Optimization by linear kinetic equations and mean-field Langevin dynamics

Probably one of the most striking examples of the close connections betweenglobal optimization processes and statistical physics is the simulatedannealing method, inspired by the famous Monte Carlo algorithm devised byMetropolis et al. in the middle of the last century. In this paper we show howthe tools of linear kinetic theory allow to describe this gradient-freealgorithm from the perspective of statistical physics and how convergence tothe global minimum can be related to classical entropy inequalities. Thisanalysis highlight the strong link between linear Boltzmann equations andstochastic optimization methods governed by Markov processes. Thanks to thisformalism we can establish the connections between the simulated annealingprocess and the corresponding mean-field Langevin dynamics characterized by astochastic gradient descent approach. Generalizations to other selectionstrategies in simulated annealing that avoid the acceptance-rejection dynamicare also provided.