title: Variational Bayesian Optimistic Sampling
abstract: We consider online sequential decision problems where an agent must balance  exploration and exploitation. We derive a set of Bayesian `optimistic' policies  which, in the stochastic multi-armed bandit case, includes the Thompson sampling  policy. We provide a new analysis showing that any algorithm producing policies in the optimistic set enjoys $\tilde O(\sqrt{AT})$ Bayesian regret for a problem with $A$ actions after $T$ rounds. We extend the regret analysis for optimistic policies to bilinear saddle-point problems which include zero-sum matrix games and constrained bandits as special cases. In this case we show that Thompson sampling can produce policies outside of the optimistic set and suffer linear regret in some instances. Finding a policy inside the optimistic set amounts to solving a convex optimization problem and we call the resulting algorithm `variational Bayesian optimistic sampling' (VBOS). The procedure works for any posteriors, \ie, it does not require the posterior to have any special properties, such as log-concavity, unimodality, or smoothness. The variational view of the problem has many useful properties, including the ability to tune the exploration-exploitation tradeoff, add regularization, incorporate constraints, and linearly parameterize the policy.
authors: Brendan O'Donoghue Tor Lattimore
hIndex: Tor Lattimore 31