Convergence of Langevin-simulated annealing algorithms with multiplicative noise

We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for V : R d →<!-- → --> R V : R^d → R a potential function to minimize, we consider the stochastic differential equation d Y t = −<!-- − --> σ<!-- σ --> σ<!-- σ --> ⊤<!-- ⊤ --> ∇<!-- ∇ --> V ( Y t ) dY_t = - σ σ ^ ∇ V(Y_t) d t + a ( t ) σ<!-- σ --> ( Y t ) d W t + a ( t ) 2 Υ<!-- Υ --> ( Y t ) d t dt + a(t)σ (Y_t)dW_t + a(t)^2Υ (Y_t)dt , where ( W t ) (W_t) is a Brownian motion, where σ<!-- σ --> : R d →<!-- → --> M d ( R ) σ : R^d → M_d(R) is an adaptive (multiplicative) noise, where a : R + →<!-- → --> R + a : R^+ → R^+ is a function decreasing to 0 0 and where Υ<!-- Υ --> Υ is a correction term. This setting can be applied to optimization problems arising in Machine Learning; allowing σ<!-- σ --> 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<!-- σ --> d W t dY_t = -∇ V(Y_t)dt + σ dW_t . The case where σ<!-- σ --> σ is a constant matrix has been extensively studied; however little attention has been paid to the general case. We prove the convergence for the L 1 L^1 -Wasserstein distance of Y t Y_t and of the associated Euler scheme Y ¯<!-- ¯ --> t Ȳ_t to some measure ν<!-- ν --> ⋆<!-- ⋆ --> ν ^ which is supported by argmin ⁡<!-- ⁡ --> ( V ) argmin(V) and give rates of convergence to the instantaneous Gibbs measure ν<!-- ν --> a ( t ) ν ₐ₍ₜ₎ of density ∝<!-- ∝ --> exp ⁡<!-- ⁡ --> ( −<!-- − --> 2 V ( x ) / a ( t ) 2 ) ∝ exp (-2V(x)/a(t)^2) . To do so, we first consider the case where a a is a piecewise constant function. We find again the classical schedule a ( t ) = A log −<!-- − --> 1 / 2 ⁡<!-- ⁡ --> ( t ) a(t) = Alog -1/2(t) . We then prove the convergence for the general case by giving bounds for the Wasserstein distance to the stepwise constant case using ergodicity properties.