Variational Bayesian Optimistic Sampling

TL;DR

This paper introduces a Bayesian optimistic sampling approach for online decision problems, providing a unified analysis of regret bounds and extending to complex settings like saddle-point problems, with a flexible, variational framework.

Contribution

It develops a new class of Bayesian optimistic policies, including a variational method that works with any posterior, and extends regret analysis to bilinear saddle-point problems.

Findings

Optimistic policies achieve $ ilde O( oot{A}{T})$ Bayesian regret.

Thompson sampling may suffer linear regret outside the optimistic set.

The variational approach allows flexible policy tuning and constraint incorporation.

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.