Optimistic optimization of a Brownian

TL;DR

This paper presents an adaptive algorithm for optimizing a Brownian motion, achieving a logarithmic squared sample complexity for approximating its maximum, improving upon previous polynomial-rate methods.

Contribution

It introduces a novel adaptive optimization algorithm based on the optimism-in-the-face-of-uncertainty principle with significantly improved sample complexity.

Findings

Sample complexity of order log^2(1/ε)

Outperforms previous polynomial-rate algorithms

Uses an adaptive, instance-specific approach

Abstract

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.