Convergence of Langevin-Simulated Annealing algorithms with multiplicative noise II: Total Variation

Abstract

We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for V:Rd→R a potential function to minimize, we consider the stochastic differential equation dYt=−σσ⊤∇V(Yt)dt+a(t)σ(Yt)dWt+a(t)2Υ(Yt)dt, where (Wt) is a Brownian motion, where σ:Rd→Md(R) is an adaptive (multiplicative) noise, where a:R+→R+ is a function decreasing to 0 and where Υ is a correction term. Allowing σ to depend on the position brings faster convergence in comparison with the classical Langevin equation dYt=−∇V(Yt)dt+σdWt. In a previous paper we established the convergence in L1-Wasserstein distance of Yt and of its associated Euler scheme Y¯t to argmin(V) with the classical schedule a(t)=Alog−1/2(t). In the present paper we prove the convergence in total variation distance. The total variation case appears more demanding to deal with and requires regularization lemmas.

Pierre Bras

PhD Student in Applied Mathematics

I am a PhD student under the direction of Gilles Pagès, interested in Machine Learning, Stochastic Optimization and Numerical Probability.