Should one compute the Temporal Difference fix point or minimize the Bellman Residual? The unified oblique projection view

TL;DR

This paper compares the effectiveness of Temporal Difference fix-point computation and Bellman Residual minimization in evaluating value functions in Markov Decision Processes, proposing a unified oblique projection framework.

Contribution

It introduces a unified oblique projection perspective that simplifies and extends the analysis of TD(0) and BR methods, highlighting their differences and performance guarantees.

Findings

BR has a performance guarantee, TD(0) does not

TD(0) often yields slightly better solutions but can be numerically unstable

Simulations show BR is more reliable overall

Abstract

We investigate projection methods, for evaluating a linear approximation of the value function of a policy in a Markov Decision Process context. We consider two popular approaches, the one-step Temporal Difference fix-point computation (TD(0)) and the Bellman Residual (BR) minimization. We describe examples, where each method outperforms the other. We highlight a simple relation between the objective function they minimize, and show that while BR enjoys a performance guarantee, TD(0) does not in general. We then propose a unified view in terms of oblique projections of the Bellman equation, which substantially simplifies and extends the characterization of (schoknecht,2002) and the recent analysis of (Yu & Bertsekas, 2008). Eventually, we describe some simulations that suggest that if the TD(0) solution is usually slightly better than the BR solution, its inherent numerical instability makes it very bad in some cases, and thus worse on average.