|^\{2\}\.

### 3\.1Parameter tracking

To establish an upper bound on the parameter tracking error of Algorithm[1](https://arxiv.org/html/2606.28879#alg1), we need the following assumptions\.

###### Assumption 1\(Bounded regressors and parameters\)

The random regressor sequence\{ϕk,ℱk\}\\\{\\phi\_\{k\},\\mathcal\{F\}\_\{k\}\\\}is adapted and uniformly bounded, i\.e\., there exists a constantC\>0C\>0such that

‖ϕk‖≤C,∀k≥0,a\.s\.\\\|\\phi\_\{k\}\\\|\\leq C,~~\\forall k\\geq 0,~\\text\{a\.s\.\}\(12\)Moreover, the time\-varying parameter satisfiesθk∈D\\theta\_\{k\}\\in Dfor allk≥0k\\geq 0, whereD⊂ℝd2D\\subset\\mathbb\{R\}^\{d\_\{2\}\}is a given compact convex set, its upper bound is denoted by a constantLL:

‖θk‖≤L,∀k≥0\.\\\|\\theta\_\{k\}\\\|\\leq L,~~\\forall k\\geq 0\.\(13\)In addition, there exists a constantCΔ≥0C\_\{\\Delta\}\\geq 0such that

𝔼​‖Δk\+1‖2≤CΔ2,∀k≥0,\\mathbb\{E\}\\\|\\Delta\_\{k\+1\}\\\|^\{2\}\\leq C\_\{\\Delta\}^\{2\},~~\\forall k\\geq 0,\(14\)whereΔk\+1\\Delta\_\{k\+1\}is the parameter variation defined in \([2](https://arxiv.org/html/2606.28879#S2.E2)\)\.

###### Assumption 2\(Gradient regularity and noise moments\)

Letℓk\+1​\(ϑ\)\\ell\_\{k\+1\}\(\\vartheta\)andgk​\(ϑ\)g\_\{k\}\(\\vartheta\)be the Clarke selections in \([5](https://arxiv.org/html/2606.28879#S2.E5)\)\. The following conditions hold uniformly for allk≥0k\\geq 0and allϑ∈D\\vartheta\\in D, a\.s\.:

1. \(i\)\(*Boundedness and noise moment*\) There exist constantsMℓ,Mg,Cw\>0M\_\{\\ell\},M\_\{g\},C\_\{w\}\>0such that \|ℓk\+1​\(ϑ\)\|≤Mℓ,‖gk​\(ϑ\)‖≤Mg,\|\\ell\_\{k\+1\}\(\\vartheta\)\|\\leq M\_\{\\ell\},~~\\\|g\_\{k\}\(\\vartheta\)\\\|\\leq M\_\{g\},\(15\)and 𝔼​\|wk\+1​\(ϑ\)\|2≤Cw2\.\\mathbb\{E\}\|w\_\{k\+1\}\(\\vartheta\)\|^\{2\}\\leq C\_\{w\}^\{2\}\.\(16\)
2. \(ii\)\(*Sector condition*\) There exists a constantμ\>0\\mu\>0such that ψk​\(ϑ\)​gk​\(ϑ\)T​\(ϑ−θk\)≥μ​\(gk​\(ϑ\)T​\(θk−ϑ\)\)2\.\\psi\_\{k\}\(\\vartheta\)g\_\{k\}\(\\vartheta\)^\{\\mathrm\{T\}\}\(\\vartheta\-\\theta\_\{k\}\)\\geq\\mu\\,\\bigl\(g\_\{k\}\(\\vartheta\)^\{\\mathrm\{T\}\}\(\\theta\_\{k\}\-\\vartheta\)\\bigr\)^\{2\}\.\(17\)wheregk​\(ϑ\)g\_\{k\}\(\\vartheta\)is defined in \([5](https://arxiv.org/html/2606.28879#S2.E5)\) andψk​\(ϑ\)\\psi\_\{k\}\(\\vartheta\)is defined in \([10](https://arxiv.org/html/2606.28879#S3.E10)\)\.
3. \(iii\)\(*Regularity*\) There exist constantsLψ,Lg\>0L\_\{\\psi\},L\_\{g\}\>0such that \|ψk​\(ϑ\)\|≤\\displaystyle\|\\psi\_\{k\}\(\\vartheta\)\|\\leqLψ​‖θk−ϑ‖,\\displaystyle L\_\{\\psi\}\\\|\\theta\_\{k\}\-\\vartheta\\\|,\(18\)‖gk​\(ϑ1\)−gk​\(ϑ2\)‖≤\\displaystyle~~\\\|g\_\{k\}\(\\vartheta\_\{1\}\)\-g\_\{k\}\(\\vartheta\_\{2\}\)\\\|\\leqLg​‖ϑ1−ϑ2‖,∀ϑ,ϑ1,ϑ2∈D\.\\displaystyle L\_\{g\}\\\|\\vartheta\_\{1\}\-\\vartheta\_\{2\}\\\|,~\\forall\\,\\vartheta,\\vartheta\_\{1\},\\vartheta\_\{2\}\\in D\.

###### Assumption 3\(Conditional excitation\)

There existγ\>0\\gamma\>0andh∈ℕh\\in\\mathbb\{N\}withh≥1h\\geq 1such that, for allj≥0j\\geq 0,

infϑ∈Dλmin​\(𝔼​\[∑i=j​h\+1\(j\+1\)​hgi​\(ϑ\)​gi​\(ϑ\)T\|ℱj​h\]\)≥γ,a\.s\.\\inf\_\{\\vartheta\\in D\}\\lambda\_\{\\min\}\\Bigl\(\\mathbb\{E\}\\Bigl\[\\sum\_\{i=jh\+1\}^\{\(j\+1\)h\}g\_\{i\}\(\\vartheta\)g\_\{i\}\(\\vartheta\)^\{\\mathrm\{T\}\}\\Big\|\\mathcal\{F\}\_\{jh\}\\Bigr\]\\Bigr\)\\geq\\gamma,~~\\text\{a\.s\.\}\(19\)

We establish theoretical guarantees for parameter tracking\.

###### Theorem 1

Suppose that Assumptions[1](https://arxiv.org/html/2606.28879#Thmassumption1)–[3](https://arxiv.org/html/2606.28879#Thmassumption3)hold\. With appropriately chosen small hyper\-parametersα\\alpha,β1\\beta\_\{1\}and1−β21\-\\beta\_\{2\}, the parameter tracking error satisfies

𝔼​‖θ~k‖2=\\displaystyle\\mathbb\{E\}\\\|\\tilde\{\\theta\}\_\{k\}\\\|^\{2\}=O​\(\[α2\+β12​δ12\+δ2\]32α​δ1\+\(α2\+β12\)​δ1​Cw2α\+CΔα​δ1\)\\displaystyle O\\left\(\\frac\{\[\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\delta\_\{1\}^\{2\}\+\\delta\_\{2\}\]^\{\\frac\{3\}\{2\}\}\}\{\\alpha\\delta\_\{1\}\}\+\\frac\{\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\)\\delta\_\{1\}C\_\{w\}^\{2\}\}\{\\alpha\}\+\\frac\{C\_\{\\Delta\}\}\{\\alpha\\delta\_\{1\}\}\\right\)\(21\)\+O​\(λη​k​𝔼​‖θ~0‖2\),\\displaystyle\+O\\left\(\\lambda^\{\\eta k\}\\mathbb\{E\}\\\|\\tilde\{\\theta\}\_\{0\}\\\|^\{2\}\\right\),whereθ~k=θk−θ^k\\tilde\{\\theta\}\_\{k\}=\\theta\_\{k\}\-\\hat\{\\theta\}\_\{k\},δ1≜1−β1\\delta\_\{1\}\\triangleq 1\-\\beta\_\{1\},δ2≜1−β2\\delta\_\{2\}\\triangleq 1\-\\beta\_\{2\},λ∈\(0,1\)\\lambda\\in\(0,1\),η≜12​h\\eta\\triangleq\\frac\{1\}\{2h\}, andhhis the excitation length defined in \([19](https://arxiv.org/html/2606.28879#S3.E19)\)\. Moreover,CΔC\_\{\\Delta\}andCwC\_\{w\}are the second\-moment bounds in \([14](https://arxiv.org/html/2606.28879#S3.E14)\) and \([16](https://arxiv.org/html/2606.28879#S3.E16)\), respectively\.

The proof of Theorem[1](https://arxiv.org/html/2606.28879#Thmtheorem1)is given in Section[4](https://arxiv.org/html/2606.28879#S4)\.

### 3\.2Prediction performance

So far, we have established parameter\-tracking guarantees for Adam, which may secure the performance of prediction for general data sequence\. These guarantees, however, rely on somewhat stringent data excitation conditions\. Fortunately, such conditions turn out to be not necessary for guaranteed adaptive prediction, as will be shown in Theorem[2](https://arxiv.org/html/2606.28879#Thmtheorem2)below\.

We need the following prediction\-oriented sector assumption which is weaker than the usual convexity condition\.

