Stochastic simultaneous optimistic optimization

TL;DR

This paper introduces StoSOO, a new algorithm for global maximization of noisy functions that adapts to local smoothness without prior knowledge, achieving near-optimal performance.

Contribution

The paper presents StoSOO, an optimistic hierarchical partitioning algorithm that does not require knowledge of the function's semi-metric, unlike previous methods.

Findings

StoSOO performs nearly as well as tuned algorithms under unknown local smoothness.

Finite-time analysis shows StoSOO's effectiveness in noisy global maximization.

The algorithm adapts to local smoothness without prior semi-metric knowledge.

Abstract

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.