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∝ 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 αd, and that when α∝ d−1 the regret scales as Ω(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.
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BibTeX
@misc{jaiswal2026posterior,
title = {{Posterior Tempering Explains Variance Inflation in Linear and Generalized Linear Thompson Sampling}},
author = {Prateek Jaiswal and Debdeep Pati and Anirban Bhattacharya and Bani K. Mallick},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01999},
eprint = {2609.01999},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01999},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Posterior Tempering Explains Variance Inflation in Linear and Generalized Linear Thompson Sampling
AU - Prateek Jaiswal
AU - Debdeep Pati
AU - Anirban Bhattacharya
AU - Bani K. Mallick
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.01999
UR - https://arxiv.org/abs/2609.01999
AB - 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∝ d^−1, our general regret bound yields the best known regret bound of O(d^3/2√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 αd, and that when α∝ d^−1 the regret scales as Ω(d^3/2√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.
ER -