We address the problem of optimizing a Brownian motion. We consider a (random) realization $W$ of a Brownian motion with input space in $[0,1]$. Given $W$, our goal is to return an $\epsilon$-approximation of its maximum using the smallest possible number of function evaluations, the sample complexity of the algorithm. We provide an algorithm with sample complexity of order $\log^2(1/\epsilon)$. This improves over previous results of Al-Mharmah and Calvin (1996) and Calvin et al. (2017) which provided only polynomial rates. Our algorithm is adaptive---each query depends on previous values---and is an instance of the optimism-in-the-face-of-uncertainty principle.
翻译:我们处理优化布朗运动的问题。我们考虑的是(随机)实现布朗运动的W美元,投入空间为$[10,1,1美元]。考虑到W$,我们的目标是用尽可能少的功能评估数量,即算法的样本复杂性,归还其最大金额的美元-欧元-美元。我们提供了一种具有序列($\log>2(1/\epsilon)样本复杂性的算法。这比以前只提供多元利率的Al-Mharmah和Calvin(1996年)和Calvin等人(2017年)的结果有所改进。我们的算法是适应性的,教程查询取决于以前的价值观,并且是乐观主义的不确定原则的例子。