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

Pierre Bras, Gilles Pagès

September 2021

### 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.

Publication

In *arXiv e-prints*

###### PhD Student in Applied Mathematics

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