Prior Diffusiveness and Regret in the Linear-Gaussian Bandit
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917184461078528 |
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| author | Zhu, Yifan Duchi, John C. Van Roy, Benjamin |
| author_facet | Zhu, Yifan Duchi, John C. Van Roy, Benjamin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02022 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Prior Diffusiveness and Regret in the Linear-Gaussian Bandit Zhu, Yifan Duchi, John C. Van Roy, Benjamin Machine Learning 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. |
| title | Prior Diffusiveness and Regret in the Linear-Gaussian Bandit |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2601.02022 |