Simple Bayesian Algorithms for Best Arm Identification

This paper considers the optimal adaptive allocation of measurement effort for identifying the best among a finite set of options or designs. An experimenter sequentially chooses designs to measure and observes noisy signals of their quality with the goal of confidently identifying the best design after a small number of measurements. I propose three simple Bayesian algorithms for adaptively allocating measurement effort. One is Top-Two Probability sampling, which computes the two designs with the highest posterior probability of being optimal, and then randomizes to select among these two. One is a variant a top-two sampling which considers not only the probability a design is optimal, but the expected amount by which its quality exceeds that of other designs. The final algorithm is a modified version of Thompson sampling that is tailored for identifying the best design. I prove that these simple algorithms satisfy a strong optimality property. In a frequestist setting where the true quality of the designs is fixed, one hopes the posterior definitively identifies the optimal design, in the sense that that the posterior probability assigned to the event that some other design is optimal converges to zero as measurements are collected. I show that under the proposed algorithms this convergence occurs at an exponential rate, and the corresponding exponent is the best possible among all allocation rules.