Almost Sure Convergence Rates of Stochastic Approximation and Reinforcement Learning via a Poisson-Moreau Drift

• URL: http://arxiv.org/abs/2605.07104v1
• Date: Fri, 08 May 2026 01:32:03 GMT
• Title: Almost Sure Convergence Rates of Stochastic Approximation and Reinforcement Learning via a Poisson-Moreau Drift
• Abstract summary: We establish almost sure convergence rates for approximation and reinforcement learning under Markovian noise.<n>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.
• Score: 19.81737958703724
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

Related papers

• Faster Diffusion Models via Higher-Order Approximation [28.824924809206255]
  We propose a principled, training-free sampling algorithm that requires only the order of $$ d1+2/K varepsilon-1/K $$ score function evaluations.<n>Our theory is robust vis-a-vis inexact score estimation, degrading gracefully as the score estimation error increases.<n>More broadly, our work develops a theoretical framework towards understanding the efficacy of high-order methods for accelerated sampling.
  arXiv  Detail & Related papers  (2025-06-30T16:49:03Z)
• $O(1/k)$ Finite-Time Bound for Non-Linear Two-Time-Scale Stochastic Approximation [0.0]
  We obtain an improved gradient bound of $O (1/k)$ for nonlinear two-time-scale approximations.<n>Our result applies to algorithms such as descent-ascent and two-time-scale Lagrangian optimization.
  arXiv  Detail & Related papers  (2025-04-27T22:45:00Z)
• Finite-Time Bounds for Two-Time-Scale Stochastic Approximation with Arbitrary Norm Contractions and Markovian Noise [5.7071219882414885]
  Two-time-scale Approximation (SA) is an iterative algorithm with applications in reinforcement learning and optimization.<n>Motivated by applications in reinforcement learning, we give the first mean square bound on non linear two-time-scale SA.
  arXiv  Detail & Related papers  (2025-03-24T07:03:23Z)
• Sign Operator for Coping with Heavy-Tailed Noise in Non-Convex Optimization: High Probability Bounds Under $(L_0, L_1)$-Smoothness [74.18546828528298]
  We show that SignSGD with Majority Voting can robustly work on the whole range of complexity with $kappakappakappakappa-1right, kappakappakappa-1right, kappakappakappa-1right, kappakappakappa-1right, kappakappakappa-1right, kappakappakappa-1right, kappakappakappa-1right, kappa
  arXiv  Detail & Related papers  (2025-02-11T19:54:11Z)
• Almost Sure Convergence Rates and Concentration of Stochastic Approximation and Reinforcement Learning with Markovian Noise [31.241889735283166]
  We provide the first almost sure convergence rate for $Q$-learning with Markovian samples without count-based learning rates. We also provide the first concentration bound for off-policy temporal difference learning with Markovian samples.
  arXiv  Detail & Related papers  (2024-11-20T21:09:09Z)
• Beyond likelihood ratio bias: Nested multi-time-scale stochastic approximation for likelihood-free parameter estimation [49.78792404811239]
  We study inference in simulation-based models where the analytical form of the likelihood is unknown.<n>We use a ratio-free nested multi-time-scale approximation (SA) method that simultaneously tracks the score and drives the parameter update.<n>We show that our algorithm can eliminate the original bias $Obig(sqrtfrac1Nbig)$ and accelerate the convergence rate from $Obig(beta_k+sqrtfracalpha_kNbig)$.
  arXiv  Detail & Related papers  (2024-11-20T02:46:15Z)
• Adaptive Federated Learning Over the Air [108.62635460744109]
  We propose a federated version of adaptive gradient methods, particularly AdaGrad and Adam, within the framework of over-the-air model training. Our analysis shows that the AdaGrad-based training algorithm converges to a stationary point at the rate of $mathcalO( ln(T) / T 1 - frac1alpha ).
  arXiv  Detail & Related papers  (2024-03-11T09:10:37Z)
• Under-bagging Nearest Neighbors for Imbalanced Classification [63.026765294759876]
  We propose an ensemble learning algorithm called textitunder-bagging $k$-NN (textitunder-bagging $k$-NN) for imbalanced classification problems.
  arXiv  Detail & Related papers  (2021-09-01T14:10:38Z)
• Fast Rates for the Regret of Offline Reinforcement Learning [69.23654172273085]
  We study the regret of reinforcement learning from offline data generated by a fixed behavior policy in an infinitehorizon discounted decision process (MDP) We show that given any estimate for the optimal quality function $Q*$, the regret of the policy it defines converges at a rate given by the exponentiation of the $Q*$-estimate's pointwise convergence rate.
  arXiv  Detail & Related papers  (2021-01-31T16:17:56Z)
• Finite-Time Analysis for Double Q-learning [50.50058000948908]
  We provide the first non-asymptotic, finite-time analysis for double Q-learning. We show that both synchronous and asynchronous double Q-learning are guaranteed to converge to an $epsilon$-accurate neighborhood of the global optimum.
  arXiv  Detail & Related papers  (2020-09-29T18:48:21Z)
• Single-Timescale Stochastic Nonconvex-Concave Optimization for Smooth Nonlinear TD Learning [145.54544979467872]
  We propose two single-timescale single-loop algorithms that require only one data point each step. Our results are expressed in a form of simultaneous primal and dual side convergence.
  arXiv  Detail & Related papers  (2020-08-23T20:36:49Z)
• Finite Time Analysis of Linear Two-timescale Stochastic Approximation with Markovian Noise [28.891930079358954]
  We provide a finite-time analysis for linear two timescale SA scheme. Our bounds show that there is no discrepancy in the convergence rate between Markovian and martingale noise. We present an expansion of the expected error with a matching lower bound.
  arXiv  Detail & Related papers  (2020-02-04T13:03:17Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.