Thompson Sampling for 1-Dimensional Exponential Family Bandits
This work addresses a theoretical gap in bandit algorithms for researchers, though it is incremental as it builds on prior work.
The authors tackled the limited theoretical guarantees for Thompson Sampling beyond Bernoulli bandits by proving its asymptotic optimality for 1-dimensional exponential family bandits using the Jeffreys prior, with a finite-time exponential concentration inequality for posterior distributions.
Thompson Sampling has been demonstrated in many complex bandit models, however the theoretical guarantees available for the parametric multi-armed bandit are still limited to the Bernoulli case. Here we extend them by proving asymptotic optimality of the algorithm using the Jeffreys prior for 1-dimensional exponential family bandits. Our proof builds on previous work, but also makes extensive use of closed forms for Kullback-Leibler divergence and Fisher information (and thus Jeffreys prior) available in an exponential family. This allow us to give a finite time exponential concentration inequality for posterior distributions on exponential families that may be of interest in its own right. Moreover our analysis covers some distributions for which no optimistic algorithm has yet been proposed, including heavy-tailed exponential families.