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

arXiv: https://arxiv.org/abs/2609.01999 · HTML: https://mlsanity.com/paper/2609.01999/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 29 references that are in the map.
- First posted on arXiv 2 Sep 2026.

## Builds on (most cited first, 7 of 7)

- [Convergence Rates of Variational Posterior Distributions](https://mlsanity.com/paper/1712.02519/) (2017)
- [Action Centered Contextual Bandits](https://mlsanity.com/paper/1711.03596/) (2017)
- [Thompson Sampling with a Mixture Prior](https://mlsanity.com/paper/2106.05608/) (2021)
- [Thompson Sampling for High-Dimensional Sparse Linear Contextual Bandits](https://mlsanity.com/paper/2211.05964/) (2022)
- [On Frequentist Regret of Linear Thompson Sampling](https://mlsanity.com/paper/2006.06790/) (2020)
- [Double Doubly Robust Thompson Sampling for Generalized Linear Contextual Bandits](https://mlsanity.com/paper/2209.06983/) (2022)
- [Geometry-Aware Approaches for Balancing Performance and Theoretical Guarantees in Linear Bandits](https://mlsanity.com/paper/2306.14872/) (2023)

## Abstract (arXiv)

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.