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

• URL: http://arxiv.org/abs/2609.01999v1
• Date: Wed, 02 Sep 2026 02:17:05 GMT
• Title: Posterior Tempering Explains Variance Inflation in Linear and Generalized Linear Thompson Sampling
• Abstract summary: We study a variant of the Thompson Sampling (TS) algorithm, called $$-TS, for solving generalized linear bandit problems.<n>We formalize the idea of variance inflation by introducing $$-TS that uses a fractional or $$-posterior instead of the standard posterior.<n>Our main contribution is to identify general regularity conditions on the prior and reward distributions that enable a regret analysis of $$-TS without assuming any tractable approximation of the posterior distribution.
• Score: 6.1259398119902
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $α\propto d^{-1}$, 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 $α\propto d^{-1}$ the regret scales as $Ω(d^{3/2}\sqrt{T})$, explaining the origin of the $d^{3/2}$ 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.

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