An Improved Regret Bound for Thompson Sampling in the Gaussian Linear Bandit Setting

Thompson sampling has been of significant recent interest due to its wide range of applicability to online learning problems and its good empirical and theoretical performance. In this paper, we analyze the performance of Thompson sampling in the canonical Gaussian linear bandit setting. We prove that the Bayesian regret of Thompson sampling in this setting is bounded by$O(\sqrt {T\log (T)} )$ improving on an earlier bound of $O(\sqrt T \log (T))$ n the literature for the case of the infinite, and compact action set. Our proof relies on a Cauchy–Schwarz type inequality which can be of interest in its own right.