First-Order Regret Analysis of Thompson Sampling
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We address online combinatorial optimization when the player has a prior over the adversary's sequence of losses. In this framework, Russo and Van Roy proposed an information-theoretic analysis of Thompson Sampling based on the information ratio, resulting in optimal worst-case regret bounds. In this paper we introduce three novel ideas to this line of work. First we propose a new quantity, the scale-sensitive information ratio, which allows us to obtain more refined first-order regret bounds (i.e., bounds of the form √(L^*) where L^* is the loss of the best combinatorial action). Second we replace the entropy over combinatorial actions by a coordinate entropy, which allows us to obtain the first optimal worst-case bound for Thompson Sampling in the combinatorial setting. Finally, we introduce a novel link between Bayesian agents and frequentist confidence intervals. Combining these ideas we show that the classical multi-armed bandit first-order regret bound Õ(√(d L^*)) still holds true in the more challenging and more general semi-bandit scenario. This latter result improves the previous state of the art bound Õ(√((d+m^3)L^*)) by Lykouris, Sridharan and Tardos.
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