Stochastic Approximation

Recent finite-time analyses of nonlinear two-time-scale stochastic approximation show that under contractive assumptions the slow iterate $Y_k$ with stepsizes $β_k=Θ(k^{-1})$ and $α_k=Θ(k^{-a})$, $a\in(1/2,1)$, generally satisfies a mean-square rate of order $k^{-a}$; decoupled $k^{-1}$ rates require strong local linearity. We identify a sharp regularity-dependent boundary. In a rate-determining normal form where the slow drift contains a locally linear leakage and a nonlinear remainder of order $1+ρ$ ($ρ\in[0,1]$), the uncorrected recursion satisfies $$\mathbb{E}\|Y_k\|^2 \le C\bigl(k^{-1}+k^{-a(1+ρ)}\bigr),$$ and a matching scalar Gaussian lower bound shows that the slower term is unavoidable without modifying the update. Thus the decoupled $k^{-1}$ rate is guaranteed for the uncorrected recursion exactly when $a(1+ρ)\ge 1$. This lower bound concerns only the naive update; it is not an information-theoretic obstruction. We demonstrate this by equipping the normal-form recursion with an auxiliary online bias estimator $$M_{k+1}=M_k+γ_k(R(X_k)-M_k),\qquad β_k\llγ_k\llα_k,$$ and subtracting $M_k$ from the slow update. Under the same stability, moment, and remainder assumptions, the corrected recursion achieves $\mathbb{E}\|\widetilde Y_k\|^2=O(k^{-1})$ for every $ρ\in[0,1]$, including regimes where the uncorrected update provably suffers the slower rate. Finally, we prove localized transfer theorems that extend the phase-transition mechanism to general nonlinear TTSA in fast-manifold coordinates. The proofs are non-asymptotic and rely on two Abel-transform cancellations: one for the locally linear fast-error leakage, and one for the tracked nonlinear bias.

In this paper, we study the finite-time behavior of the TD(0) temporal-difference method with linear function approximation (LFA). We consider on-policy independent and identically distributed (i.i.d.) samples, a constant learning step, and the Polyak-Juditsky averaging method. We establish a new convergence rate, for the Mean-Square Error (MSE) on the approximated function, that is (i) fast in the sense that it admits an optimal dependency in the number of iterations k (i.e., of order 1/k), (ii) robust to ill-conditioning: it only depends on an initial error and modelindependent constants and (iii) sharp up to a multiplicative constant lower than 11. In particular, it does not depend on the smallest eigenvalue of the uncentered covariance matrix of the linear parametrization, unlike all pre-existing O(1/k) rates in the TD(0) literature. We also introduce PCTD(0), a variant of TD(0), which benefits from better convergence properties under an additional assumption of strong mixing on the Markov Chain.

Coupled gradient descent - where the update of one parameter depends on another - arises naturally in bilevel optimization, two-time-scale stochastic approximation, and generative adversarial networks. When the coupled Jacobian is block-triangular, asymptotic stability is determined by the spectral radii of the diagonal blocks, yet transient amplification before convergence can be arbitrarily large due to non-normality. We develop a sharp pseudospectral theory for block-triangular Jacobians J = [[A, 0], [C, D]], proving Kreiss-constant bounds of the form K(J) <= 2/(1-γ) + ||C||/(4(1-γ)) when ρ(A), ρ(D) <= γ< 1 and A, D are symmetric, and establishing matching minimax lower bounds. We characterize the critical coupling threshold for spectral instability and extend the theory to nearly self-referential systems via a Neumann-series perturbation framework. As a consequence, we obtain a finite-horizon O(K(J)^2 log(1/δ)) iteration complexity bound. Framed as scaling laws for stochastic two-time-scale optimization, our results expose a non-asymptotic, instance-dependent regime of high-dimensional learning dynamics that is invisible to spectral-radius analysis. Experiments on linear-quadratic problems, IQC-based comparisons, and neural-network training confirm the theory.