###### Assumption 4

Letgk​\(ϑ\)g\_\{k\}\(\\vartheta\)be the Clarke selection in \([5](https://arxiv.org/html/2606.28879#S2.E5)\) andψk​\(ϑ\)\\psi\_\{k\}\(\\vartheta\)andwk\+1​\(ϑ\)w\_\{k\+1\}\(\\vartheta\)be defined in \([10](https://arxiv.org/html/2606.28879#S3.E10)\)\. Under Assumption[1](https://arxiv.org/html/2606.28879#Thmassumption1), there exist constantsMψ,Mg,Cw,κ\>0M\_\{\\psi\},M\_\{g\},C\_\{w\},\\kappa\>0, such that for allk≥0k\\geq 0and allϑ∈D\\vartheta\\in D, a\.s\.,

\|ψk​\(ϑ\)\|≤Mψ,‖gk​\(ϑ\)‖≤Mg,𝔼​\|wk\+1​\(ϑ\)\|2≤Cw2,\|\\psi\_\{k\}\(\\vartheta\)\|\\leq M\_\{\\psi\},~~\\\|g\_\{k\}\(\\vartheta\)\\\|\\leq M\_\{g\},~~\\mathbb\{E\}\|w\_\{k\+1\}\(\\vartheta\)\|^\{2\}\\leq C\_\{w\}^\{2\},\(28\)and, for somep∈\[1,2\]p\\in\[1,2\],

ψk​\(ϑ\)​gk​\(ϑ\)T​\(ϑ−θk\)≥κ​\(ℒ​\(fk​\(ϕk,θk\),fk​\(ϕk,ϑ\)\)\)2p,\\psi\_\{k\}\(\\vartheta\)\\,g\_\{k\}\(\\vartheta\)^\{\\mathrm\{T\}\}\(\\vartheta\-\\theta\_\{k\}\)\\geq\\kappa\\Bigl\(\\mathcal\{L\}\\bigl\(f\_\{k\}\(\\phi\_\{k\},\\theta\_\{k\}\),\\,f\_\{k\}\(\\phi\_\{k\},\\vartheta\)\\bigr\)\\Bigr\)^\{\\frac\{2\}\{p\}\},\(29\)

###### Theorem 2

Suppose that Assumptions[1](https://arxiv.org/html/2606.28879#Thmassumption1)and[4](https://arxiv.org/html/2606.28879#Thmassumption4)hold\. Then,

1n​∑k=1n𝔼​ℒ​\(fk​\(ϕk,θk\),fk​\(ϕk,θ^k\)\)\\displaystyle\\frac\{1\}\{n\}\\sum\_\{k=1\}^\{n\}\\mathbb\{E\}\\mathcal\{L\}\\\!\\left\(f\_\{k\}\(\\phi\_\{k\},\\theta\_\{k\}\),\\,f\_\{k\}\(\\phi\_\{k\},\\hat\{\\theta\}\_\{k\}\)\\right\)\(30\)=\\displaystyle=O\(\[α2\+β12​δ12\+δ2α​δ1\+\(α2\+δ2\)​Cwα​δ1\\displaystyle O\\Biggl\(\\Biggl\[\\frac\{\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\delta\_\{1\}^\{2\}\+\\sqrt\{\\delta\_\{2\}\}\}\{\\alpha\\delta\_\{1\}\}\+\\frac\{\(\\alpha^\{2\}\+\\sqrt\{\\delta\_\{2\}\}\)C\_\{w\}\}\{\\alpha\\delta\_\{1\}\}\+\(α2\+β12\)​δ1​Cw2α\+\(1\+Cw\)​CΔα​δ1\]p2\)\\displaystyle\\qquad\+\\frac\{\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\)\\delta\_\{1\}C\_\{w\}^\{2\}\}\{\\alpha\}\+\\frac\{\(1\+C\_\{w\}\)C\_\{\\Delta\}\}\{\\alpha\\delta\_\{1\}\}\\Biggr\]^\{\\frac\{p\}\{2\}\}\\Biggr\)\+O​\(\(α​δ1​n\)−p2\)\.\\displaystyle\\quad\+O\\\!\\left\(\(\\alpha\\delta\_\{1\}n\)^\{\-\\frac\{p\}\{2\}\}\\right\)\.provided thatα2\+β12<β2,\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}<\\sqrt\{\\beta\_\{2\}\},whereδ1=1−β1\\delta\_\{1\}=1\-\\beta\_\{1\},δ2=1−β2\\delta\_\{2\}=1\-\\beta\_\{2\},CΔC\_\{\\Delta\}andCwC\_\{w\}are the constants defined in \([14](https://arxiv.org/html/2606.28879#S3.E14)\) and \([28](https://arxiv.org/html/2606.28879#S3.E28)\) respectively,p∈\[1,2\]p\\in\[1,2\]is defined in Assumption[4](https://arxiv.org/html/2606.28879#Thmassumption4)\.

The proof of Theorem[2](https://arxiv.org/html/2606.28879#Thmtheorem2)is given in Section[4](https://arxiv.org/html/2606.28879#S4)\.

## 4Proofs of the main theorems

In this section, we present the proofs of the main theorems of this paper\. For convenience of analysis, we first introduce some notation\. Let

ℐj≜\{j​h¯,…,\(j\+1\)​h¯−1\},j≥0,h¯≜2​h,a≜α​δ1,\\mathcal\{I\}\_\{j\}\\triangleq\\\{j\\bar\{h\},\\dots,\(j\+1\)\\bar\{h\}\-1\\\},~j\\geq 0,~\\bar\{h\}\\triangleq 2h,~a\\triangleq\\alpha\\delta\_\{1\},\(31\)wherehhis defined in Assumption[3](https://arxiv.org/html/2606.28879#Thmassumption3),δ1=1−β1\\delta\_\{1\}=1\-\\beta\_\{1\}\. We further introduce the following quantities, fork≥0k\\geq 0,

ψk\\displaystyle\\psi\_\{k\}≜ψk​\(θ^k\),\\displaystyle\\triangleq\\psi\_\{k\}\(\\hat\{\\theta\}\_\{k\}\),wk\+1\\displaystyle w\_\{k\+1\}≜wk\+1​\(θ^k\),\\displaystyle\\triangleq w\_\{k\+1\}\(\\hat\{\\theta\}\_\{k\}\),\(32\)Φk\+1\\displaystyle\\Phi\_\{k\+1\}≜ℓk\+12​diag⁡\{gk⊙gk\},\\displaystyle\\triangleq\\ell\_\{k\+1\}^\{2\}\\operatorname\{diag\}\\\{g\_\{k\}\\odot g\_\{k\}\\\},Sk\\displaystyle S\_\{k\}≜μ​V¯k−14​gk​gkT​V¯k−14\.\\displaystyle\\triangleq\\mu\\,\\bar\{V\}\_\{k\}^\{\-\\frac\{1\}\{4\}\}g\_\{k\}g\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\}^\{\-\\frac\{1\}\{4\}\}\.
The purpose of this section is to present the main proof structure\. Several technical estimates are stated as lemmas\. Their proofs rely on lengthy but standard perturbation bounds for the adaptive scaling matrix, the non\-expansiveness of the weighted projection, and the conditional excitation condition\. To keep this short version concise, we omit these constant\-level details and focus on the Lyapunov construction and the resulting block\-recursive estimates\.

###### Lemma 1

\(\[[34](https://arxiv.org/html/2606.28879#bib.bib34)\]\) LetΠQ​\(⋅\)\\Pi\_\{Q\}\(\\cdot\)be the projection mapping defined in Definition[1](https://arxiv.org/html/2606.28879#Thmdefinition1)\. Then, for anyx1,x2∈ℝd2x\_\{1\},x\_\{2\}\\in\\mathbb\{R\}^\{d\_\{2\}\},

‖ΠQ​\(x1\)−ΠQ​\(x2\)‖Q≤‖x1−x2‖Q\.\\\|\\Pi\_\{Q\}\(x\_\{1\}\)\-\\Pi\_\{Q\}\(x\_\{2\}\)\\\|\_\{Q\}\\;\\leq\\;\\\|x\_\{1\}\-x\_\{2\}\\\|\_\{Q\}\.\(33\)

###### Lemma 2

Under the conditions of Theorem[1](https://arxiv.org/html/2606.28879#Thmtheorem1), one can deduce that

𝔼\[∑i∈ℐjSi\|ℱj​h¯\]⪰γ0I⪰𝟎,j≥0,a\.s\.,\\mathbb\{E\}\\Bigl\[\\sum\_\{i\\in\\mathcal\{I\}\_\{j\}\}S\_\{i\}\\Big\\rvert\\mathcal\{F\}\_\{j\\bar\{h\}\}\\Bigr\]\\succeq\\gamma\_\{0\}I\\succeq\\mathbf\{0\},~j\\geq 0,~\\text\{ a\.s\.\},\(34\)whereγ0=μ​γ4​\(Mℓ​Mg\+c0\)\\gamma\_\{0\}=\\frac\{\\mu\\gamma\}\{4\(M\_\{\\ell\}M\_\{g\}\+c\_\{0\}\)\},h¯=2​h\\bar\{h\}=2h\.

