We study the problem of global maximization of a function f given a finite number of evaluations perturbed by noise. We consider a very weak assumption on the function, namely that it is locally smooth (in some precise sense) with respect to some semi-metric, around one of its global maxima. Compared to previous works on bandits in general spaces (Kleinberg et al., 2008; Bubeck et al., 2011a) our algorithm does not require the knowledge of this semi-metric. Our algorithm, StoSOO, follows an optimistic strategy to iteratively construct upper confidence bounds over the hierarchical partitions of the function domain to decide which point to sample next. A finite-time analysis of StoSOO shows that it performs almost as well as the best specifically-tuned algorithms even though the local smoothness of the function is not known.

翻译：我们研究在给定有限个受噪声干扰的评估点的情况下，实现函数f全局最大化的问题。我们假设函数具有非常弱的条件，即在其某个全局最大值附近，函数相对于某个半度量是局部光滑的（在精确意义下）。与以往在一般空间中的多臂老虎机研究工作（Kleinberg 等，2008; Bubeck 等，2011a）相比，我们的算法无需知晓该半度量。所提出的算法 StoSOO 采用乐观策略，通过迭代构建函数定义域层次划分上的上置信界，以决定下一个采样点。对 StoSOO 的有限时间分析表明，即使函数的局部光滑性未知，其表现几乎与专门调优的最佳算法相当。