Risk and optimal policies in bandit experiments

This paper provides a decision theoretic analysis of bandit experiments. The bandit setting corresponds to a dynamic programming problem, but solving this directly is typically infeasible. Working within the framework of diffusion asymptotics, we define a suitable notion of asymptotic Bayes risk for bandit settings. For normally distributed rewards, the minimal Bayes risk can be characterized as the solution to a nonlinear second-order partial differential equation (PDE). Using a limit of experiments approach, we show that this PDE characterization also holds asymptotically under both parametric and non-parametric distribution of the rewards. The approach further describes the state variables it is asymptotically sufficient to restrict attention to, and therefore suggests a practical strategy for dimension reduction. The upshot is that we can approximate the dynamic programming problem defining the bandit setting with a PDE which can be efficiently solved using sparse matrix routines. We derive near-optimal policies from the numerical solutions to these equations. The proposed policies substantially dominate existing methods such Thompson sampling. The framework also allows for substantial generalizations to the bandit problem such as time discounting and pure exploration motives.