###### Lemma 3

Under Assumptions[1](https://arxiv.org/html/2606.28879#Thmassumption1)–[2](https://arxiv.org/html/2606.28879#Thmassumption2), for anyj\>i≥0j\>i\\geq 0,

𝔼​‖Θ~j−Θ~i‖2≤\\displaystyle\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{j\}\-\\tilde\{\\Theta\}\_\{i\}\\\|^\{2\}\\leq3​\(Mℓ​Mg\+c0\)​\(j−i\)2​\(CΔ2\+α2c02​Mℓ2​Mg2\)\\displaystyle 3\(M\_\{\\ell\}M\_\{g\}\+c\_\{0\}\)\(j\-i\)^\{2\}\\Bigl\(C\_\{\\Delta\}^\{2\}\+\\tfrac\{\\alpha^\{2\}\}\{c\_\{0\}^\{2\}\}M\_\{\\ell\}^\{2\}M\_\{g\}^\{2\}\\Bigr\)\(35\)\+\(1−β2j−i\)​Mℓ2​Mg2​\(3​L2c0\+β12​Mℓ2​Mg22​c03\)\\displaystyle\+\(1\-\\beta\_\{2\}^\{j\-i\}\)M\_\{\\ell\}^\{2\}M\_\{g\}^\{2\}\\Bigl\(\\tfrac\{3L^\{2\}\}\{c\_\{0\}\}\+\\beta\_\{1\}^\{2\}\\tfrac\{M\_\{\\ell\}^\{2\}M\_\{g\}^\{2\}\}\{2c\_\{0\}^\{3\}\}\\Bigr\)\+β12​\(1−β1j−i\)2​8​Mℓ2​Mg2c0,\\displaystyle\+\\beta\_\{1\}^\{2\}\(1\-\\beta\_\{1\}^\{j\-i\}\)^\{2\}\\,\\tfrac\{8M\_\{\\ell\}^\{2\}M\_\{g\}^\{2\}\}\{c\_\{0\}\},whereΘ~j\\tilde\{\\Theta\}\_\{j\}is defined in \([11](https://arxiv.org/html/2606.28879#S3.E11)\)\.

###### Lemma 4

Under the conditions of Theorem[1](https://arxiv.org/html/2606.28879#Thmtheorem1), there exist constantsCb\>0C\_\{b\}\>0andγ0\>0\\gamma\_\{0\}\>0such that, for allj≥0j\\geq 0,

𝔼​‖Θ~\(j\+1\)​h¯‖2≤\\displaystyle\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{\(j\+1\)\\bar\{h\}\}\\\|^\{2\}\\leq\(1−a​γ0\)​𝔼​‖Θ~j​h¯‖2\\displaystyle\(1\-a\\gamma\_\{0\}\)\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{j\\bar\{h\}\}\\\|^\{2\}\(36\)\+Cb\[\(α\+β12\+δ2\)\(CΔ2\+α2\+β12δ12\+δ2\)\\displaystyle\+C\_\{b\}\\Bigl\[\(\\alpha\+\\beta\_\{1\}^\{2\}\+\\sqrt\{\\delta\_\{2\}\}\)\(C\_\{\\Delta\}^\{2\}\+\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\delta\_\{1\}^\{2\}\+\\delta\_\{2\}\)\+CΔ\+CΔ2\+\(α2\+β12\)δ12Cw2\],\\displaystyle\\qquad\+C\_\{\\Delta\}\+C\_\{\\Delta\}^\{2\}\+\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\)\\delta\_\{1\}^\{2\}C\_\{w\}^\{2\}\\Bigr\],whereh¯=2​h\\bar\{h\}=2h,a=α​δ1a=\\alpha\\delta\_\{1\},δ1=1−β1\\delta\_\{1\}=1\-\\beta\_\{1\}, andδ2=1−β2\\delta\_\{2\}=1\-\\beta\_\{2\}\.

Proof of Theorem[1](https://arxiv.org/html/2606.28879#Thmtheorem1):Using \([8](https://arxiv.org/html/2606.28879#S2.E8)\), \([9](https://arxiv.org/html/2606.28879#S2.E9)\), together with Algorithm[1](https://arxiv.org/html/2606.28879#alg1), it follows under Assumption[2](https://arxiv.org/html/2606.28879#Thmassumption2)that, for allk≥0k\\geq 0,

\[V¯k\+114​\(θk\+1−θ^k\+α​V¯k\+1−12​mk\+1\)β1​V¯k\+1−14​mk\+1\]\\displaystyle\\begin\{bmatrix\}\\bar\{V\}\_\{k\+1\}^\{\\frac\{1\}\{4\}\}\(\\theta\_\{k\+1\}\-\\hat\{\\theta\}\_\{k\}\+\\alpha\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{2\}\}m\_\{k\+1\}\)\\\\ \\beta\_\{1\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{4\}\}m\_\{k\+1\}\\end\{bmatrix\}\(37\)=\\displaystyle=\[V¯k\+114​θ~k\+a​V¯k\+1−14​gk​ψk\+α​β1​V¯k\+1−14​mkβ12​V¯k\+1−14​mk\+β1​δ1​V¯k\+1−14​gk​ψk\]⏟𝒯k\+1\\displaystyle\\underbrace\{\\begin\{bmatrix\}\\bar\{V\}\_\{k\+1\}^\{\\frac\{1\}\{4\}\}\\tilde\{\\theta\}\_\{k\}\+a\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{4\}\}g\_\{k\}\\psi\_\{k\}\+\\alpha\\beta\_\{1\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{4\}\}m\_\{k\}\\\\ \\beta\_\{1\}^\{2\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{4\}\}m\_\{k\}\+\\beta\_\{1\}\\delta\_\{1\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{4\}\}g\_\{k\}\\psi\_\{k\}\\end\{bmatrix\}\}\_\{\\mathcal\{T\}\_\{k\+1\}\}\+\[a​V¯k\+1−14​gkβ1​δ1​V¯k\+1−14​gk\]⏟𝒩k\+1​wk\+1\+\[V¯k\+114𝟎\]⏟𝒫k\+1​Δk\+1,\\displaystyle\+\\underbrace\{\\begin\{bmatrix\}a\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{4\}\}g\_\{k\}\\\\ \\beta\_\{1\}\\delta\_\{1\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{4\}\}g\_\{k\}\\end\{bmatrix\}\}\_\{\\mathcal\{N\}\_\{k\+1\}\}w\_\{k\+1\}\+\\underbrace\{\\begin\{bmatrix\}\\bar\{V\}\_\{k\+1\}^\{\\frac\{1\}\{4\}\}\\\\ \\mathbf\{0\}\\end\{bmatrix\}\}\_\{\\mathcal\{P\}\_\{k\+1\}\}\\Delta\_\{k\+1\},wherea=α​δ1a=\\alpha\\delta\_\{1\},δ1=1−β1\\delta\_\{1\}=1\-\\beta\_\{1\}, andδ2=1−β2\\delta\_\{2\}=1\-\\beta\_\{2\}\.

We choose the stochastic Lyapunov function

VkΘ≜‖Θ~k‖2,k≥0\.V\_\{k\}^\{\\Theta\}\\triangleq\\\|\\tilde\{\\Theta\}\_\{k\}\\\|^\{2\},\\qquad k\\geq 0\.Then, by \([8](https://arxiv.org/html/2606.28879#S2.E8)\), \([9](https://arxiv.org/html/2606.28879#S2.E9)\), \([11](https://arxiv.org/html/2606.28879#S3.E11)\), \([37](https://arxiv.org/html/2606.28879#S4.E37)\), and Lemma[1](https://arxiv.org/html/2606.28879#Thmlemma1), we obtain

‖Θ~k\+1‖2≤\\displaystyle\\\|\\tilde\{\\Theta\}\_\{k\+1\}\\\|^\{2\}\\leq𝒯k\+1T​𝒯k\+1\+2​𝒯k\+1T​𝒩k\+1​wk\+1\+2​𝒯k\+1T​𝒫k\+1​Δk\+1\\displaystyle\\;\\mathcal\{T\}\_\{k\+1\}^\{\\mathrm\{T\}\}\\mathcal\{T\}\_\{k\+1\}\+2\\mathcal\{T\}\_\{k\+1\}^\{\\mathrm\{T\}\}\\mathcal\{N\}\_\{k\+1\}w\_\{k\+1\}\+2\\mathcal\{T\}\_\{k\+1\}^\{\\mathrm\{T\}\}\\mathcal\{P\}\_\{k\+1\}\\Delta\_\{k\+1\}\(38\)\+2​𝒩k\+1T​𝒩k\+1​wk\+12\+2​Δk\+1T​𝒫k\+1T​𝒫k\+1​Δk\+1\.\\displaystyle\+2\\mathcal\{N\}\_\{k\+1\}^\{\\mathrm\{T\}\}\\mathcal\{N\}\_\{k\+1\}w\_\{k\+1\}^\{2\}\+2\\Delta\_\{k\+1\}^\{\\mathrm\{T\}\}\\mathcal\{P\}\_\{k\+1\}^\{\\mathrm\{T\}\}\\mathcal\{P\}\_\{k\+1\}\\Delta\_\{k\+1\}\.Here the last inequality follows by expanding the square and applying2​a​b≤a2\+b22ab\\leq a^\{2\}\+b^\{2\}to the cross term between𝒩k\+1​wk\+1\\mathcal\{N\}\_\{k\+1\}w\_\{k\+1\}and𝒫k\+1​Δk\+1\\mathcal\{P\}\_\{k\+1\}\\Delta\_\{k\+1\}\.

