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

arXiv:2605.07104v1 Announce Type: new 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 - 2eta}). 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}loglog 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.

Source: arXiv cs.LG | 2026-05-11