Convergence of the kinetic annealing for general potentials

TL;DR

This paper proves the convergence of kinetic Langevin simulated annealing under slow cooling schedules for general potentials, and shows non-convergence under faster schedules, using advanced hypocoercive estimates.

Contribution

It extends convergence analysis to non-elliptic, non-reversible kinetic settings with general potentials, adapting localization strategies and Sobolev hypocoercive estimates.

Findings

Convergence under slow logarithmic cooling schedules.

Non-convergence under fast cooling schedules.

Application of hypocoercive estimates to kinetic annealing.

Abstract

The convergence of the kinetic Langevin simulated annealing is proven under mild assumptions on the potential $U$ for slow logarithmic cooling schedules. Moreover, non-convergence for fast logarithmic and non-logarithmic cooling schedules is established. The results are based on an adaptation to non-elliptic non-reversible kinetic settings of a localization/local convergence strategy developed by Fournier and Tardif in the overdamped elliptic case, and on precise quantitative high order Sobolev hypocoercive estimates.