Convergence of Langevin-simulated annealing algorithms with multiplicative noise II: Total variation

Abstract

Abstract We study the convergence of Langevin-simulated annealing type algorithms with multiplicative noise, i.e. for V : R d → R V\colon\mathbb{R}^{d}\to\mathbb{R} a potential function to minimize, we consider the stochastic differential equation d ⁢ Y t = − σ ⁢ σ ⊤ ⁢ ∇ V ⁢ ( Y t ) ⁢ d ⁢ t + a ⁢ ( t ) ⁢ σ ⁢ ( Y t ) ⁢ d ⁢ W t + a ⁢ ( t ) 2 ⁢ Υ ⁢ ( Y t ) ⁢ d ⁢ t 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 ) (W_{t}) is a Brownian motion, σ : R d → M d ⁢ ( R ) \sigma\colon\mathbb{R}^{d}\to\mathcal{M}_{d}(\mathbb{R}) is an adaptive (multiplicative) noise, a : R + → R + a\colon\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 d ⁢ Y t = − ∇ V ⁢ ( Y t ) ⁢ d ⁢ t + σ ⁢ d ⁢ W t dY_{t}=-\nabla V(Y_{t})\,dt+\sigma\,dW_{t} . In a previous paper, we established the convergence in L 1 L^{1} -Wasserstein distance of Y t Y_{t} and of its associated Euler scheme Y ¯ t \bar{Y}_{t} to argmin ⁡ ( V ) \operatorname{argmin}(V) with the classical schedule a ⁢ ( t ) = A ⁢ log − 1 / 2 ⁡ ( t ) 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.