Tight Regret Bounds for Bayesian Optimization in One Dimension
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866913823234981888 |
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| author | Scarlett, Jonathan |
| author_facet | Scarlett, Jonathan |
| contents | We consider the problem of Bayesian optimization (BO) in one dimension, under a Gaussian process prior and Gaussian sampling noise. We provide a theoretical analysis showing that, under fairly mild technical assumptions on the kernel, the best possible cumulative regret up to time $T$ behaves as $Ω(\sqrt{T})$ and $O(\sqrt{T\log T})$. This gives a tight characterization up to a $\sqrt{\log T}$ factor, and includes the first non-trivial lower bound for noisy BO. Our assumptions are satisfied, for example, by the squared exponential and Matérn-$ν$ kernels, with the latter requiring $ν> 2$. Our results certify the near-optimality of existing bounds (Srinivas {\em et al.}, 2009) for the SE kernel, while proving them to be strictly suboptimal for the Matérn kernel with $ν> 2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1805_11792 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Tight Regret Bounds for Bayesian Optimization in One Dimension Scarlett, Jonathan Machine Learning Information Theory Optimization and Control We consider the problem of Bayesian optimization (BO) in one dimension, under a Gaussian process prior and Gaussian sampling noise. We provide a theoretical analysis showing that, under fairly mild technical assumptions on the kernel, the best possible cumulative regret up to time $T$ behaves as $Ω(\sqrt{T})$ and $O(\sqrt{T\log T})$. This gives a tight characterization up to a $\sqrt{\log T}$ factor, and includes the first non-trivial lower bound for noisy BO. Our assumptions are satisfied, for example, by the squared exponential and Matérn-$ν$ kernels, with the latter requiring $ν> 2$. Our results certify the near-optimality of existing bounds (Srinivas {\em et al.}, 2009) for the SE kernel, while proving them to be strictly suboptimal for the Matérn kernel with $ν> 2$. |
| title | Tight Regret Bounds for Bayesian Optimization in One Dimension |
| topic | Machine Learning Information Theory Optimization and Control |
| url | https://arxiv.org/abs/1805.11792 |