Bridging the Gap Between Average and Discounted TD Learning

TL;DR

This paper introduces a new algorithm for policy evaluation in average-reward TD learning that guarantees convergence and improves sample complexity, bridging the gap with discounted learning methods.

Contribution

A novel average-reward TD learning algorithm that guarantees convergence and reduces sample complexity from quartic to quadratic, applicable to both linear and tabular cases.

Findings

Guarantees convergence to the solution of a projected Bellman equation.

Achieves quadratic sample complexity, matching discounted setting efficiency.

Applicable to both linear function approximation and tabular settings.

Abstract

The analysis of Temporal Difference (TD) learning in the average-reward setting faces notable theoretical difficulties because the Bellman operator is not contractive with respect to any norm. This complicates standard analyses of stochastic updates that are effective in discounted settings. Although a considerable body of literature addresses these challenges, existing theoretical approaches come with limitations. We introduce a novel algorithm designed explicitly for policy evaluation in the average-reward setting, utilizing sampling from two Markovian trajectories. Our proposed method overcomes previous limitations by guaranteeing convergence to the unique solution of a properly defined projected Bellman equation. Notably, and in contrast to earlier work, our convergence analysis is uniformly applicable to both linear function approximation and tabular settings and does not involve explicit dimension-dependent terms in its convergence bounds. These results align with what is known to hold in the discounted setting. Furthermore, our algorithm achieves improved dependence on the problem's condition number, reducing the sample complexity from quartic, as in prior literature, to quadratic scaling, and thus matching the efficiency seen in the discounted setting.