On the Statistical Benefits of Temporal Difference Learning

TL;DR

This paper provides an asymptotic theory demonstrating the statistical advantages of temporal difference learning over direct estimation in Markov chains, highlighting potential error reductions and improved difference estimates.

Contribution

It introduces a theoretical framework quantifying when and how TD learning reduces estimation error compared to direct methods, based on problem structure.

Findings

Inverse trajectory pooling coefficient characterizes error reduction.

Error bounds depend on trajectory crossing time, often much smaller than horizon.

Potential for significant improvements in value difference estimates.

Abstract

Given a dataset on actions and resulting long-term rewards, a direct estimation approach fits value functions that minimize prediction error on the training data. Temporal difference learning (TD) methods instead fit value functions by minimizing the degree of temporal inconsistency between estimates made at successive time-steps. Focusing on finite state Markov chains, we provide a crisp asymptotic theory of the statistical advantages of this approach. First, we show that an intuitive inverse trajectory pooling coefficient completely characterizes the percent reduction in mean-squared error of value estimates. Depending on problem structure, the reduction could be enormous or nonexistent. Next, we prove that there can be dramatic improvements in estimates of the difference in value-to-go for two states: TD's errors are bounded in terms of a novel measure - the problem's trajectory crossing time - which can be much smaller than the problem's time horizon.