We survey Lyapunov-based techniques for the finite-time analysis of stochastic iterative algorithms, also known as stochastic approximation (SA) algorithms, for solving fixed-point equations $\bar{F}(x)=x$, where the operator $\bar{F}(\cdot)$ can only be accessed through a noisy oracle. We first focus on the standard setting in which $\bar{F}(\cdot)$ is contractive with respect to some norm and the noise is i.i.d., and explain how generalized Moreau envelopes serve as universal Lyapunov functions, regardless of the underlying norm. We then show how this framework yields mean-square convergence guarantees and applies to stochastic gradient descent, linear SA, and value-based reinforcement learning algorithms such as Q-learning and temporal-difference learning. Finally, we discuss extensions to Markovian noise, seminorm-contractive operators, dissipative operators, and high-probability bounds, and conclude with open problems. The goal is to present a unified and self-contained roadmap for the finite-time analysis of SA and its applications, especially in reinforcement learning.

This work studies the convergence of two-timescale stochastic approximations (SA), a class of iterative algorithms that update two sets of parameters in fast and slow timescales respectively. Notable examples of two-timescale SA in reinforcement learning (RL) include temporal difference learning with gradient correction (TDC) and actor-critic methods. Previously, the stability (i.e., boundedness) and convergence of two-timescale SA were only established under i.i.d. noise. This work instead establishes the stability and convergence of two-timescale SA under Markovian noise, a setup that is more realistic in RL. Notably, we do not need to use any projection operator and the noise does not need to live in a compact space. Our key technical novelty is to control the fast timescale parameter with the running max of the slow timescale parameter, instead of with the current slow timescale parameter, as most prior works do. As a key application, we establish the first almost sure convergence of TDC with eligibility traces under off-policy learning with linear function approximation.

The loss and the norm of its gradient separate the healthy and the pathological regimes of neural-network training only weakly, whilst the curvature of the empirical risk differs qualitatively between them but is inaccessible explicitly at parameter counts $P\sim 10^{6}-10^{8}$. We present a stochastic estimator of the trace of the diagonal blocks of the Hessian matrix of the empirical risk of a neural network. The procedure combines the Hutchinson stochastic trace estimator with a single Hessian-vector product over the whole parameter vector and recovers unbiased estimates of every per-layer trace in one backward pass through the computational graph. We show that correctness under weight sharing requires the layer-wise Hessian to be assembled before the second differentiation: unrolling shared weights into independent coordinates introduces a systematic bias whose sign and magnitude are governed by the cross-instance blocks of the unrolled Hessian. A closed-form expression for the variance of the estimator at a fixed Hessian is derived, together with a decomposition of the total variance under the mini-batch sampling distribution. This decomposition yields a critical probe count $K^{\star}$ that balances the two sources of randomness and supports the practical recommendation $K\in[5,10]$ in the on-line monitoring regime. The estimator is applied to the detection of the label-memorisation regime of ResNet-18, ResNet-34, and VGG-11 on CIFAR-10 and CIFAR-100, where a calibrated cumulative-sum decision rule attains an empirical detection power of $179/180$ at a false-alarm rate of $16/120$.

We establish maximal concentration bounds for the iterates generated by stochastic approximation algorithms with general step sizes, where the noise has a finite-state Markovian component plus a Martingale-difference component. When the Martingale-difference noise is bounded, we show that the tail of the error can be sub-Gaussian, sub-Weibull, or something lighter than any Pareto but heavier than any Weibull, depending on the step size sequence and on whether the random operator is almost surely contractive, almost surely non-expansive, or expansive with positive probability. Our analysis relies on a novel Lyapunov function involving the moment-generating function of the solution to a Poisson equation, together with an auxiliary projected algorithm. We complement the upper bounds with worst-case examples showing that qualitatively sharper bounds are impossible. We further study the case of unbounded Martingale-difference noise when the average operator is contractive, and the step sizes are of order $1/k$. In this setting, we show that if the random operator is almost surely non-expansive, then the error tail is at most three times heavier than the noise tail, whereas if the random operator is expansive with positive probability, then the error may have substantially heavier tails. These results are obtained through a novel black-box truncation argument that reduces the unbounded-noise setting to the bounded-noise case.