By applying the block contraction estimate in Lemma[4](https://arxiv.org/html/2606.28879#Thmlemma4)to the one\-step Lyapunov inequality \([38](https://arxiv.org/html/2606.28879#S4.E38)\), we obtain, for allj≥0j\\geq 0,

𝔼​‖Θ~\(j\+1\)​h¯‖2≤\\displaystyle\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{\(j\+1\)\\bar\{h\}\}\\\|^\{2\}\\leq\(1−a​γ0\)​𝔼​‖Θ~j​h¯‖2\\displaystyle\(1\-a\\gamma\_\{0\}\)\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{j\\bar\{h\}\}\\\|^\{2\}\(39\)\+Cb\[\(α\+β12\+δ2\)\(CΔ2\+α2\+β12δ12\+δ2\)\\displaystyle\+C\_\{b\}\\Bigl\[\(\\alpha\+\\beta\_\{1\}^\{2\}\+\\sqrt\{\\delta\_\{2\}\}\)\(C\_\{\\Delta\}^\{2\}\+\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\delta\_\{1\}^\{2\}\+\\delta\_\{2\}\)\+CΔ\+CΔ2\+\(α2\+β12\)δ12Cw2\]\.\\displaystyle\\qquad\+C\_\{\\Delta\}\+C\_\{\\Delta\}^\{2\}\+\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\)\\delta\_\{1\}^\{2\}C\_\{w\}^\{2\}\\Bigr\]\.For brevity, the constant\-level perturbation estimates leading to \([39](https://arxiv.org/html/2606.28879#S4.E39)\) are omitted in this short version\.

Denote the second term on the right\-hand side of \([39](https://arxiv.org/html/2606.28879#S4.E39)\) byBα,β,Δ,wB\_\{\\alpha,\\beta,\\Delta,w\}, i\.e\.,

Bα,β,Δ,w≜Cb\[\\displaystyle B\_\{\\alpha,\\beta,\\Delta,w\}\\triangleq C\_\{b\}\\Bigl\[\(α\+β12\+δ2\)​\(CΔ2\+α2\+β12​δ12\+δ2\)\\displaystyle\(\\alpha\+\\beta\_\{1\}^\{2\}\+\\sqrt\{\\delta\_\{2\}\}\)\(C\_\{\\Delta\}^\{2\}\+\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\delta\_\{1\}^\{2\}\+\\delta\_\{2\}\)\(40\)\+CΔ\+CΔ2\+\(α2\+β12\)δ12Cw2\]\.\\displaystyle\+C\_\{\\Delta\}\+C\_\{\\Delta\}^\{2\}\+\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\)\\delta\_\{1\}^\{2\}C\_\{w\}^\{2\}\\Bigr\]\.Since1−a​γ0∈\(0,1\)1\-a\\gamma\_\{0\}\\in\(0,1\), iterating \([39](https://arxiv.org/html/2606.28879#S4.E39)\) gives, for anyj≥1j\\geq 1,

𝔼​‖Θ~j​h¯‖2≤\\displaystyle\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{j\\bar\{h\}\}\\\|^\{2\}\\leq\(1−a​γ0\)j​𝔼​‖Θ~0‖2\+Bα,β,Δ,w​∑t=0j−1\(1−a​γ0\)t\\displaystyle\(1\-a\\gamma\_\{0\}\)^\{j\}\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{0\}\\\|^\{2\}\+B\_\{\\alpha,\\beta,\\Delta,w\}\\sum\_\{t=0\}^\{j\-1\}\(1\-a\\gamma\_\{0\}\)^\{t\}\(41\)≤\\displaystyle\\leq\(1−a​γ0\)j​𝔼​‖Θ~0‖2\+Bα,β,Δ,wa​γ0\.\\displaystyle\(1\-a\\gamma\_\{0\}\)^\{j\}\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{0\}\\\|^\{2\}\+\\frac\{B\_\{\\alpha,\\beta,\\Delta,w\}\}\{a\\gamma\_\{0\}\}\.
For a general timek≥0k\\geq 0, let

j1≜max⁡\{j≥0:j​h¯≤k\}\.j\_\{1\}\\triangleq\\max\\\{j\\geq 0:j\\bar\{h\}\\leq k\\\}\.By Young’s inequality, for anyν\>0\\nu\>0,

𝔼​‖Θ~k‖2\\displaystyle\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{k\}\\\|^\{2\}=𝔼​‖Θ~j1​h¯\+\(Θ~k−Θ~j1​h¯\)‖2\\displaystyle=\\mathbb\{E\}\\bigl\\\|\\tilde\{\\Theta\}\_\{j\_\{1\}\\bar\{h\}\}\+\(\\tilde\{\\Theta\}\_\{k\}\-\\tilde\{\\Theta\}\_\{j\_\{1\}\\bar\{h\}\}\)\\bigr\\\|^\{2\}\(42\)≤\(1\+ν\)​𝔼​‖Θ~j1​h¯‖2\+\(1\+1ν\)​𝔼​‖Θ~k−Θ~j1​h¯‖2\.\\displaystyle\\leq\(1\+\\nu\)\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{j\_\{1\}\\bar\{h\}\}\\\|^\{2\}\+\\Bigl\(1\+\\frac\{1\}\{\\nu\}\\Bigr\)\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{k\}\-\\tilde\{\\Theta\}\_\{j\_\{1\}\\bar\{h\}\}\\\|^\{2\}\.Since0≤k−j1​h¯≤h¯−10\\leq k\-j\_\{1\}\\bar\{h\}\\leq\\bar\{h\}\-1, Lemma[3](https://arxiv.org/html/2606.28879#Thmlemma3)implies that there exists a constantCΘ\>0C\_\{\\Theta\}\>0such that

𝔼​‖Θ~k−Θ~j1​h¯‖2≤CΘ​\(CΔ2\+α2\+β12​δ12\+δ2\)\.\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{k\}\-\\tilde\{\\Theta\}\_\{j\_\{1\}\\bar\{h\}\}\\\|^\{2\}\\leq C\_\{\\Theta\}\\bigl\(C\_\{\\Delta\}^\{2\}\+\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\delta\_\{1\}^\{2\}\+\\delta\_\{2\}\\bigr\)\.\(43\)Combining \([41](https://arxiv.org/html/2606.28879#S4.E41)\)–\([43](https://arxiv.org/html/2606.28879#S4.E43)\), and takingν=1\\nu=1for simplicity, yields

𝔼​‖Θ~k‖2=\\displaystyle\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{k\}\\\|^\{2\}=O​\(\(1−a​γ0\)j1​𝔼​‖Θ~0‖2\)\+O​\(Bα,β,Δ,wa\)\\displaystyle O\\\!\\left\(\(1\-a\\gamma\_\{0\}\)^\{j\_\{1\}\}\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{0\}\\\|^\{2\}\\right\)\+O\\\!\\left\(\\frac\{B\_\{\\alpha,\\beta,\\Delta,w\}\}\{a\}\\right\)\(44\)\+O​\(CΔ2\+α2\+β12​δ12\+δ2\)\.\\displaystyle\+O\\\!\\left\(C\_\{\\Delta\}^\{2\}\+\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\delta\_\{1\}^\{2\}\+\\delta\_\{2\}\\right\)\.
By \([8](https://arxiv.org/html/2606.28879#S2.E8)\) and \([11](https://arxiv.org/html/2606.28879#S3.E11)\), and sincem0=0m\_\{0\}=0andV0=𝟎V\_\{0\}=\\mathbf\{0\}, we have

‖Θ~0‖2=c0​‖θ~0‖2,‖Θ~k‖2≥c0​‖θ~k‖2\.\\\|\\tilde\{\\Theta\}\_\{0\}\\\|^\{2\}=c\_\{0\}\\\|\\tilde\{\\theta\}\_\{0\}\\\|^\{2\},\\qquad\\\|\\tilde\{\\Theta\}\_\{k\}\\\|^\{2\}\\geq c\_\{0\}\\\|\\tilde\{\\theta\}\_\{k\}\\\|^\{2\}\.\(45\)Moreover, for the admissible hyperparameter range considered in the theorem,β1\\beta\_\{1\}is bounded away from11, and hence

α\+β12\+δ2=O​\(α2\+β12​δ12\+δ2\)\.\\alpha\+\\beta\_\{1\}^\{2\}\+\\sqrt\{\\delta\_\{2\}\}=O\\\!\\left\(\\sqrt\{\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\delta\_\{1\}^\{2\}\+\\delta\_\{2\}\}\\right\)\.Sincea=α​δ1≤1a=\\alpha\\delta\_\{1\}\\leq 1, the last term in \([44](https://arxiv.org/html/2606.28879#S4.E44)\) is absorbed into the correspondingO\(⋅/a\)O\(\\cdot/a\)terms\. In addition, becauseθk∈D\\theta\_\{k\}\\in DandDDis compact, the drift level is bounded, so the termCΔ2C\_\{\\Delta\}^\{2\}can be absorbed intoCΔC\_\{\\Delta\}up to a change of constants\. Therefore, usinga=α​δ1a=\\alpha\\delta\_\{1\}and setting

λ≜1−a​γ0∈\(0,1\),\\lambda\\triangleq 1\-a\\gamma\_\{0\}\\in\(0,1\),we obtain

𝔼​‖θ~k‖2≤\\displaystyle\\mathbb\{E\}\\\|\\tilde\{\\theta\}\_\{k\}\\\|^\{2\}\\leqO​\(\[α2\+β12​δ12\+δ2\]32α​δ1\)\\displaystyle O\\left\(\\frac\{\[\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\delta\_\{1\}^\{2\}\+\\delta\_\{2\}\]^\{\\frac\{3\}\{2\}\}\}\{\\alpha\\delta\_\{1\}\}\\right\)\(46\)\+O​\(\(α2\+β12\)​δ1α​Cw2\)\+O​\(CΔα​δ1\)\\displaystyle\+O\\left\(\\frac\{\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\)\\delta\_\{1\}\}\{\\alpha\}C\_\{w\}^\{2\}\\right\)\+O\\left\(\\frac\{C\_\{\\Delta\}\}\{\\alpha\\delta\_\{1\}\}\\right\)\+O​\(λj1​𝔼​‖θ~0‖2\)\.\\displaystyle\+O\\left\(\\lambda^\{j\_\{1\}\}\\mathbb\{E\}\\\|\\tilde\{\\theta\}\_\{0\}\\\|^\{2\}\\right\)\.This completes the proof\.■\\hfill\\blacksquare

