Tight Regret Bounds for Bayesian Optimization in One Dimension

• URL: http://arxiv.org/abs/1805.11792v3
• Date: Wed, 07 May 2025 04:10:27 GMT
• Title: Tight Regret Bounds for Bayesian Optimization in One Dimension
• Authors: Jonathan Scarlett,
• Abstract summary: We consider the problem of Bayesian optimization (BO) in one dimension, under a Gaussian process prior and Gaussian sampling noise.<n>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 $Omega(sqrtT)$ and $O(sqrtTlog T)$.
• Score: 47.51554144092745
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\Omega(\sqrt{T})$ 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 Mat\'ern-$\nu$ kernels, with the latter requiring $\nu > 2$. 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 Mat\'ern kernel with $\nu > 2$.

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