Variational Bayesian Optimistic Sampling
This provides a general framework for improving regret bounds in decision problems, though it is incremental as it builds on existing Bayesian methods like Thompson sampling.
The paper tackles the problem of balancing exploration and exploitation in online sequential decision-making by introducing a set of Bayesian optimistic policies, showing that algorithms in this set achieve $ ilde O(\sqrt{AT})$ Bayesian regret for multi-armed bandits and extending this to bilinear saddle-point problems like zero-sum games.
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.