The Uncertainty Bellman Equation and Exploration
This addresses the exploration challenge in reinforcement learning for agents in complex environments, offering a scalable method with significant performance gains, though it builds on existing Bellman equation concepts.
The paper tackles the exploration/exploitation problem in reinforcement learning by introducing an uncertainty Bellman equation (UBE) that extends exploratory benefits across time-steps, proving it provides a tighter upper bound on Q-value variance than traditional methods. Substituting UBE-exploration for ε-greedy improved DQN performance on 51 out of 57 Atari games.
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 $ε$-greedy improves DQN performance on 51 out of 57 games in the Atari suite.