Proof of Theorem[2](https://arxiv.org/html/2606.28879#Thmtheorem2):Under the same Lyapunov function‖Θ~k‖2\\\|\\tilde\{\\Theta\}\_\{k\}\\\|^\{2\},k≥0k\\geq 0, and following the same line of analysis as in \([38](https://arxiv.org/html/2606.28879#S4.E38)\), we have

‖Θ~k\+1‖2\\displaystyle\\\|\\tilde\{\\Theta\}\_\{k\+1\}\\\|^\{2\}\(47\)≤\\displaystyle\\leqθ~kT​V¯k\+112​θ~k\+2​a​ψk​gkT​θ~k\+2​α​β1​θ~kT​mk\\displaystyle\\tilde\{\\theta\}\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\\frac\{1\}\{2\}\}\\tilde\{\\theta\}\_\{k\}\+2a\\psi\_\{k\}g\_\{k\}^\{\\mathrm\{T\}\}\\tilde\{\\theta\}\_\{k\}\+2\\alpha\\beta\_\{1\}\\tilde\{\\theta\}\_\{k\}^\{\\mathrm\{T\}\}m\_\{k\}\+\(α2\+β12\)​δ12​ψk2​gkT​V¯k\+1−12​gk\+\(α2\+β12\)​β12​mkT​V¯k\+1−12​mk\\displaystyle\+\\bigl\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\bigr\)\\delta\_\{1\}^\{2\}\\psi\_\{k\}^\{2\}g\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{2\}\}g\_\{k\}\+\\bigl\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\bigr\)\\beta\_\{1\}^\{2\}m\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{2\}\}m\_\{k\}\+2​\(α2\+β12\)​β1​δ1​ψk​gkT​V¯k\+1−12​mk\+2​a​gkT​θ~k​wk\+1\\displaystyle\+2\\bigl\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\bigr\)\\beta\_\{1\}\\delta\_\{1\}\\psi\_\{k\}g\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{2\}\}m\_\{k\}\+2ag\_\{k\}^\{\\mathrm\{T\}\}\\tilde\{\\theta\}\_\{k\}w\_\{k\+1\}\+2​\(α2\+β12\)​δ12​ψk​gkT​V¯k\+1−12​gk​wk\+1\\displaystyle\+2\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\)\\delta\_\{1\}^\{2\}\\psi\_\{k\}g\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{2\}\}g\_\{k\}w\_\{k\+1\}\+2​\(α2\+β12\)​β1​δ1​mkT​V¯k\+1−12​gk​wk\+1\\displaystyle\+2\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\)\\beta\_\{1\}\\delta\_\{1\}m\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{2\}\}g\_\{k\}w\_\{k\+1\}\+2​θ~kT​V¯k\+112​Δk\+1\+2​a​ψk​gkT​Δk\+1\+2​α​β1​mkT​Δk\+1\\displaystyle\+2\\tilde\{\\theta\}\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\\frac\{1\}\{2\}\}\\Delta\_\{k\+1\}\+2a\\psi\_\{k\}g\_\{k\}^\{\\mathrm\{T\}\}\\Delta\_\{k\+1\}\+2\\alpha\\beta\_\{1\}m\_\{k\}^\{\\mathrm\{T\}\}\\Delta\_\{k\+1\}\+2​\(α2\+β12\)​δ12​gkT​V¯k\+1−12​gk​wk\+12\+2​Δk\+1T​V¯k\+112​Δk\+1\.\\displaystyle\+2\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\)\\delta\_\{1\}^\{2\}g\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{2\}\}g\_\{k\}w\_\{k\+1\}^\{2\}\+2\\Delta\_\{k\+1\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\\frac\{1\}\{2\}\}\\Delta\_\{k\+1\}\.
We analyze each term on the RHS of \([47](https://arxiv.org/html/2606.28879#S4.E47)\) separately, so as to establish a recursive Lyapunov inequality without requiring any data excitation condition\. The overall argument parallels that of the proof of Theorem[1](https://arxiv.org/html/2606.28879#Thmtheorem1)\.

Sinceα2\+β12<β2\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}<\\sqrt\{\\beta\_\{2\}\}, we can choose a sufficiently large constantr\>0r\>0such that

1r\+\(r\+2\)​\(α2\+β12\)r​β2≤1\.\\frac\{1\}\{r\}\+\\frac\{\(r\+2\)\\bigl\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\bigr\)\}\{r\\sqrt\{\\beta\_\{2\}\}\}\\leq 1\.Then we have

‖Θ~k\+1‖2\\displaystyle\\\|\\tilde\{\\Theta\}\_\{k\+1\}\\\|^\{2\}\(48\)≤\\displaystyle\\leq‖Θ~k‖2−2​a​κ​ℒk2p\+δ2​θ~kT​\(Φk\+112\+c0​I\)​θ~k\\displaystyle\\\|\\tilde\{\\Theta\}\_\{k\}\\\|^\{2\}\-2a\\kappa\\mathcal\{L\}\_\{k\}^\{\\frac\{2\}\{p\}\}\+\\sqrt\{\\delta\_\{2\}\}\\tilde\{\\theta\}\_\{k\}^\{\\mathrm\{T\}\}\(\\Phi\_\{k\+1\}^\{\\frac\{1\}\{2\}\}\+c\_\{0\}I\)\\tilde\{\\theta\}\_\{k\}\+r​α2​θ~kT​V¯k12​θ~k\+\(r\+2\)​\(α2\+β12\)​δ12​ψk2​gkT​V¯k\+1−12​gk\\displaystyle\+r\\alpha^\{2\}\\tilde\{\\theta\}\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\}^\{\\frac\{1\}\{2\}\}\\tilde\{\\theta\}\_\{k\}\+\(r\+2\)\\bigl\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\bigr\)\\delta\_\{1\}^\{2\}\\psi\_\{k\}^\{2\}g\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{2\}\}g\_\{k\}\+2​a​gkT​θ~k​wk\+1\+\(r\+3\)​\(α2\+β12\)​δ12​gkT​V¯k\+1−12​gk​wk\+12\\displaystyle\+2ag\_\{k\}^\{\\mathrm\{T\}\}\\tilde\{\\theta\}\_\{k\}w\_\{k\+1\}\+\(r\+3\)\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\)\\delta\_\{1\}^\{2\}g\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{2\}\}g\_\{k\}w\_\{k\+1\}^\{2\}\+2​θ~kT​V¯k\+112​Δk\+1\+2​a​ψk​gkT​Δk\+1\+2​α​β1​mkT​Δk\+1\\displaystyle\+2\\tilde\{\\theta\}\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\\frac\{1\}\{2\}\}\\Delta\_\{k\+1\}\+2a\\psi\_\{k\}g\_\{k\}^\{\\mathrm\{T\}\}\\Delta\_\{k\+1\}\+2\\alpha\\beta\_\{1\}m\_\{k\}^\{\\mathrm\{T\}\}\\Delta\_\{k\+1\}\+2​Δk\+1T​V¯k\+112​Δk\+1,a\.s\.,\\displaystyle\+2\\Delta\_\{k\+1\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\\frac\{1\}\{2\}\}\\Delta\_\{k\+1\},~\\text\{a\.s\.\},where we have used the fact that\[1r\+\(r\+2\)​\(α2\+β12\)r​β2\]≤1\\Bigl\[\\frac\{1\}\{r\}\+\\frac\{\(r\+2\)\\bigl\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\\bigr\)\}\{r\\sqrt\{\\beta\_\{2\}\}\}\\Bigr\]\\leq 1\.

To analyze \([48](https://arxiv.org/html/2606.28879#S4.E48)\), we further introduce several inequalities in expectation\.

With Assumption[4](https://arxiv.org/html/2606.28879#Thmassumption4), \([10](https://arxiv.org/html/2606.28879#S3.E10)\) and \([32](https://arxiv.org/html/2606.28879#S4.E32)\), and noticing that\{wk,ℱk\}\\\{w\_\{k\},\\mathcal\{F\}\_\{k\}\\\}is the martingale difference sequence andψk\\psi\_\{k\}isℱk\\mathcal\{F\}\_\{k\}\-measurable, we have

𝔼​‖Φk\+112‖≤Mg​Mψ2\+Cw2\.\\displaystyle\\mathbb\{E\}\\\|\\Phi\_\{k\+1\}^\{\\frac\{1\}\{2\}\}\\\|\\leq M\_\{g\}\\sqrt\{M\_\{\\psi\}^\{2\}\+C\_\{w\}^\{2\}\}\.\(49\)Then, combining‖θ~k‖≤2​L\\\|\\tilde\{\\theta\}\_\{k\}\\\|\\leq 2Lunder Assumption[1](https://arxiv.org/html/2606.28879#Thmassumption1), we have

𝔼​θ~kT​\(Φk\+112\+c0​I\)​θ~k≤4​L2​\(Mg​Mψ2\+Cw2\+c0\),\\mathbb\{E\}\\tilde\{\\theta\}\_\{k\}^\{\\mathrm\{T\}\}\(\\Phi\_\{k\+1\}^\{\\frac\{1\}\{2\}\}\+c\_\{0\}I\)\\tilde\{\\theta\}\_\{k\}\\leq 4L^\{2\}\\Bigl\(M\_\{g\}\\sqrt\{M\_\{\\psi\}^\{2\}\+C\_\{w\}^\{2\}\}\+c\_\{0\}\\Bigr\),\(50\)where we used thatΦk\\Phi\_\{k\}is diagonal\.

