Global Optimality and Finite Sample Analysis of Softmax Off-Policy Actor Critic under State Distribution Mismatch

TL;DR

This paper proves the global optimality and convergence rate of a softmax off-policy actor-critic algorithm in tabular settings without density ratio correction, using stochastic, approximate updates aligned with practical methods.

Contribution

It introduces a finite sample analysis for an off-policy actor-critic method that does not rely on density ratio correction, extending theoretical understanding to more practical scenarios.

Findings

Proves global optimality of the algorithm.

Establishes convergence rate under finite samples.

Removes restrictive assumptions from previous analyses.

Abstract

In this paper, we establish the global optimality and convergence rate of an off-policy actor critic algorithm in the tabular setting without using density ratio to correct the discrepancy between the state distribution of the behavior policy and that of the target policy. Our work goes beyond existing works on the optimality of policy gradient methods in that existing works use the exact policy gradient for updating the policy parameters while we use an approximate and stochastic update step. Our update step is not a gradient update because we do not use a density ratio to correct the state distribution, which aligns well with what practitioners do. Our update is approximate because we use a learned critic instead of the true value function. Our update is stochastic because at each step the update is done for only the current state action pair. Moreover, we remove several restrictive assumptions from existing works in our analysis. Central to our work is the finite sample analysis of a generic stochastic approximation algorithm with time-inhomogeneous update operators on time-inhomogeneous Markov chains, based on its uniform contraction properties.