We derive the first finite-time logarithmic Bayes regret upper bounds for Bayesian bandits. In a multi-armed bandit, we obtain O(c logn)and O(ch log2 n)upper bounds for an upper confidence bound algorithm, where ch and c are constants depending on the prior distribution and the gaps of bandit instances sampled from it, respectively. The latter bound asymptotically matches the lower bound of Lai (1987). Our proofs are a major technical departure from prior works, while being simple and general. To show the generality of our techniques, we apply them to linear bandits. Our results provide insights on the value of prior in the Bayesian setting, both in the objective and as a side information given to the learner. They significantly improve upon existing O( n)bounds, which have become standard in the literature despite the logarithmic lower bound of Lai (1987).

Peter L learning UAI, pages Dimitri Dynamic, Belmont, Ronen I optimal Journal Aditya processes.

Therefore, we help address the open problem on agnosticQ-learning proposed in [Wen and Van Roy,2013].

Itisoften the responsibility of the agent designer toconstruct thistargetwhich,inrichandcomplexenvironments,constitutesaonerousburden; without full knowledge of the environment itself, a designer may forge a suboptimal learning target that poorly balances the amount ofinformation an agent must acquire to identify the target against the target's associated performance shortfall.