Prior Diffusiveness and Regret in the Linear-Gaussian Bandit

• URL: http://arxiv.org/abs/2601.02022v1
• Date: Mon, 05 Jan 2026 11:30:08 GMT
• Title: Prior Diffusiveness and Regret in the Linear-Gaussian Bandit
• Authors: Yifan Zhu, John C. Duchi, Benjamin Van Roy,
• Abstract summary: We prove that Thompson sampling exhibits $tildeO(d sqrtT + d r sqrtmathrmTr(_0))$ Bayesian regret in the linear-Gaussian bandit.<n>We establish these results via a new elliptical potential'' lemma, and also provide a lower bound indicating that the burn-in term is unavoidable.
• Score: 25.66105507730649
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We prove that Thompson sampling exhibits $\tilde{O}(σd \sqrt{T} + d r \sqrt{\mathrm{Tr}(Σ_0)})$ Bayesian regret in the linear-Gaussian bandit with a $\mathcal{N}(μ_0, Σ_0)$ prior distribution on the coefficients, where $d$ is the dimension, $T$ is the time horizon, $r$ is the maximum $\ell_2$ norm of the actions, and $σ^2$ is the noise variance. In contrast to existing regret bounds, this shows that to within logarithmic factors, the prior-dependent ``burn-in'' term $d r \sqrt{\mathrm{Tr}(Σ_0)}$ decouples additively from the minimax (long run) regret $σd \sqrt{T}$. Previous regret bounds exhibit a multiplicative dependence on these terms. We establish these results via a new ``elliptical potential'' lemma, and also provide a lower bound indicating that the burn-in term is unavoidable.

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