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

arXiv:2605.07104v3 Announce Type: replace-cross 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