Objective Improvement in Information-Geometric Optimization

TL;DR

This paper provides a theoretical foundation for the IGO framework, proving monotone improvement guarantees and extending the analysis to block updates and fitness-proportional selection.

Contribution

It offers a formal proof that IGO guarantees monotone improvement with step sizes up to 1, and extends the analysis to block parameter updates and fitness-proportional selection.

Findings

Monotone improvement guaranteed for step sizes ≤ 1.

Range of step sizes is independent of the objective function.

Expected fitness improves with fitness-proportional selection.

Abstract

Information-Geometric Optimization (IGO) is a unified framework of stochastic algorithms for optimization problems. Given a family of probability distributions, IGO turns the original optimization problem into a new maximization problem on the parameter space of the probability distributions. IGO updates the parameter of the probability distribution along the natural gradient, taken with respect to the Fisher metric on the parameter manifold, aiming at maximizing an adaptive transform of the objective function. IGO recovers several known algorithms as particular instances: for the family of Bernoulli distributions IGO recovers PBIL, for the family of Gaussian distributions the pure rank-mu CMA-ES update is recovered, and for exponential families in expectation parametrization the cross-entropy/ML method is recovered. This article provides a theoretical justification for the IGO framework, by proving that any step size not greater than 1 guarantees monotone improvement over the course of optimization, in terms of q-quantile values of the objective function f. The range of admissible step sizes is independent of f and its domain. We extend the result to cover the case of different step sizes for blocks of the parameters in the IGO algorithm. Moreover, we prove that expected fitness improves over time when fitness-proportional selection is applied, in which case the RPP algorithm is recovered.