In this paper, we establish Berry-Esseen-type bounds for federated linear stochastic approximation (LSA). Our results provide the first federated Gaussian approximations for LSA that explicitly capture communication-computation trade-offs and heterogeneity-aware error terms, quantifying the effects of local step size, number of local updates, and heterogeneity on convergence rates. We present results for both (i) constant step size regime and (ii) decreasing step size with an increasing number of local iterations, recovering the recent rates of Bonnerjee et al. [2025] as a special case. As a primary application of our results, we develop an online multiplier bootstrap procedure for inference on the last iterate, which avoids explicit estimation of the asymptotic covariance matrix, and obtain non-asymptotic validity guarantees for this procedure.

In this paper, we derive rates of convergence in the high-dimensional central limit theorem for Polyak--Ruppert averaged iterates generated by entropy-regularized asynchronous Q-learning with linear function approximation and a polynomial stepsize $k^{-ω}$, $ω\in (1/2,1)$. Assuming that the sequence of observed triples $(s_k,a_k,s_{k+1})_{k \geq 0}$ forms a uniformly geometrically ergodic Markov chain, and under suitable regularity conditions for the projected soft Bellman equation, we establish a Gaussian approximation bound in the convex distance with rate of order $n^{-1/4}$, up to polylogarithmic factors in $n$, where $n$ is the number of samples used by the algorithm. To obtain this result, we combine a linearization of the soft Bellman recursion with a Gaussian approximation for the leading martingale term. Finally, we derive high-order moment bounds for the algorithm's last iterate, which might be of independent interest.

We study online estimation in latent-variable models by recasting the problem as tracking a moving empirical equilibrium. Standard online EM and stochastic approximation analyses primarily study convergence toward the population parameter and typically do not isolate the empirical batch optimum from the online tracking error at finite horizon. Our framework decomposes the online estimate into the frozen batch equilibrium at the current running statistic and a tracking lag that captures the algorithm's delay behind this moving target. We prove a batch-to-online transfer theorem: provided $\lVert e_T \rVert_{L^{2}} = o(T^{-1/2})$, the online estimator inherits the batch central limit theorem and the sharp first-order risk constant. Our key observation is that the empirical optimum evolves on a smooth equilibrium manifold indexed by the running statistic. An $m$-th order equilibrium-jet predictor combined with an order-$ν$ frozen corrector yields localized tracking rates $O(T^{-ν(m+1)})$. We formalize EM-compressibility and EM-jet$^R$-compressibility as the structural conditions that make the equilibrium response and the Newton corrector evaluable from a retained streaming statistic. The theory is instantiated in latent linear Gaussian covariance estimation, where the first-order scheme operates on a compressed $d \times d$ statistic with explicit finite-sample risk envelopes and a certified restart rule.

Designing model-free algorithms for distributionally robust reinforcement learning (DRRL) poses fundamental challenges. The robust Bellman operator is nonlinear in the transition kernel, which makes one-sample Bellman updates biased, while the adversarial optimization underlying robustness makes robust evaluation computationally demanding. To address these difficulties, we consider the natural small-ambiguity regime under Kullback--Leibler ambiguity sets and propose an approximate DRRL framework based on a first-order expansion of the relevant robust functional. This yields an approximate robust Bellman equation that removes the adversarial optimization while remaining first-order accurate in the ambiguity radius. To learn the fixed point of this approximate equation, we propose Mean-Variance Stochastic Approximation (MVSA), a model-free algorithm that uses only one-sample updates. This is achieved via a lifted stochastic approximation dynamics and a two-time-scale design. We then prove convergence and a central limit theorem for MVSA: its main iterate satisfies a central limit theorem at the canonical $n^{-1/2}$ scale, with explicitly characterized asymptotic covariances. Finally, we validate our theoretical findings with a numerical experiment.

