A Finite Sample Analysis of Distributional TD Learning with Linear Function Approximation

TL;DR

This paper provides a finite-sample analysis of distributional TD learning with linear function approximation, showing it has similar sample complexity to classic TD learning, thus efficiently estimating return distributions.

Contribution

It is the first to derive sharp finite-sample rates for distributional TD learning with linear function approximation, matching the complexity of traditional TD methods.

Findings

Sample complexity matches that of classic linear TD learning.

Distributional TD learning is statistically as efficient as learning the value function.

Provides theoretical insights into the statistical efficiency of distributional reinforcement learning.

Abstract

In this paper, we study the finite-sample statistical rates of distributional temporal difference (TD) learning with linear function approximation. The aim of distributional TD learning is to estimate the return distribution of a discounted Markov decision process for a given policy {\pi}. Previous works on statistical analysis of distributional TD learning mainly focus on the tabular case. In contrast, we first consider the linear function approximation setting and derive sharp finite-sample rates. Our theoretical results demonstrate that the sample complexity of linear distributional TD learning matches that of classic linear TD learning. This implies that, with linear function approximation, learning the full distribution of the return from streaming data is no more difficult than learning its expectation (value function). To derive tight sample complexity bounds, we conduct a fine-grained analysis of the linear-categorical Bellman equation and employ the exponential stability arguments for products of random matrices. Our results provide new insights into the statistical efficiency of distributional reinforcement learning algorithms.