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STOCHASTIC PROXIMAL LANGEVIN ALGORITHM
Adil Salim Joint work with Dmitry Kovalev and Peter Richtárik
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STOCHASTIC PROXIMAL LANGEVIN ALGORITHM Adil Salim Joint work with - - PowerPoint PPT Presentation
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 Richtrik 1 SAMPLING PROBLEM (d x ) exp( U ( x ))d x , U : d convex where . 2 LANGEVIN MONTE CARLO (LMC) W k Assume
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Typical non asymptotic result: . KL(μk|μ⋆) = 𝒫(1/
k)
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LANGEVIN MONTE CARLO (LMC)
Assume smooth, i.i.d standard gaussian and ,
U Wk γ > 0
xk+1 = xk − γ∇U(xk) + 2γWk+1 .
Gaussian noise Gradient descent
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FIRST INTUITION FOR LMC
dXt = − ∇U(Xt)dt + 2dWt .
LMC can be seen as a Euler discretization of the Langevin equation: Non asymptotic results using this intuition in [Dalalyan 2017], [Durmus Moulines 2017].
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LMC can be seen as an (inexact) Gradient Descent for:
μ⋆ = argmin ∫ Udμ(x) + ∫ μ(x)log(μ(x))dx μ⋆ = argmin KL(μ|μ⋆) .
SECOND INTUITION FOR LMC
Non asymptotic results using this intuition (+ extensions of LMC beyond GD) in [Durmus et al. 2018], [Wibisono 2018], [Bernton 2018].
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CONTRIBUTION: STOCHASTIC PROXIMAL LANGEVIN
xk+1 = proxγg(⋅,ξk+1)(xk) +
2γWk+1 .
Stochastic Prox
Case 1: U(x) = Eξ(g(x, ξ))
Nonsmooth
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Case 2:
U(x) = Eξ(f(x, ξ)) + ∑ Eξ(gi(x, ξ))
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CONTRIBUTION: STOCHASTIC PROXIMAL LANGEVIN
i
Smooth Nonsmooth
See our Poster #161.
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STOCHASTIC SUBGRADIENT VS STOCHASTIC PROX
Stochastic subgradients [Durmus et al. 2018] Sampling .
μ⋆(dx) ∝ exp( − |x|)dx
Stochastic proximal [Us]
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