Information-Geometric Optimization Algorithms: A Unifying Picture via Invariance Principles

Ludovic Arnold (1, 2) , Anne Auger (2) , Nikolaus Hansen (1, 2, 3) , Yann Ollivier (1, 2)

1 LRI - Laboratoire de Recherche en Informatique
2 TAO - Machine Learning and Optimisation
3 MSR - INRIA - Microsoft Research - Inria Joint Centre

We present the information-geometric optimization (IGO) method, which turns any smooth parametric family of probability distributions on an arbitrary search space $X$ into a continuous-time black-box optimization method on $X$. Invariance as a design principle keeps the number of arbitrary choices to a minimum. IGO conducts a natural gradient ascent using an adaptive, time-dependent transformation of the objective function. The cross-entropy method is recovered in a particular case with a large time step. From specific families of distributions on discrete or continuous spaces, IGO naturally recovers versions of known algorithms: CMA-ES for Gaussian distributions, and PBIL for Bernoulli distributions. IGO is invariant under reparametrization of the search space $X$, under a change of parameters of the probability distribution, and under increasing transformation of the function to be optimized. Theoretical considerations suggest that IGO achives minimal diversity loss through optimization. First experiments using restricted Boltzmann machines show that IGO may be able to spontaneously perform multimodal optimization.

Mots clés: Optimization; Fisher information; Natural gradient; CMA-ES; Boltzmann machines

Domaines: Optimisation et contrôle [math.OC]; Recherche opérationnelle [math.OC]

Ludovic Arnold, Anne Auger, Nikolaus Hansen, Yann Ollivier. Information-Geometric Optimization Algorithms: A Unifying Picture via Invariance Principles. 2011. ⟨hal-00601503v1⟩