Bivariate Estimation-of-Distribution Algorithms Can Find an Exponential Number of Optima

TL;DR

This paper demonstrates that bivariate estimation-of-distribution algorithms can efficiently find exponentially many optima in multimodal landscapes, outperforming univariate models, with implications for optimization and sampling strategies.

Contribution

It introduces the EBOM test function and proves that bivariate EDAs can effectively model and sample from exponentially large optima sets, unlike univariate models.

Findings

Bivariate EDAs quickly approximate an ideal model for EBOM.

Univariate models cannot efficiently sample all optima, especially with inverse-polynomial probabilities.

Bivariate models achieve uniform sampling of exponentially many optima.

Abstract

Finding a large set of optima in a multimodal optimization landscape is a challenging task. Classical population-based evolutionary algorithms typically converge only to a single solution. While this can be counteracted by applying niching strategies, the number of optima is nonetheless trivially bounded by the population size. Estimation-of-distribution algorithms (EDAs) are an alternative, maintaining a probabilistic model of the solution space instead of a population. Such a model is able to implicitly represent a solution set far larger than any realistic population size. To support the study of how optimization algorithms handle large sets of optima, we propose the test function EqualBlocksOneMax (EBOM). It has an easy fitness landscape with exponentially many optima. We show that the bivariate EDA mutual-information-maximizing input clustering, without any problem-specific modification, quickly generates a model that behaves very similarly to a theoretically ideal model for EBOM, which samples each of the exponentially many optima with the same maximal probability. We also prove via mathematical means that no univariate model can come close to having this property: If the probability to sample an optimum is at least inverse-polynomial, there is a Hamming ball of logarithmic radius such that, with high probability, each sample is in this ball.