Binary Interaction Methods for High Dimensional Global Optimization and Machine Learning

Applied Mathematics & Optimization, vol. 86

Abstract

Abstract In this work we introduce a new class of gradient-free global optimization methods based on a binary interaction dynamics governed by a Boltzmann type equation. In each interaction the particles act taking into account both the best microscopic binary position and the best macroscopic collective position. For the resulting kinetic optimization methods, convergence to the global minimizer is guaranteed for a large class of functions under appropriate parameter constraints that do not depend on the dimension of the problem. In the mean-field limit we show that the resulting Fokker-Planck partial differential equations generalize the current class of consensus based optimization (CBO) methods. Algorithmic implementations inspired by the well-known direct simulation Monte Carlo methods in kinetic theory are derived and discussed. Several examples on prototype test functions for global optimization are reported including an application to machine learning.

Authors 3

1. Alessandro Benfenati

   University of Milan

   Affiliation as printed

   Department of Environmental Science and Policy, University of Milan, Milan, Italy

2. Giacomo Borghi Aachen Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics, RWTH Aachen University, Aachen, Germany

3. Lorenzo Pareschi corresponding

   University of Ferrara

   Affiliation as printed

   Department of Mathematics and Computer Science, University of Ferrara, Ferrara, Italy