Similarly, we have

𝔼​gkT​θ~k​wk\+1=𝔼​\[gkT​θ~k​𝔼​\[wk\+1∣ℱk\]\]=0\.\\mathbb\{E\}g\_\{k\}^\{\\mathrm\{T\}\}\\tilde\{\\theta\}\_\{k\}\\,w\_\{k\+1\}=\\mathbb\{E\}\\bigl\[g\_\{k\}^\{\\mathrm\{T\}\}\\tilde\{\\theta\}\_\{k\}\\,\\mathbb\{E\}\\\!\\left\[w\_\{k\+1\}\\mid\\mathcal\{F\}\_\{k\}\\right\]\\bigr\]=0\.\(51\)
SinceVk=β2​Vk−1\+δ2​ΦkV\_\{k\}=\\beta\_\{2\}V\_\{k\-1\}\+\\delta\_\{2\}\\Phi\_\{k\}withV0=0V\_\{0\}=0, it follows from \([32](https://arxiv.org/html/2606.28879#S4.E32)\) that, similarly to \([49](https://arxiv.org/html/2606.28879#S4.E49)\), we have

𝔼​‖Vk‖≤Mg2​\(Mψ2\+Cw2\)\.\\displaystyle\\mathbb\{E\}\\\|V\_\{k\}\\\|\\leq M\_\{g\}^\{2\}\(M\_\{\\psi\}^\{2\}\+C\_\{w\}^\{2\}\)\.\(52\)Consequently, sinceV¯k12=Vk12\+c0​I\\bar\{V\}\_\{k\}^\{\\frac\{1\}\{2\}\}=V\_\{k\}^\{\\frac\{1\}\{2\}\}\+c\_\{0\}Iis diagonal, we have

𝔼​‖V¯k12‖≤𝔼​‖Vk12‖\+c0≤Mg​Mψ2\+Cw2\+c0\.\\mathbb\{E\}\\\|\\bar\{V\}\_\{k\}^\{\\frac\{1\}\{2\}\}\\\|\\leq\\mathbb\{E\}\\\|V\_\{k\}^\{\\frac\{1\}\{2\}\}\\\|\+c\_\{0\}\\leq M\_\{g\}\\sqrt\{M\_\{\\psi\}^\{2\}\+C\_\{w\}^\{2\}\}\+c\_\{0\}\.\(53\)Since‖V¯k‖≤\(‖Vk‖12\+c0\)2\\\|\\bar\{V\}\_\{k\}\\\|\\leq\\bigl\(\\\|V\_\{k\}\\\|^\{\\frac\{1\}\{2\}\}\+c\_\{0\}\\bigr\)^\{2\}, taking expectations and using Jensen’s inequality yield

𝔼​‖V¯k‖≤\(\(𝔼​‖Vk‖\)12\+c0\)2≤\(Mg​Mψ2\+Cw2\+c0\)2\.\\mathbb\{E\}\\\|\\bar\{V\}\_\{k\}\\\|\\leq\\Bigl\(\(\\mathbb\{E\}\\\|V\_\{k\}\\\|\)^\{\\frac\{1\}\{2\}\}\+c\_\{0\}\\Bigr\)^\{2\}\\leq\\bigl\(M\_\{g\}\\sqrt\{M\_\{\\psi\}^\{2\}\+C\_\{w\}^\{2\}\}\+c\_\{0\}\\bigr\)^\{2\}\.\(54\)With \([54](https://arxiv.org/html/2606.28879#S4.E54)\) and‖θ~k‖≤2​L\\\|\\tilde\{\\theta\}\_\{k\}\\\|\\leq 2L, we have

𝔼​θ~kT​V¯k12​θ~k≤𝔼​‖V¯k12‖​‖θ~k‖2≤4​L2​\(Mg​Mψ2\+Cw2\+c0\)\.\\mathbb\{E\}\\tilde\{\\theta\}\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\}^\{\\frac\{1\}\{2\}\}\\tilde\{\\theta\}\_\{k\}\\leq\\mathbb\{E\}\\\|\\bar\{V\}\_\{k\}^\{\\frac\{1\}\{2\}\}\\\|\\\|\\tilde\{\\theta\}\_\{k\}\\\|^\{2\}\\leq 4L^\{2\}\\bigl\(M\_\{g\}\\sqrt\{M\_\{\\psi\}^\{2\}\+C\_\{w\}^\{2\}\}\+c\_\{0\}\\bigr\)\.\(55\)
Moreover, under Assumption[4](https://arxiv.org/html/2606.28879#Thmassumption4), and with \([8](https://arxiv.org/html/2606.28879#S2.E8)\),giT​V¯i\+1−12​gi≤Mg2c0g\_\{i\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{i\+1\}^\{\-\\frac\{1\}\{2\}\}g\_\{i\}\\leq\\frac\{M\_\{g\}^\{2\}\}\{c\_\{0\}\}a\.s\.,

𝔼​ψk2​gkT​V¯k\+1−12​gk≤Mψ2​Mg2c0\.\\mathbb\{E\}\\psi\_\{k\}^\{2\}g\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{2\}\}g\_\{k\}\\leq\\frac\{M\_\{\\psi\}^\{2\}M\_\{g\}^\{2\}\}\{c\_\{0\}\}\.\(56\)𝔼​gkT​V¯k\+1−12​gk​wk\+12≤Mg2c0​𝔼​wk\+12≤Mg2​Cw2c0\.\\mathbb\{E\}g\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\-\\frac\{1\}\{2\}\}g\_\{k\}w\_\{k\+1\}^\{2\}\\leq\\frac\{M\_\{g\}^\{2\}\}\{c\_\{0\}\}\\mathbb\{E\}w\_\{k\+1\}^\{2\}\\leq\\frac\{M\_\{g\}^\{2\}C\_\{w\}^\{2\}\}\{c\_\{0\}\}\.\(57\)
With \([54](https://arxiv.org/html/2606.28879#S4.E54)\), using‖θ~k‖≤2​L\\\|\\tilde\{\\theta\}\_\{k\}\\\|\\leq 2L,𝔼​‖Δk\+1‖2≤CΔ2\\mathbb\{E\}\\\|\\Delta\_\{k\+1\}\\\|^\{2\}\\leq C\_\{\\Delta\}^\{2\}, and noting thatV¯k12\\bar\{V\}\_\{k\}^\{\\frac\{1\}\{2\}\}is diagonal and positive definite, we have

𝔼​θ~kT​V¯k\+112​Δk\+1≤2​L​CΔ​\(Mg​Mψ2\+Cw2\+c0\)\.\\displaystyle\\mathbb\{E\}\\tilde\{\\theta\}\_\{k\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\\frac\{1\}\{2\}\}\\Delta\_\{k\+1\}\\leq 2LC\_\{\\Delta\}\\bigl\(M\_\{g\}\\sqrt\{M\_\{\\psi\}^\{2\}\+C\_\{w\}^\{2\}\}\+c\_\{0\}\\bigr\)\.\(58\)
Similarly, we have

𝔼​ψk​gkT​Δk\+1≤Mψ​Mg​CΔ\.\\mathbb\{E\}\\psi\_\{k\}g\_\{k\}^\{\\mathrm\{T\}\}\\Delta\_\{k\+1\}\\leq M\_\{\\psi\}M\_\{g\}C\_\{\\Delta\}\.\(59\)
Similar to the analysis of \([52](https://arxiv.org/html/2606.28879#S4.E52)\), we have

𝔼​‖mk‖2≤Mg2​\(Mψ2\+Cw2\),\\displaystyle\\mathbb\{E\}\\\|m\_\{k\}\\\|^\{2\}\\leq M\_\{g\}^\{2\}\(M\_\{\\psi\}^\{2\}\+C\_\{w\}^\{2\}\),\(60\)where we use the weighted Cauchy–Schwarz inequality\.

