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

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 ∈ (1/2, 1), we obtain an almost sure convergence rate arbitrarily close to o(n1 - 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-1loglog 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.