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

TL;DR

This paper proves the convergence in total variation distance of Langevin-Simulated Annealing algorithms with multiplicative noise, showing faster convergence due to adaptive noise, and extends previous work from Wasserstein to total variation metrics.

Contribution

It establishes the total variation convergence of Langevin-Simulated Annealing algorithms with multiplicative noise, improving upon prior Wasserstein convergence results.

Findings

Convergence in total variation distance is achieved.

Adaptive multiplicative noise accelerates convergence.

Regularization lemmas are developed for the analysis.

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 $dY_t = - \sigma \sigma^\top \nabla V(Y_t) dt + a(t)\sigma(Y_t)dW_t + a(t)^2\Upsilon(Y_t)dt$, 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 $\Upsilon$ is a correction term. Allowing $\sigma$ to depend on the position brings faster convergence in comparison with the classical Langevin equation $dY_t = -\nabla V(Y_t)dt + \sigma dW_t$. 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.