The Uncertainty Bellman Equation and Exploration

TL;DR

This paper introduces an uncertainty Bellman equation (UBE) that propagates uncertainty in reinforcement learning, providing a scalable exploration method that improves performance on Atari games.

Contribution

It formulates a novel uncertainty Bellman equation whose fixed point bounds posterior variance, enabling scalable and effective exploration in complex RL environments.

Findings

UBE yields a tighter variance bound than count-based bonuses.

Replacing epsilon-greedy with UBE improves DQN performance on Atari.

The method scales well to large, complex systems.

Abstract

We consider the exploration/exploitation problem in reinforcement learning. For exploitation, it is well known that the Bellman equation connects the value at any time-step to the expected value at subsequent time-steps. In this paper we consider a similar \textit{uncertainty} Bellman equation (UBE), which connects the uncertainty at any time-step to the expected uncertainties at subsequent time-steps, thereby extending the potential exploratory benefit of a policy beyond individual time-steps. We prove that the unique fixed point of the UBE yields an upper bound on the variance of the posterior distribution of the Q-values induced by any policy. This bound can be much tighter than traditional count-based bonuses that compound standard deviation rather than variance. Importantly, and unlike several existing approaches to optimism, this method scales naturally to large systems with complex generalization. Substituting our UBE-exploration strategy for $\epsilon$-greedy improves DQN performance on 51 out of 57 games in the Atari suite.