Then we have

𝔼​mkT​Δk\+1≤\\displaystyle\\mathbb\{E\}m\_\{k\}^\{\\mathrm\{T\}\}\\Delta\_\{k\+1\}\\leq𝔼​‖mk‖2​𝔼​‖Δk\+1‖2\\displaystyle\\sqrt\{\\mathbb\{E\}\\\|m\_\{k\}\\\|^\{2\}\\mathbb\{E\}\\\|\\Delta\_\{k\+1\}\\\|^\{2\}\}\(61\)≤\\displaystyle\\leqMg​CΔ​Mψ2\+Cw2\.\\displaystyle M\_\{g\}C\_\{\\Delta\}\\sqrt\{M\_\{\\psi\}^\{2\}\+C\_\{w\}^\{2\}\}\.
Moreover,

𝔼​Δk\+1T​V¯k\+112​Δk\+1≤\\displaystyle\\mathbb\{E\}\\Delta\_\{k\+1\}^\{\\mathrm\{T\}\}\\bar\{V\}\_\{k\+1\}^\{\\frac\{1\}\{2\}\}\\Delta\_\{k\+1\}\\leq\(E​‖V¯k\+1‖\)12​\(E​‖Δk\+1‖4\)12\\displaystyle\(E\\\|\\bar\{V\}\_\{k\+1\}\\\|\)^\{\\frac\{1\}\{2\}\}\(E\\\|\\Delta\_\{k\+1\}\\\|^\{4\}\)^\{\\frac\{1\}\{2\}\}\(62\)≤\\displaystyle\\leq2​L​CΔ​\(Mg​Mψ2\+Cw2\+c0\),\\displaystyle 2LC\_\{\\Delta\}\\bigl\(M\_\{g\}\\sqrt\{M\_\{\\psi\}^\{2\}\+C\_\{w\}^\{2\}\}\+c\_\{0\}\\bigr\),where we used the fact that‖Δk\+1‖4≤4​L2​‖Δk\+1‖2\\\|\\Delta\_\{k\+1\}\\\|^\{4\}\\leq 4L^\{2\}\\\|\\Delta\_\{k\+1\}\\\|^\{2\}\.

Combining \([48](https://arxiv.org/html/2606.28879#S4.E48)\)–\([62](https://arxiv.org/html/2606.28879#S4.E62)\), taking expectations on both sides of \([48](https://arxiv.org/html/2606.28879#S4.E48)\) and summing the resulting inequality fromk=1k=1tonn, we have

𝔼​‖Θ~n\+1‖2≤\\displaystyle\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{n\+1\}\\\|^\{2\}\\leq𝔼​‖Θ~1‖2−∑i=1n2​a​κ​𝔼​ℒi2p\\displaystyle\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{1\}\\\|^\{2\}\-\\sum\_\{i=1\}^\{n\}2a\\kappa\\mathbb\{E\}\\mathcal\{L\}\_\{i\}^\{\\frac\{2\}\{p\}\}\(63\)\+4​n​L2​ℳ0​\(δ2\+r​α2\)\\displaystyle\+4nL^\{2\}\\mathcal\{M\}\_\{0\}\\Bigl\(\\sqrt\{\\delta\_\{2\}\}\+r\\alpha^\{2\}\\Bigr\)\+n​\(α2\+β12\)​δ12​Mg2c0​\[\(r\+2\)​Mψ2\+\(r\+3\)​Cw2\]\\displaystyle\+n\(\\alpha^\{2\}\+\\beta\_\{1\}^\{2\}\)\\delta\_\{1\}^\{2\}\\,\\frac\{M\_\{g\}^\{2\}\}\{c\_\{0\}\}\\Bigl\[\(r\+2\)M\_\{\\psi\}^\{2\}\+\(r\+3\)C\_\{w\}^\{2\}\\Bigr\]\+2​n​\(α​β1\+4​L\)​ℳ0​CΔ\+2​n​a​Mψ​Mg​CΔ,\\displaystyle\+2n\(\\alpha\\beta\_\{1\}\+4L\)\\mathcal\{M\}\_\{0\}C\_\{\\Delta\}\+2naM\_\{\\psi\}M\_\{g\}C\_\{\\Delta\},whereℳ0=Mg​Mψ2\+Cw2\+c0\.\\mathcal\{M\}\_\{0\}=M\_\{g\}\\sqrt\{M\_\{\\psi\}^\{2\}\+C\_\{w\}^\{2\}\}\+c\_\{0\}\.

Moreover, one can deduce that

∑i=1n𝔼​ℒi≤n1−p2​𝔼​\(∑i=1nℒi2p\)p2≤n1−p2​\(𝔼​∑i=1nℒi2p\)p2,\\sum\_\{i=1\}^\{n\}\\mathbb\{E\}\\mathcal\{L\}\_\{i\}\\leq n^\{1\-\\frac\{p\}\{2\}\}\\mathbb\{E\}\\Bigl\(\\sum\_\{i=1\}^\{n\}\\mathcal\{L\}\_\{i\}^\{\\frac\{2\}\{p\}\}\\Bigr\)^\{\\frac\{p\}\{2\}\}\\leq n^\{1\-\\frac\{p\}\{2\}\}\\Bigl\(\\mathbb\{E\}\\sum\_\{i=1\}^\{n\}\\mathcal\{L\}\_\{i\}^\{\\frac\{2\}\{p\}\}\\Bigr\)^\{\\frac\{p\}\{2\}\},\(64\)wherep∈\[1,2\]p\\in\[1,2\]is defined in Assumption[4](https://arxiv.org/html/2606.28879#Thmassumption4)\.

Combining \([63](https://arxiv.org/html/2606.28879#S4.E63)\) and \([64](https://arxiv.org/html/2606.28879#S4.E64)\), using\(x1\+x2\)p2≤x1p2\+x2p2\(x\_\{1\}\+x\_\{2\}\)^\{\\frac\{p\}\{2\}\}\\leq x\_\{1\}^\{\\frac\{p\}\{2\}\}\+x\_\{2\}^\{\\frac\{p\}\{2\}\}forx1,x2≥0x\_\{1\},x\_\{2\}\\geq 0andp∈\[1,2\]p\\in\[1,2\], and noting that𝔼​‖Θ~1‖2\\mathbb\{E\}\\\|\\tilde\{\\Theta\}\_\{1\}\\\|^\{2\}is uniformly bounded, we obtain \([30](https://arxiv.org/html/2606.28879#S3.E30)\)\.■\\hfill\\blacksquare

## 5Experiments

We illustrate the theoretical findings through experiments on both synthetic and real\-world datasets\.

Numerical Experiments\.

We consider an online linear regression problem with drifting parameters and nonstationary regressors\. The data\-generating process is given by

\{ϕk\+1=Ak​ϕk\+vk,θk=Π\[−5,5\]5​\(θk−1\+ηk\),yk=ϕkT​θk\+εk,\\left\\\{\\begin\{aligned\} &\\phi\_\{k\+1\}=A\_\{k\}\\phi\_\{k\}\+v\_\{k\},\\\\\[2\.84526pt\] &\\theta\_\{k\}=\\Pi\_\{\[\-5,5\]^\{5\}\}\(\\theta\_\{k\-1\}\+\\eta\_\{k\}\),\\\\\[2\.84526pt\] &y\_\{k\}=\\phi\_\{k\}^\{\\mathrm\{T\}\}\\theta\_\{k\}\+\\varepsilon\_\{k\},\\end\{aligned\}\\right\.\(65\)whereΠ\[−5,5\]5​\(⋅\)\\Pi\_\{\[\-5,5\]^\{5\}\}\(\\cdot\)denotes the projection onto the hypercube\[−5,5\]5\[\-5,5\]^\{5\}, andθ0∼N​\(0,I5\),ηk​∼i\.i\.d\.​N​\(0,0\.012​I5\),\\theta\_\{0\}\\sim\\mathrm\{N\}\(0,I\_\{5\}\),~\\eta\_\{k\}\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}\\mathrm\{N\}\(0,0\.01^\{2\}I\_\{5\}\),whereN​\(⋅,⋅\)\\mathrm\{N\}\(\\cdot,\\cdot\)denotes the Gaussian distribution\. This model introduces gradual temporal drift in the parameters\. The regressor\{ϕk\}\\\{\\phi\_\{k\}\\\}is generated by a time\-varying autoregressive process with diagonal matrixAk=diag​\(0\.68\+0\.3​sin⁡\(2​π​k50000\),0\.14,0\.11,0\.85,0\.62\)A\_\{k\}=\\mathrm\{diag\}\(0\.68\+0\.3\\sin\\left\(\\frac\{2\\pi k\}\{50000\}\\right\),0\.14,0\.11,0\.85,0\.62\),vk​∼i\.i\.d\.​𝒩​\(0,0\.25​I5\)v\_\{k\}\\overset\{\\mathrm\{i\.i\.d\.\}\}\{\\sim\}\\mathcal\{N\}\\left\(0,0\.25I\_\{5\}\\right\)\. The observation noiseεk∼𝒩​\(0,0\.36\)\.\\varepsilon\_\{k\}\\sim\\mathcal\{N\}\(0,0\.36\)\.It can be verified that the regressor sequence\{ϕk\}\\\{\\phi\_\{k\}\\\}is nonstationary but satisfies the proposed data excitation condition \([19](https://arxiv.org/html/2606.28879#S3.E19)\) withh=1h=1andγ=0\.25\\gamma=0\.25\. The presence of drifting parameters together with correlated nonstationary regressors results in a challenging online learning setting\.

