Risk and optimal policies in bandit experiments
This work addresses the problem of improving decision-making efficiency in bandit experiments for researchers and practitioners, offering a novel theoretical framework with practical applications.
The paper tackles the problem of optimizing policies in bandit experiments by analyzing asymptotic Bayes and minimax risk using diffusion processes and PDEs, resulting in optimal policies that substantially dominate existing methods like Thompson sampling, with the latter's risk often being twice as high.
We provide a decision theoretic analysis of bandit experiments under local asymptotics. Working within the framework of diffusion processes, we define suitable notions of asymptotic Bayes and minimax risk for these experiments. For normally distributed rewards, the minimal Bayes risk can be characterized as the solution to a 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 distributions of the rewards. The approach further describes the state variables it is asymptotically sufficient to restrict attention to, and thereby suggests a practical strategy for dimension reduction. The PDEs characterizing minimal Bayes risk can be solved efficiently using sparse matrix routines or Monte-Carlo methods. We derive the optimal Bayes and minimax policies from their numerical solutions. These optimal policies substantially dominate existing methods such as Thompson sampling; the risk of the latter is often twice as high.