Papers › Convergence of Langevin-Simulated Annealing algorithms with multiplicative noise
Convergence of Langevin-Simulated Annealing algorithms with multiplicative noise
Pierre Bras, Gilles Pagès
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We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for V : ℝᵈ →ℝ a potential function to minimize, we consider the stochastic equation dYₜ = - σσ^⊤ ∇V(Yₜ) dt + a(t)σ(Yₜ)dWₜ + a(t)²Υ(Yₜ)dt, where (Wₜ) is a Brownian motion, where σ: ℝᵈ →ℳ_d(ℝ) is an adaptive (multiplicative) noise, where a : ℝ^+ →ℝ^+ is a function decreasing to 0 and where Υ is a correction term. This setting can be applied to optimization problems arising in Machine Learning. The case where 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 L¹-Wasserstein distance of Yₜ and of the associated Euler-scheme Y̅ₜ to some measure ν^⋆ which is supported by argmin(V) and give rates of convergence to the instantaneous Gibbs measure νₐ₍ₜ₎ of density ∝exp(-2V(x)/a(t)²). To do so, we first consider the case where a is a piecewise constant function. We find again the classical schedule 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.
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