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

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-1, our general regret bound yields the best known regret bound of O(d3/2√Tlog 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 α d, and that when α propto d-1 the regret scales as Omega(d3/2√T), explaining the origin of the d3/2 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.