![Refer to caption](https://arxiv.org/html/2606.28879v1/x1.png)Figure 1:Parameter tracking error and prediction error under different online optimization algorithms, and sensitivity of Adam to hyperparameters\(β1,β2\)\(\\beta\_\{1\},\\beta\_\{2\}\)\.We compare five standard online optimization methods \(see, e\.g\.,\[[28](https://arxiv.org/html/2606.28879#bib.bib28)\]\):*SGD*with stepsizeα=10−2\\alpha=10^\{\-2\},*SGD with Momentum*\(α=10−2,β=0\.9\\alpha=10^\{\-2\},\\beta=0\.9\),*AdaGrad*\(α=0\.9,c0=10−8\\alpha=0\.9,c\_\{0\}=10^\{\-8\}\),*RMSProp*\(α=10−2,β=0\.99,c0=10−8\\alpha=10^\{\-2\},\\beta=0\.99,c\_\{0\}=10^\{\-8\}\), and*Adam*\(α=10−2,β1=0\.1,β2=0\.99,c0=10−8\\alpha=10^\{\-2\},\\beta\_\{1\}=0\.1,\\beta\_\{2\}=0\.99,c\_\{0\}=10^\{\-8\}\)\. For Adam, the parameter update is followed by a projection onto the admissible setD=\[−10,10\]5D=\[\-10,10\]^\{5\}, which guarantees bounded estimates\. All methods perform updates based on the instantaneous squared loss, i\.e\.,\(yk\+1−y^k\+1\)2\(y\_\{k\+1\}\-\\hat\{y\}\_\{k\+1\}\)^\{2\}, without mini\-batches or offline training\. Performance is evaluated using two metrics: \(i\) the parameter tracking error‖θ~k‖2\\\|\\tilde\{\\theta\}\_\{k\}\\\|^\{2\}, and \(ii\) a rolling\-window average of the squared prediction loss over the most recentW=5000W=5000samples\. As shown in Fig\.[1](https://arxiv.org/html/2606.28879#S5.F1)\(a\), Adam achieves the lowest parameter tracking error throughout the time horizon, indicating effective tracking of the drifting parameters\. Figure[1](https://arxiv.org/html/2606.28879#S5.F1)\(b\) shows that Adam also achieves the lowest rolling mean prediction error compared to the other methods\. In contrast, AdaGrad exhibits larger fluctuations and higher regret, indicating its sensitivity to nonstationarity\. It should be noted that the hyperparameter choice used for Adam, namely the step sizeα=CΔ=10−2\\alpha=\\sqrt\{C\_\{\\Delta\}\}=10^\{\-2\}together with a smallβ1\\beta\_\{1\}and a largeβ2\\beta\_\{2\}, is consistent with the intuitive guidelines in Remark[7](https://arxiv.org/html/2606.28879#Thmremark7)and aligns with our theoretical results\.

We further investigate the sensitivity of Adam to its hyperparameters\(β1,β2\)\(\\beta\_\{1\},\\beta\_\{2\}\)from a prediction\-error perspective\. Fixingβ1=0\.1\\beta\_\{1\}=0\.1and varyingβ2∈\(0\.1,0\.3,0\.5,0\.7,0\.9,0\.95,0\.99\)\\beta\_\{2\}\\in\{\(0\.1,0\.3,0\.5,0\.7,0\.9,0\.95,0\.99\)\}, Fig\.[1](https://arxiv.org/html/2606.28879#S5.F1)\(c\) exhibits a clear performance stratification: largerβ2\\beta\_\{2\}consistently leads to lower prediction error and noticeably smoother trajectories\. In contrast, whenβ2\\beta\_\{2\}is fixed at0\.990\.99, varyingβ1\\beta\_\{1\}over\(0\.1,0\.3,0\.5,0\.7,0\.9\)\{\(0\.1,0\.3,0\.5,0\.7,0\.9\)\}yields curves that are largely close to one another in Fig\.[1](https://arxiv.org/html/2606.28879#S5.F1)\(d\), suggesting a weaker dependence onβ1\\beta\_\{1\}over a broad range\. However, takingβ1\\beta\_\{1\}large \(e\.g\.,β1=0\.9\\beta\_\{1\}=0\.9\) degrades prediction accuracy in time\-varying and nonstationary regimes\. Overall, these empirical observations align with our theoretical results in time\-varying scenarios with nonstationary data and offer practical guidance for hyperparameter selection: choosing a sufficiently largeβ2\\beta\_\{2\}together with a comparatively smallerβ1\\beta\_\{1\}yields lower prediction error\.

Real\-World Data Experiments\.

Having verified the performance of Adam under a synthetic drifting\-parameter model, we next evaluate the practical impact of Adam’s momentum parameters\(β1,β2\)\(\\beta\_\{1\},\\beta\_\{2\}\)in a real\-world nonstationary time\-series prediction task\. Experiments are conducted on the UCI Air Quality dataset\[[35](https://arxiv.org/html/2606.28879#bib.bib35)\], which exhibits pronounced temporal variability, where the goal is one\-step\-ahead prediction of the carbon monoxide \(CO\) concentration using an online linear model\. At each time step, the input feature vector includes a bias term, the past1212lagged values of the target variable, sinusoidal hour\-of\-day features capturing daily periodicity, and contemporaneous measurements from other sensors\. All features are standardized prior to training\.

Model parameters are updated online via the Adam algorithm with stepsizeα=10−3\\alpha=10^\{\-3\}andc0=10−8c\_\{0\}=10^\{\-8\}\. We perform a grid search overβ1∈\{0,0\.02,0\.05,0\.1,0\.2,0\.3,0\.5,0\.7\}\\beta\_\{1\}\\in\\\{0,0\.02,0\.05,0\.1,0\.2,0\.3,0\.5,0\.7\\\}andβ2∈\{0\.9,0\.95,0\.99,0\.995,0\.999,0\.9995\}\\beta\_\{2\}\\in\\\{0\.9,0\.95,0\.99,0\.995,0\.999,0\.9995\\\}\. After each update, the parameter vectorθk∈ℝd\\theta\_\{k\}\\in\\mathbb\{R\}^\{d\}is projected onto a bounded admissible setD=\[−10,10\]dD=\[\-10,10\]^\{d\}\. Performance is evaluated using the squared prediction loss\(yk\+1−y^k\+1\)2\(y\_\{k\+1\}\-\\hat\{y\}\_\{k\+1\}\)^\{2\}\. The prediction error is defined as the average loss, i\.e\.,1n​∑k=1n\(yk\+1−y^k\+1\)2\\frac\{1\}\{n\}\\sum\_\{k=1\}^\{n\}\(y\_\{k\+1\}\-\\hat\{y\}\_\{k\+1\}\)^\{2\}, without mini\-batching or offline recalibration\. Figure[2](https://arxiv.org/html/2606.28879#S5.F2)shows that prediction performance exhibits a clear dependence on the second\-moment parameterβ2\\beta\_\{2\}\. The left heat map shows a clear monotone dependence onβ2\\beta\_\{2\}across allβ1\\beta\_\{1\}\. Largerβ2\\beta\_\{2\}consistently yields lower average prediction error, andβ2≥0\.999\\beta\_\{2\}\\geq 0\.999performs best\. By comparison, sensitivity toβ1\\beta\_\{1\}is limited over a moderate range, but degrades for largeβ1\\beta\_\{1\}\.

![Refer to caption](https://arxiv.org/html/2606.28879v1/x2.png)Figure 2:Left: heatmap of average prediction error as a function of\(β1,β2\)\(\\beta\_\{1\},\\beta\_\{2\}\)\. Right: average prediction error trajectories under different online optimization algorithms\.The right panel compares Adam with four standard online optimization methods using fixed parameter configurations\. Specifically, we consider:*SGD*with stepsizeα=5×10−3\\alpha=5\\times 10^\{\-3\};*SGD with momentum*\(α=5×10−3,β=0\.1\)\(\\alpha=5\\times 10^\{\-3\},\\beta=0\.1\);*AdaGrad*\(α=0\.5,c0=10−8\)\(\\alpha=0\.5,c\_\{0\}=10^\{\-8\}\);*RMSProp*\(α=10−2,β=0\.9995,c0=10−8\)\(\\alpha=10^\{\-2\},\\beta=0\.9995,c\_\{0\}=10^\{\-8\}\);*Adam*\(α=2×10−2,β1=0\.01,β2=0\.9995,c0=10−8\)\(\\alpha=2\\times 10^\{\-2\},\\beta\_\{1\}=0\.01,\\beta\_\{2\}=0\.9995,c\_\{0\}=10^\{\-8\}\)\. For Adam, each parameter update is followed by a projection onto the admissible setD=\[−10,10\]dD=\[\-10,10\]^\{d\}, wheredddenotes the feature dimension, ensuring bounded parameter estimates during the online learning process\. While previous studies have demonstrated the advantages of Adam in the stationary settings, our experiment focuses on its performance under online nonstationary data streams\. The results show that Adam produces significantly smoother and more stable prediction\-error trajectories in this dynamic environment, highlighting its adaptation to distributional changes and time\-varying data characteristics with suitable hyperparameters\.

## 6Conclusion

Despite the well\-known Adam’s widespread practical success, providing rigorous guarantees for both parameter estimation and adaptive prediction remain challenging, particularly when data are generated by time\-varying, nonstationary dynamical systems\. By combining Lyapunov\-based techniques with new tools for analyzing products of random matrices in Adam’s dynamics, we established explicit parameter\-tracking and prediction guarantees for a broad class of nonlinear stochastic systems\. Our results rely on a conditional excitation condition on the data, which substantially relaxes the commonly imposed i\.i\.d\. assumptions and therefore applies to feedback\-driven and temporally dependent data streams\. Moreover, compared with existing related results, our bounds also feature a milder and better\-conditioned dependence on the hyperparameters, yield sharper scalings with respect to momentum and second\-moment adaptation, and explicitly quantify the combined impact of gradient noise and parameter drift\. In this way, we provide theoretical guarantees that extend well beyond i\.i\.d\. and constant parameter learning settings\. These findings offer a theoretical foundation of Adam\-type algorithm under general datasets and also suggest guidelines for hyperparameter selection, which are further supported by two experiments\. Interesting directions for future work include extending the analysis to broader nonlinear model classes and developing systematic hyperparameter design principles\.

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