On Regret Bounds of Thompson Sampling for Bayesian Optimization
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917328084533248 |
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| author | Takeno, Shion Iwazaki, Shogo |
| author_facet | Takeno, Shion Iwazaki, Shogo |
| contents | We study a widely used Bayesian optimization method, Gaussian process Thompson sampling (GP-TS), under the assumption that the objective function is a sample path from a GP. Compared with the GP upper confidence bound (GP-UCB) with established high-probability and expected regret bounds, most analyses of GP-TS have been limited to expected regret. Moreover, whether the recent analyses of GP-UCB for the lenient regret and the improved cumulative regret upper bound can be applied to GP-TS remains unclear. To fill these gaps, this paper shows several regret bounds: (i) a regret lower bound for GP-TS, which implies that GP-TS suffers from a polynomial dependence on $1/δ$ with probability $δ$, (ii) an upper bound of the second moment of cumulative regret, which directly suggests an improved regret upper bound on $δ$, (iii) expected lenient regret upper bounds, and (iv) an improved cumulative regret upper bound on the time horizon $T$. Along the way, we provide several useful lemmas, including a relaxation of the necessary condition from recent analysis to obtain improved regret upper bounds on $T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_09276 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Regret Bounds of Thompson Sampling for Bayesian Optimization Takeno, Shion Iwazaki, Shogo Machine Learning We study a widely used Bayesian optimization method, Gaussian process Thompson sampling (GP-TS), under the assumption that the objective function is a sample path from a GP. Compared with the GP upper confidence bound (GP-UCB) with established high-probability and expected regret bounds, most analyses of GP-TS have been limited to expected regret. Moreover, whether the recent analyses of GP-UCB for the lenient regret and the improved cumulative regret upper bound can be applied to GP-TS remains unclear. To fill these gaps, this paper shows several regret bounds: (i) a regret lower bound for GP-TS, which implies that GP-TS suffers from a polynomial dependence on $1/δ$ with probability $δ$, (ii) an upper bound of the second moment of cumulative regret, which directly suggests an improved regret upper bound on $δ$, (iii) expected lenient regret upper bounds, and (iv) an improved cumulative regret upper bound on the time horizon $T$. Along the way, we provide several useful lemmas, including a relaxation of the necessary condition from recent analysis to obtain improved regret upper bounds on $T$. |
| title | On Regret Bounds of Thompson Sampling for Bayesian Optimization |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2603.09276 |