Finite Sample Analysis of Two-Time-Scale Natural Actor-Critic Algorithm

TL;DR

This paper provides a theoretical analysis of the convergence rates and sample complexity of a two-time-scale natural actor-critic algorithm in reinforcement learning, under general ergodic Markov decision process assumptions.

Contribution

It offers the first finite sample convergence analysis for this class of algorithms in the tabular setting with a single sample trajectory.

Findings

Convergence rate of (1/T^{1/4}) with fixed exploration.

Sample complexity of (1/\u03b4^{8}) to reach near-optimal policy.

Improved sample complexity of (1/4^{6}) when decreasing exploration over time.

Abstract

Actor-critic style two-time-scale algorithms are one of the most popular methods in reinforcement learning, and have seen great empirical success. However, their performance is not completely understood theoretically. In this paper, we characterize the \emph{global} convergence of an online natural actor-critic algorithm in the tabular setting using a single trajectory of samples. Our analysis applies to very general settings, as we only assume ergodicity of the underlying Markov decision process. In order to ensure enough exploration, we employ an $\epsilon$-greedy sampling of the trajectory. For a fixed and small enough exploration parameter $\epsilon$, we show that the two-time-scale natural actor-critic algorithm has a rate of convergence of $\tilde{\mathcal{O}}(1/T^{1/4})$, where $T$ is the number of samples, and this leads to a sample complexity of $\Tilde{\mathcal{O}}(1/\delta^{8})$ samples to find a policy that is within an error of $\delta$ from the \emph{global optimum}. Moreover, by carefully decreasing the exploration parameter $\epsilon$ as the iterations proceed, we present an improved sample complexity of $\Tilde{\mathcal{O}}(1/\delta^{6})$ for convergence to the global optimum.