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.
翻译:我们考虑了巴伊西亚优化一维的布朗运动的问题,在这种运动中,适应性选择的美元观测被高森噪音腐蚀。我们显示,作为最小可能的预期累积遗憾和最小的预期简单遗憾规模,分别是$(Omega)(gma\ sqrt{T/\log(T)})\ cap\mathcal{O}(gmath\sqrt{T}\cdort\log T)和$\Omega(sigma/ sqrt{T\log(T)})\ cap\mathcal{O}(sigmag\log T/\\sqrt{T}),其中$\ sgmag_2$是噪音差异。因此,我们的上下界和下界紧贴紧到$\mathcal{O}((log T)\\ 1.5}。上界使用基于信任框的算法,而Markov 的属性以布朗运动的下层假设为下层。