Monotone Improvement of Information-Geometric Optimization Algorithms with a Surrogate Function

TL;DR

This paper provides a theoretical analysis of how surrogate functions can ensure monotonic improvement in information-geometric optimization algorithms, specifically demonstrating conditions under which the expected objective decreases monotonically.

Contribution

It introduces a theoretical framework showing that surrogate functions maintaining high rank correlation lead to monotonic optimization progress, with analysis for Gaussian distributions in CMA-ES.

Findings

Surrogate functions with high Kendall's rank correlation ensure monotonic decrease in expected objective.

The acceptable correlation threshold for monotonic improvement is characterized.

Alternative correlation measures like Pearson can also be used to maintain surrogate quality.

Abstract

A surrogate function is often employed to reduce the number of objective function evaluations for optimization. However, the effect of using a surrogate model in evolutionary approaches has not been theoretically investigated. This paper theoretically analyzes the information-geometric optimization framework using a surrogate function. The value of the expected objective function under the candidate sampling distribution is used as the measure of progress of the algorithm. We assume that the surrogate function is maintained so that the population version of the Kendall's rank correlation coefficient between the surrogate function and the objective function under the candidate sampling distribution is greater than or equal to a predefined threshold. We prove that information-geometric optimization using such a surrogate function leads to a monotonic decrease in the expected objective function value if the threshold is sufficiently close to one. The acceptable threshold value is analyzed for the case of the information-geometric optimization instantiated with Gaussian distributions, i.e., the rank-$\mu$ update CMA-ES, on a convex quadratic objective function. As an alternative to the Kendall's rank correlation coefficient, we investigate the use of the Pearson correlation coefficient between the weights assigned to candidate solutions based on the objective function and the surrogate function.