Establishing almost sure convergence rates for stochastic approximation and reinforcement learning under Markovian noise is a fundamental theoretical challenge. We make progress towards this challenge for a class of stochastic approximation algorithms whose expected updates are contractive, a setting that arises in many reinforcement learning algorithms such as $Q$-learning and linear temporal difference learning. Specifically, for a power-law learning rate $O(n^{-η})$ with $η\in (1/2, 1)$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{1 - 2η})$. For a harmonic learning rate $O(n^{-1})$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{-1})$, which we argue is a strong result because it is close to the optimal rate $O(n^{-1}\log\log n)$ given by the law of the iterated logarithm (for a special case of i.i.d. noise). Key to our analysis is a novel Lyapunov drift construction that applies a Poisson-equation based correction for Markovian noise to the well-established Moreau-envelope smoothing for the contractive mapping.

We study finite-iteration behavior of asynchronous categorical distributional temporal-difference methods, covering scalar categorical TD (CTD) in the Cramér geometry and multivariate signed-categorical TD (MTD) in the maximum mean discrepancy (MMD) geometry. We establish high-probability guarantees under i.i.d. sampling and under a continuing Markovian trajectory. For CTD, the Cramér sample complexity to the true return law has leading term $\tilde O(ρ_{\min}^{-1}(1-γ)^{-2}\varepsilon^{-2})$ in the i.i.d. and $\tilde O(μ_{\min}^{-1}(1-γ)^{-2}\varepsilon^{-2}+μ_{\min}^{-1}t_{\mathrm{mix}})$ in the Markovian setting, without using variance reduction or data dropping. For MTD, the MMD sample complexity has leading terms $\tilde O(ρ_{\min}^{-1}(1-γ)^{-1-c}\varepsilon^{-2})$ and $\tilde O(μ_{\min}^{-1}(1-γ)^{-1-c}\varepsilon^{-2}+μ_{\min}^{-1}t_{\mathrm{mix}})$, where $c\in(0,2)$ is the homogeneity exponent of the MMD kernel, and they reduce to the CTD rates when $c=1$. These are, to our knowledge, the first finite-iteration rates for MTD. For undiscounted fixed-horizon policy evaluation, the same analysis applies to fixed-horizon versions of CTD and MTD under i.i.d. and episodic sampling, and the rates match the discounted ones with the effective horizon $(1-γ)^{-1}$ replaced by the horizon $H$ in the leading term. Matching minimax lower bounds show that the leading terms, and the implied $1$-Wasserstein rates, are optimal under i.i.d., Markovian, and episodic sampling. Together, these results provide a unified non-asymptotic analysis of asynchronous categorical distributional TD across scalar, multivariate, discounted, and fixed-horizon settings, with rates that are minimax optimal in their leading terms.

The analysis of Temporal Difference (TD) learning in the average-reward setting faces notable theoretical difficulties because the Bellman operator is not contractive with respect to any norm. This complicates standard analyses of stochastic updates that are effective in discounted settings. Although a considerable body of literature addresses these challenges, existing theoretical approaches come with limitations. We introduce a novel algorithm designed explicitly for policy evaluation in the average-reward setting, utilizing sampling from two Markovian trajectories. Our proposed method overcomes previous limitations by guaranteeing convergence to the unique solution of a properly defined projected Bellman equation. Notably, and in contrast to earlier work, our convergence analysis is uniformly applicable to both linear function approximation and tabular settings and does not involve explicit dimension-dependent terms in its convergence bounds. These results align with what is known to hold in the discounted setting. Furthermore, our algorithm achieves improved dependence on the problem's condition number, reducing the sample complexity from quartic, as in prior literature, to quadratic scaling, and thus matching the efficiency seen in the discounted setting.

