Expected Improvement versus Predicted Value in Surrogate-Based Optimization

Abstract

Surrogate-based optimization relies on so-called infill criteria (acquisition functions) to decide which point to evaluate next. When Kriging is used as the surrogate model of choice (also called Bayesian optimization), one of the most frequently chosen criteria is expected improvement. We argue that the popularity of expected improvement largely relies on its theoretical properties rather than empirically validated performance. Few results from the literature show evidence, that under certain conditions, expected improvement may perform worse than something as simple as the predicted value of the surrogate model. We benchmark both infill criteria in an extensive empirical study on the 'BBOB' function set. This investigation includes a detailed study of the impact of problem dimensionality on algorithm performance. The results support the hypothesis that exploration loses importance with increasing problem dimensionality. A statistical analysis reveals that the purely exploitative search with the predicted value criterion performs better on most problems of five or higher dimensions. Possible reasons for these results are discussed. In addition, we give an in-depth guide for choosing the infill criteria based on prior knowledge about the problem at hand, its dimensionality, and the available budget.

$ε$-shotgun: $ε$-greedy Batch Bayesian Optimisation

“…ϵ = 0) is competitive with the best-performing methods. This result was recently confirmed by Rehbach et al [33], who empirically show that solely using the surrogate model's predicted value performs better than EI on most problems with a dimensionality of 5 or more.…”

“…Motivated by the success of the sequential exploitative and ϵ-greedy approaches [10,33], we invert the local penalisation strategy and, instead, present a method that samples from within the region that would usually be penalised (6). We empirically show that this approach out-performs recent BBO methods on a range of synthetic functions and two real-world problems.…”

“…The selection of a good set of locations to evaluate at each batch iteration is a non-trivial problem. In sequential BO, techniques which favour greedy exploitation of the surrogate model have been shown to be preferable to the more traditional acquisition functions [10,33]. De Ath et al [10], for example, show that using an variety of synthetic and real-world problems.…”

Continuous surrogate-based optimization algorithms are well-suited for expensive discrete problems

“…It typically uses a Gaussian process as a surrogate to approximate the expensive objective. Several acquisition functions exist to guide the search, such as Expected Improvement, Upper Confidence Bound, or Thompson sampling [22], information-theoretic approaches such as Predictive Entropy Search [11], or simply the surrogate itself [5,19]. Though Gaussian processes are typically used on continuous problems, they can be adapted for problems with discrete variables as well.…”

“…On top of that, it has been shown that a large dimensionality reduces the importance of choosing a complicated acquisition function [19], which helps us doing a fair comparison between surrogates.…”