STOCHASTIC PROXIMAL LANGEVIN ALGORITHM Adil Salim Joint work with - PowerPoint PPT Presentation

STOCHASTIC PROXIMAL LANGEVIN ALGORITHM Adil Salim Joint work with Dmitry Kovalev and Peter Richtárik 1

SAMPLING PROBLEM μ ⋆ (d x ) ∝ exp( − U ( x ))d x , U : ℝ d → ℝ convex where . 2

LANGEVIN MONTE CARLO (LMC) W k Assume smooth, i.i.d standard gaussian and , U γ > 0 x k +1 = x k − γ ∇ U ( x k ) + 2 γ W k +1 . Gradient descent Gaussian noise KL ( μ k | μ ⋆ ) = 𝒫 (1/ Typical non asymptotic result: . k ) 3

FIRST INTUITION FOR LMC LMC can be seen as a Euler discretization of the Langevin equation: d X t = − ∇ U ( X t )d t + 2d W t . Non asymptotic results using this intuition in [Dalalyan 2017], [Durmus Moulines 2017]. 4

SECOND INTUITION FOR LMC LMC can be seen as an (inexact) Gradient Descent for: μ ⋆ = argmin ∫ U d μ ( x ) + ∫ μ ( x )log( μ ( x )) d x μ ⋆ = argmin KL ( μ | μ ⋆ ) . Non asymptotic results using this intuition (+ extensions of LMC beyond GD) in [Durmus et al. 2018], [Wibisono 2018], [Bernton 2018]. 5

CONTRIBUTION: STOCHASTIC PROXIMAL LANGEVIN Case 1: U ( x ) = E ξ ( g ( x , ξ )) Nonsmooth x k +1 = prox γ g ( ⋅ , ξ k +1 ) ( x k ) + 2 γ W k +1 . Stochastic Prox 6

CONTRIBUTION: STOCHASTIC PROXIMAL LANGEVIN E ξ ( f ( x , ξ )) + ∑ E ξ ( g i ( x , ξ )) Case 2: U ( x ) = i Smooth Nonsmooth See our Poster #161. 7

STOCHASTIC SUBGRADIENT VS STOCHASTIC PROX μ ⋆ ( d x ) ∝ exp( − | x | ) d x Sampling . Stochastic proximal Stochastic subgradients [Us] [Durmus et al. 2018] 8

Thanks for your attention. See us at poster #161. 9