Robust Losses for Learning Value Functions

TL;DR

This paper introduces robust Bellman error losses, like Huber and Absolute Bellman errors, reformulated as saddlepoint problems, leading to more stable and less sensitive reinforcement learning algorithms for value function estimation.

Contribution

It formalizes robust Bellman errors as saddlepoint problems and develops gradient-based algorithms that improve stability and robustness over traditional squared Bellman error methods.

Findings

Robust losses outperform mean squared errors in stability and sensitivity.

Proposed algorithms show improved performance in both prediction and control tasks.

Solutions of robust losses are better suited for outlier-prone environments.

Abstract

Most value function learning algorithms in reinforcement learning are based on the mean squared (projected) Bellman error. However, squared errors are known to be sensitive to outliers, both skewing the solution of the objective and resulting in high-magnitude and high-variance gradients. To control these high-magnitude updates, typical strategies in RL involve clipping gradients, clipping rewards, rescaling rewards, or clipping errors. While these strategies appear to be related to robust losses -- like the Huber loss -- they are built on semi-gradient update rules which do not minimize a known loss. In this work, we build on recent insights reformulating squared Bellman errors as a saddlepoint optimization problem and propose a saddlepoint reformulation for a Huber Bellman error and Absolute Bellman error. We start from a formalization of robust losses, then derive sound gradient-based approaches to minimize these losses in both the online off-policy prediction and control settings. We characterize the solutions of the robust losses, providing insight into the problem settings where the robust losses define notably better solutions than the mean squared Bellman error. Finally, we show that the resulting gradient-based algorithms are more stable, for both prediction and control, with less sensitivity to meta-parameters.