Sampling from Non-smooth Distributions Through Langevin Diffusion

In this paper, we propose proximal splitting-type algorithms for sampling from distributions whose densities are not necessarily smooth nor log-concave. Our approach brings together tools from, on the one hand, variational analysis and non-smooth optimization, and on the other hand, stochastic diffu...

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Bibliographic Details
Published in:Methodology and computing in applied probability Vol. 23; no. 4; pp. 1173 - 1201
Main Authors: Luu, Tung Duy, Fadili, Jalal, Chesneau, Christophe
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2021
Springer Nature B.V
Springer Verlag
Subjects:
Algorithms
Business and Management
Computer Science
Diffusion
Economics
Electrical Engineering
Fines & penalties
Life Sciences
Machine Learning
Mathematics
Mathematics and Statistics
Numerical Analysis
Optimization
Optimization and Control
Sampling
Signal and Image Processing
Statistics
Statistics Theory
65C30
Langevin diffusion
Exponentially weighted aggregation
Proximal splitting
Monte-Carlo
65C05
Non-smooth distributions
65C60
65C50
ISSN:1387-5841, 1573-7713
Online Access:Get full text
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Summary:In this paper, we propose proximal splitting-type algorithms for sampling from distributions whose densities are not necessarily smooth nor log-concave. Our approach brings together tools from, on the one hand, variational analysis and non-smooth optimization, and on the other hand, stochastic diffusion equations, and in particular the Langevin diffusion. We establish in particular consistency guarantees of our algorithms seen as discretization schemes in this context. These algorithms are then applied to compute the exponentially weighted aggregates for regression problems involving non-smooth penalties that are commonly used to promote some notion of simplicity/complexity. Some popular penalties are detailed and implemented on some numerical experiments.
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ISSN:1387-5841
1573-7713
DOI:10.1007/s11009-020-09809-7