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

TL;DR

This paper derives almost sure convergence rates for stochastic approximation algorithms with Markovian noise, including reinforcement learning methods like Q-learning, using a novel Lyapunov drift approach.

Contribution

It introduces a new Lyapunov drift technique combining Poisson-equation correction and Moreau-envelope smoothing for analyzing convergence rates.

Findings

Achieves convergence rate close to o(n^{1 - 2ta}) for power-law learning rates

Obtains near-optimal convergence rate close to o(n^{-1}) for harmonic learning rates

Provides a theoretical framework applicable to reinforcement learning algorithms with Markovian noise

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 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 $O(n^{-1}\log\log n)$ 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.