Posterior Tempering Explains Variance Inflation in Linear and Generalized Linear Thompson Sampling

Abstract: We study a variant of the Thompson Sampling (TS) algorithm, called $α$-TS, for solving stochastic generalized linear bandit problems. Existing analyses of TS require inflating the posterior variance to derive near-optimal regret guarantees. We formalize the idea of variance inflation by introducing $α$-TS that uses a fractional or $α$-posterior instead of the standard posterior. Our main contribution is to identify general regularity conditions on the prior and reward distributions that enable a regret analysis of $α$-TS without assuming any tractable approximation of the posterior distribution, unlike previous works. For a specific choice of $α\propto d<sup>{-1}$, our general regret bound yields the best known regret bound of $O(d^{{3/2}\sqrt{T}\log} T)$ for both the exponential and sub-Gaussian families of reward distributions. We further provide an $α$-dependent lower bound showing that the regret constant depends on the product $αd$, and that when $α\propto d<sup>{-1}$ the regret scales as $Ω(d<sup>{3/2}\sqrt{T})$, explaining the origin of the $d<sup>{3/2}$ factor in the upper bound. Our proof technique adapts and combines recent advancements in the analysis of linear bandit problems with first- and second-order posterior concentration theory from the Bayesian statistics literature.