Improving the Expected Improvement Algorithm

TL;DR

This paper analyzes the limitations of the standard expected improvement algorithm in Bayesian optimization and introduces a simple modification that achieves asymptotic optimality in best-arm identification tasks.

Contribution

The paper provides a rigorous analysis of EI's shortcomings and proposes a straightforward modification that makes it asymptotically optimal for Gaussian best-arm identification.

Findings

Standard EI is far from optimal in finite-grid Bayesian optimization.

A simple modification to EI significantly improves its asymptotic performance.

The modified algorithm outperforms standard EI by an order of magnitude.

Abstract

The expected improvement (EI) algorithm is a popular strategy for information collection in optimization under uncertainty. The algorithm is widely known to be too greedy, but nevertheless enjoys wide use due to its simplicity and ability to handle uncertainty and noise in a coherent decision theoretic framework. To provide rigorous insight into EI, we study its properties in a simple setting of Bayesian optimization where the domain consists of a finite grid of points. This is the so-called best-arm identification problem, where the goal is to allocate measurement effort wisely to confidently identify the best arm using a small number of measurements. In this framework, one can show formally that EI is far from optimal. To overcome this shortcoming, we introduce a simple modification of the expected improvement algorithm. Surprisingly, this simple change results in an algorithm that is asymptotically optimal for Gaussian best-arm identification problems, and provably outperforms standard EI by an order of magnitude.