  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.
