Almost Sure Convergence Rates of Stochastic Approximation and Reinforcement Learning via a Poisson-Moreau Drift
Year: 2026
Type: preprint
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 the harmonic learning rate $α_n=α/(n+1)$, the almost sure squared-error rate is $o(n^{-ζ})$ for every $ζ<\min\{1,2(1-κ)α\}$, where $κ$ is the contraction factor. Thus exponents arbitrarily close to one are available when $2(1-κ)α\geq1$. 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.
Source: arXiv (Cornell University)
Authors: Liu, Xinyu, Xie, Zixuan, Zhang, Shangtong
Institution: University of Virginia
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7160852816