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

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 $dY_t = - σσ^\top \nabla V(Y_t) dt + a(t)σ(Y_t)dW_t + a(t)^2Υ(Y_t)dt$, where $(W_t)$ is a Brownian motion, where $σ: \mathbb{R}^d \to \mathcal{M}_d(\mathbb{R})$ is an adaptive (multiplicative) noise, where $a : \mathbb{R}^+ \to \mathbb{R}^+$ is a function decreasing to $0$ and where $Υ$ is a correction term. Allowing $σ$ to depend on the position brings faster convergence in comparison with the classical Langevin equation $dY_t = -\nabla V(Y_t)dt + σdW_t$. 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.

No reviews yet.

No comments yet.