aiqbench analysis
This paper is a theoretical analysis of almost sure convergence rates for stochastic approximation methods under Markovian noise, with direct applicability to algorithms like Q-learning and linear temporal difference learning. Its main contribution is a proof technique combining a Poisson-equation correction with Moreau-envelope smoothing, yielding rate bounds that approach the classical n^-1 benchmark under specific learning-rate conditions. For ML readers, the result matters as a sharpening of convergence guarantees for RL algorithms, though it is a mathematical result rather than an empirical one and its practical impact depends on how tight these bounds prove in real training settings.
Almost Sure Convergence Rates of Stochastic Approximation and Reinforcement Learning via a Poisson-Moreau Drift
arXiv:2605.07104v2 Announce Type: replace 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 $O(n^{-\eta})$ with $\eta \in (1/2, 1)$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{1 - 2\eta})$. For the harmonic learning rate $\alpha_n=\alpha/(n+1)$, the almost sure squared-error rate is $o(n^{-\zeta})$ for every $\zeta<\min\{1,2(1-\kappa)\alpha\}$, where $\kappa$ is the contraction factor. Thus exponents arbitrarily close to one are available when $2(1-\kappa)\alpha\geq1$. This approaches the $n^{-1}$ polynomial exponent of classical law-of-the-iterated-logarithm benchmarks. 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.
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 $O(n^{-\eta})$ with $\eta \in (1/2, 1)$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{1 - 2\eta})$. For the harmonic learning rate $\alpha_n=\alpha/(n+1)$, the almost sure squared-error rate is $o(n^{-\zeta})$ for every $\zeta<\min\{1,2(1-\kappa)\alpha\}$, where $\kappa$ is the contraction factor. Thus exponents arbitrarily close to one are available when $2(1-\kappa)\alpha\geq1$. This approaches the $n^{-1}$ polynomial exponent of classical law-of-the-iterated-logarithm benchmarks. 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.
Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)
arXiv:2605.07104 [cs.LG]
(or arXiv:2605.07104v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.07104 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 248 words
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