An Information-Theoretic Analysis for Thompson Sampling with Many Actions
This work addresses a theoretical bottleneck in bandit algorithms for researchers, offering incremental improvements in regret analysis.
The paper tackles the problem of Thompson sampling's regret bounds scaling poorly with many actions by introducing new information-theoretic bounds based on rate-distortion, leading to near-optimal bounds for linear bandits and a dramatic improvement for logistic bandits, though the latter requires computational quantification.
Information-theoretic Bayesian regret bounds of Russo and Van Roy capture the dependence of regret on prior uncertainty. However, this dependence is through entropy, which can become arbitrarily large as the number of actions increases. We establish new bounds that depend instead on a notion of rate-distortion. Among other things, this allows us to recover through information-theoretic arguments a near-optimal bound for the linear bandit. We also offer a bound for the logistic bandit that dramatically improves on the best previously available, though this bound depends on an information-theoretic statistic that we have only been able to quantify via computation.