Global Convergence of Two-timescale Actor-Critic for Solving Linear Quadratic Regulator

TL;DR

This paper proves the global convergence of a single-sample two-timescale actor-critic algorithm for solving the linear quadratic regulator problem, providing the first finite-time analysis with improved sample complexity.

Contribution

It introduces a novel analysis framework establishing global convergence and finite-sample complexity for the practical single-sample two-timescale actor-critic in LQR.

Findings

First finite-time convergence proof for this setting

Sample complexity of (^{-2.5})

Validated through extensive simulations

Abstract

The actor-critic (AC) reinforcement learning algorithms have been the powerhouse behind many challenging applications. Nevertheless, its convergence is fragile in general. To study its instability, existing works mostly consider the uncommon double-loop variant or basic models with finite state and action space. We investigate the more practical single-sample two-timescale AC for solving the canonical linear quadratic regulator (LQR) problem, where the actor and the critic update only once with a single sample in each iteration on an unbounded continuous state and action space. Existing analysis cannot conclude the convergence for such a challenging case. We develop a new analysis framework that allows establishing the global convergence to an $\epsilon$-optimal solution with at most an $\mathcal{O}(\epsilon^{-2.5})$ sample complexity. To our knowledge, this is the first finite-time convergence analysis for the single sample two-timescale AC for solving LQR with global optimality. The sample complexity improves those of other variants by orders, which sheds light on the practical wisdom of single sample algorithms. We also further validate our theoretical findings via comprehensive simulation comparisons.