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 $V:{\mathbb{R}}^d\to \mathbb{R}$ a potential function to minimize, we consider the stochastic differential equation $d{Y}_t=-\sigma {\sigma }^{\top }\nabla V(Y_t)dt+a(t)\sigma (Y_t)d{W}_t+a(t{)}^2{\rm Y}(Y_t)dt$, where $(W_t)$ is a Brownian motion, where $\sigma :{\mathbb{R}}^d\to {\mathcal{M}}_d(\mathbb{R})$ is an adaptive (multiplicative) noise, where $a:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a function decreasing to $0$ and where ${\rm Y}$ is a correction term. Allowing $\sigma$ to depend on the position brings faster convergence in comparison with the classical Langevin equation $d{Y}_t=-\nabla V(Y_t)dt+\sigma d{W}_t$. In a previous paper we established the convergence in $L^1$-Wasserstein distance of $Y_t$ and of its associated Euler scheme ${\bar{Y}}_t$ to $\text{argmin}(V)$ with the classical schedule $a(t)=A{\log }^{-1/2}(t)$. 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.

18 pages. arXiv admin note: text overlap with arXiv:2109.11669