Tight Regret Bounds for Noisy Optimization of a Brownian Motion

• URL: http://arxiv.org/abs/2001.09327v2
• Date: Sat, 15 Jan 2022 12:02:33 GMT
• Title: Tight Regret Bounds for Noisy Optimization of a Brownian Motion
• Authors: Zexin Wang, Vincent Y. F. Tan, Jonathan Scarlett
• Abstract summary: We consider the problem of Bayesian optimization of a one-dimensional Brownian motion in which the $T$ adaptively chosen observations are corrupted by Gaussian noise. We show that as the smallest possible expected cumulative regret and the smallest possible expected simple regret scale as $Omega(sigmasqrtT cdot log T)$.
• Score: 118.65407541895851
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the problem of Bayesian optimization of a one-dimensional Brownian motion in which the $T$ adaptively chosen observations are corrupted by Gaussian noise. We show that as the smallest possible expected cumulative regret and the smallest possible expected simple regret scale as $\Omega(\sigma\sqrt{T / \log (T)}) \cap \mathcal{O}(\sigma\sqrt{T} \cdot \log T)$ and $\Omega(\sigma / \sqrt{T \log (T)}) \cap \mathcal{O}(\sigma\log T / \sqrt{T})$ respectively, where $\sigma^2$ is the noise variance. Thus, our upper and lower bounds are tight up to a factor of $\mathcal{O}( (\log T)^{1.5} )$. The upper bound uses an algorithm based on confidence bounds and the Markov property of Brownian motion (among other useful properties), and the lower bound is based on a reduction to binary hypothesis testing.

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