Finite-Sample Analysis of Nonlinear Stochastic Approximation with Applications in Reinforcement Learning

TL;DR

This paper provides finite-sample convergence analysis for nonlinear stochastic approximation algorithms with Markovian noise, including applications to reinforcement learning methods like Q-learning, demonstrating exponential and sublinear convergence rates.

Contribution

It establishes finite-sample bounds for nonlinear SA with Markovian noise, applicable to RL algorithms without requiring i.i.d. samples or projection steps.

Findings

Constant stepsize yields exponential convergence to a neighborhood.

Diminishing stepsize achieves O(log(k)/k) convergence rate.

Numerical results support theoretical convergence bounds.

Abstract

Motivated by applications in reinforcement learning (RL), we study a nonlinear stochastic approximation (SA) algorithm under Markovian noise, and establish its finite-sample convergence bounds under various stepsizes. Specifically, we show that when using constant stepsize (i.e., $\alpha_k\equiv \alpha$), the algorithm achieves exponential fast convergence to a neighborhood (with radius $O(\alpha\log(1/\alpha))$) around the desired limit point. When using diminishing stepsizes with appropriate decay rate, the algorithm converges with rate $O(\log(k)/k)$. Our proof is based on Lyapunov drift arguments, and to handle the Markovian noise, we exploit the fast mixing of the underlying Markov chain. To demonstrate the generality of our theoretical results on Markovian SA, we use it to derive the finite-sample bounds of the popular $Q$-learning with linear function approximation algorithm, under a condition on the behavior policy. Importantly, we do not need to make the assumption that the samples are i.i.d., and do not require an artificial projection step in the algorithm to maintain the boundedness of the iterates. Numerical simulations corroborate our theoretical results.