Reliable decision-making with streaming data requires principled uncertainty quantification of online methods. While first-order methods enable efficient iterate updates, their inference procedures still require updating proper (covariance) matrices, incurring $O(d^2)$ time and memory complexity, and are sensitive to ill-conditioning and noise heterogeneity of the problem. This costly inference task offers an opportunity for more robust second-order methods, which are, however, bottlenecked by solving Newton systems with $O(d^3)$ complexity. In this paper, we address this gap by studying an online Newton method with Hessian averaging, where the Newton direction at each step is approximately computed using a sketch-and-project solver with Nesterov's acceleration, matching $O(d^2)$ complexity of first-order methods. For the proposed method, we quantify its uncertainty arising from both random data and randomized computation. Under standard smoothness and moment conditions, we establish global almost-sure convergence, prove asymptotic normality of the last iterate with a limiting covariance characterized by a Lyapunov equation, and develop a fully online covariance estimator with non-asymptotic convergence guarantees. We also connect the resulting uncertainty quantification to that of exact and sketched Newton methods without Nesterov's acceleration. Extensive experiments on regression models demonstrate the superiority of the proposed method for online inference.

We consider a stochastic optimization problem involving two random variables: a context variable $X$ and a dependent variable $Y$. The objective is to minimize the expected value of a nonlinear loss functional applied to the conditional expectation $\mathbb{E}[f(X, Y,β) \mid X]$, where $f$ is a nonlinear function and $β$ represents the decision variables. We focus on the practically important setting in which direct sampling from the conditional distribution of $Y \mid X$ is infeasible, and only a stream of i.i.d. observation pairs $\{(X^k, Y^k)\}_{k=0,1,2,\ldots}$ is available. In our approach, the conditional expectation is approximated within a prespecified parametric function class. We analyze a simultaneous learning-and-optimization algorithm that jointly estimates the conditional expectation and optimizes the outer objective. Using a specially designed measure of non-optimality, combining the squared norm of the objective function's gradient and the mean square error of the auxiliary parametric model, we establish that the method achieves a convergence rate of order $\mathcal{O}\big(1/\sqrt{N}\big)$, where $N$ denotes the number of observed pairs.

Constant-stepsize stochastic approximation (SA) is widely used in learning for computational efficiency, yet the distribution of the iterates is typically intractable. Classical asymptotics results give $X_k^{(α)} \approx X^{(α)} \approx x^\star+\sqrtαY$, where $X^{(α)}$ is the steady state and $Y$ is an appropriate Gaussian limit, by progressively taking the time $k\uparrow\infty$ and stepsize $α\downarrow0$. Such limit results, however, do not quantify finite-time, finite-stepsize errors. We develop an explicit pre-limit characterization for SA with i.i.d.\ and Markovian noise. We establish existence and uniqueness of the stationary law, a geometric Wasserstein convergence to stationarity, and almost-sure and $L^3$ convergence of the steady state to the root $x^\star$, identifying the scale $\sqrtα$ as first-order fluctuation. At this scale, we derive a higher-order quantitative Gaussian approximation with a Wasserstein error, using Stein's method and Poisson equation techniques. We further obtain non-uniform Berry--Esseen-type tail bounds, incorporating both steady-state approximation and finite-time convergence errors. We instantiate the theory for strongly convex smooth SGD, linear SA, and nonlinear contractive SA. Beyond strong convexity, for general convex SGD, we identify a Gibbs limiting law and prove a pre-limit Wasserstein approximation error under stability and Stein-equation hypothesis, which are validated numerically.

