Achieving $\epsilon^{-2}$ Sample Complexity for Single-Loop Actor-Critic under Minimal Assumptions

TL;DR

This paper proves the first $ ilde{ ext{O}}( ext{epsilon}^{-2})$ sample complexity for off-policy actor-critic methods with a single-loop, single-timescale implementation under minimal assumptions, using a novel Lyapunov drift analysis.

Contribution

It introduces a new analytical framework for coupled Lyapunov drift to establish convergence rates for single-loop actor-critic algorithms under minimal assumptions.

Findings

First $ ilde{ ext{O}}( ext{epsilon}^{-2})$ sample complexity for off-policy actor-critic.

Geometric convergence rate for the actor and $ ilde{ ext{O}}(1/T)$ for the critic.

Analysis applicable to other coupled iterative algorithms with unbounded iterates.

Abstract

In this paper, we establish last-iterate convergence rates for off-policy actor--critic methods in reinforcement learning. In particular, under a single-loop, single-timescale implementation and a broad class of policy updates, including approximate policy iteration and natural policy gradient methods, we prove the first $\tilde{\mathcal{O}}(\epsilon^{-2})$ sample complexity guarantee for finding an $\epsilon$-optimal policy under minimal assumptions, namely, the existence of a policy that induces an irreducible Markov chain. This stands in stark contrast to the existing literature, where an $\tilde{\mathcal{O}}(\epsilon^{-2})$ sample complexity is achieved only through nested-loop updates and/or under strong, algorithm-dependent assumptions on the policies, such as uniform mixing and uniform exploration. Technically, to address the challenges posed by the coupled update equations arising from the single-loop implementation, as well as the potentially unbounded iterates induced by off-policy learning, our analysis is based on a coupled Lyapunov drift framework. Specifically, we establish a geometric convergence rate for the actor and an $\tilde{\mathcal{O}}(1/T)$ convergence rate for the critic, and combine the two Lyapunov drift inequalities through a cross-domination property. We believe this analytical framework is of independent interest and may be applicable to other coupled iterative algorithms with unbounded