Non-Asymptotic Analysis for Two Time-scale TDC with General Smooth Function Approximation

TL;DR

This paper provides the first finite-sample analysis of two time-scale TDC with general smooth function approximation, establishing convergence rates and broad applicability in reinforcement learning.

Contribution

It introduces novel techniques to explicitly analyze finite-sample errors for two time-scale TDC with general smooth functions in off-policy settings.

Findings

Converges at a rate of O(1/√T) up to a log factor.

Applicable to a wide range of value-based RL algorithms.

Addresses challenges in non-linear, two-time-scale, and non-convex settings.

Abstract

Temporal-difference learning with gradient correction (TDC) is a two time-scale algorithm for policy evaluation in reinforcement learning. This algorithm was initially proposed with linear function approximation, and was later extended to the one with general smooth function approximation. The asymptotic convergence for the on-policy setting with general smooth function approximation was established in [bhatnagar2009convergent], however, the finite-sample analysis remains unsolved due to challenges in the non-linear and two-time-scale update structure, non-convex objective function and the time-varying projection onto a tangent plane. In this paper, we develop novel techniques to explicitly characterize the finite-sample error bound for the general off-policy setting with i.i.d.\ or Markovian samples, and show that it converges as fast as $\mathcal O(1/\sqrt T)$ (up to a factor of $\mathcal O(\log T)$). Our approach can be applied to a wide range of value-based reinforcement learning algorithms with general smooth function approximation.