We propose a continuous-time formulation of a noisy persistent contrastive divergence (PCD)-like method for maximum likelihood estimation (MLE) of unnormalised densities. Our approach couples parameter updates and sampling of the parametrised density in a multiscale system of stochastic differential equations (SDEs). From this formulation, we derive non-asymptotic bounds for weak test-function errors between the resulting numerical schemes and the MLE point target. The error is decomposed into numerical discretisation, slow-fast averaging, and finite-temperature concentration terms. We also introduce an efficient implementation based on explicit stabilized integrators and establish corresponding long-time error estimates. This leads to a novel method for training energy-based models (EBMs) with quantitative error guarantees.

We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) operator-valued kernels whose associated integral operators are compact and hence admit discrete spectral decompositions, and (ii) separable kernels of the form $K(x,x')=k(x,x')T$, where $k$ is a scalar-valued kernel and $T$ is a positive operator on the output space. This broad setting induces expressive vector-valued reproducing kernel Hilbert spaces (RKHSs) that generalize the classical $K=kI$ paradigm, thereby enabling rich structural modeling with rigorous theoretical guarantees. To address target operators lying outside the RKHS, we introduce vector-valued interpolation spaces to precisely quantify misspecification error. Within this framework, we establish non-asymptotic convergence rates for prediction, estimation, and misspecification errors in the online and finite-horizon settings. Importantly, the framework also accommodates a range of operator learning settings, from Fredholm integral operators to encoder--decoder architectures. Numerical experiments on the two-dimensional Navier--Stokes equations illustrate the proposed approach.

We revisit a classical assumption for analyzing stochastic gradient algorithms where the squared norm of the stochastic subgradient (or the variance for smooth problems) is allowed to grow as fast as the squared norm of the optimization variable. We contextualize this assumption in view of its inception in the 1960s, its seemingly independent appearance in the recent literature, its relationship to weakest-known variance assumptions for analyzing stochastic gradient algorithms, and its relevance in deterministic problems for non-Lipschitz nonsmooth convex optimization. We build on and extend a connection recently made between this assumption and the Halpern iteration. For convex nonsmooth, and potentially stochastic, optimization, we analyze horizon-free, anytime algorithms with last-iterate rates. For problems beyond simple constrained optimization, such as convex problems with functional constraints or regularized convex-concave min-max problems, we obtain rates for optimality measures that do not require boundedness of the feasible set.

We consider linear two-time-scale stochastic approximation algorithms driven by martingale noise. Recent applications in machine learning motivate the need to understand finite-time error rates, but conventional stochastic approximation analyses focus on either asymptotic convergence in distribution or finite-time bounds that are far from optimal. Prior work on asymptotic central limit theorems (CLTs) suggests that two-time-scale algorithms may be able to achieve $1/\sqrt{K}$ error in expectation, with a constant given by the expected norm of the limiting Gaussian vector. However, the best known finite-time rates are much slower. We derive the first nonasymptotic Wasserstein-1 CLT for linear two-time-scale stochastic approximation with Polyak-Ruppert averaging driven by martingale difference noise. As a corollary, we show that the expected error achieved by Polyak-Ruppert averaging decays at rate $1/\sqrt{K}$, which significantly improves on the rates of convergence in prior works.

We investigate the statistical properties of Temporal Difference (TD) learning with Polyak-Ruppert averaging, arguably one of the most widely used algorithms in reinforcement learning, for the task of estimating the parameters of the optimal linear approximation to the value function. Assuming independent samples, we make three theoretical contributions that improve upon the current state-of-the-art results: (i) we establish refined high-dimensional Berry-Esseen bounds over the class of convex sets, achieving faster rates than the best known results, and (ii) we propose and analyze a novel, computationally efficient online plug-in estimator of the asymptotic covariance matrix; (iii) we derive sharper high probability convergence guarantees that depend explicitly on the asymptotic variance and hold under weaker conditions than those adopted in the literature. These results enable the construction of confidence regions and simultaneous confidence intervals for the linear parameters of the value function approximation, with guaranteed finite-sample coverage. We demonstrate the applicability of our theoretical findings through numerical experiments.