An Efficient Sampling Algorithm for Non-smooth Composite Potentials

Mou, Wenlong; Flammarion, Nicolas; Wainwright, Martin J.

doi:10.48550/arxiv.1910.00551

Cited by 7 publications

(12 citation statements)

References 59 publications

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“…(in Theorem 3.4), which improves the dependence on accuracy ǫ. In Mou et al (2019), the Metropolis-adjusted Langevin algorithm is levaraged with a proximal sampling oracle to remove the polynomial dependence on the accuracy ǫ (in total variation distance) and achieve a Ω d log( 1 ǫ ) convergence time for a related composite posterior distribution. Unfortunately, an additional dimension dependent factor is always introduced into the overall convergence rate.…”

Section: Related Workmentioning

confidence: 99%

When is the Convergence Time of Langevin Algorithms Dimension Independent? A Composite Optimization Viewpoint

Freund,

Ma,

Zhang

2021

Preprint

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There has been a surge of works bridging MCMC sampling and optimization, with a specific focus on translating non-asymptotic convergence guarantees for optimization problems into the analysis of Langevin algorithms in MCMC sampling. A conspicuous distinction between the convergence analysis of Langevin sampling and that of optimization is that all known convergence rates for Langevin algorithms depend on the dimensionality of the problem, whereas the convergence rates for optimization are dimension-free for convex problems. Whether a dimension independent convergence rate can be achieved by Langevin algorithm is thus a long-standing open problem. This paper provides an affirmative answer to this problem for large classes of either Lipschitz or smooth convex problems with normal priors. By viewing Langevin algorithm as composite optimization, we develop a new analysis technique that leads to dimension independent convergence rates for such problems.(2004), as well as other Bayesian classification problems Sollich (2002) with Gaussian or Bayesian elastic net priors. The second case corresponds to the regression type problems, where the entire posterior is strongly log-concave and log-Lipschitz smooth. In this case, one can separate the negative log-posterior into two parts:, which is convex and β −1 L-Lipschitz smooth. We therefore directly let g(w) = m 2 w 2 in Section 6.

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Section: Related Workmentioning

confidence: 99%

When is the Convergence Time of Langevin Algorithms Dimension Independent? A Composite Optimization Viewpoint

Freund,

Ma,

Zhang

2021

Preprint

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“…RGO is a key algorithmic ingredient used in [17] together with the alternating sampling framework to improve the iteration-complexity bounds for various sampling algorithms. Examples of a convex function g that admits an computationally efficient RGO have been presented in [26,32], including coordinate-separable regularizers, ℓ 1 -norm, and group Lasso.…”

Section: Problem Formulation and Alternating Sampling Frameworkmentioning

confidence: 99%

“…Over the last few years, several new algorithms and theoretical results in sampling with nonsmooth potentials have been established. In [26], sampling for non-smooth composite potentials is considered. The algorithm needs the proximal sampling oracle that samples from the target potential regularized by a large isotropic quadratic term as well as computes the corresponding partition function.…”

Section: Introductionmentioning

confidence: 99%

“…In [9], the authors presented an optimization approach to analyze the complexity of sampling and established a complexity result for sampling with non-smooth composite potentials. Compared with [26,32], our algorithm does not require any sampling oracle. Compared with [19], we consider sampling from a distribution supported on R d instead of a convex compact set.…”

Section: Introductionmentioning

confidence: 99%

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A Proximal Algorithm for Sampling from Non-smooth Potentials

Liang

Chen

2021

Preprint

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Markov chain Monte Carlo (MCMC) is an effective and dominant method to sample from high-dimensional complex distributions. Yet, most existing MCMC methods are only applicable to settings with smooth potentials (log-densities). In this work, we examine sampling problems with non-smooth potentials. We propose a novel MCMC algorithm for sampling from non-smooth potentials. We provide a non-asymptotical analysis of our algorithm and establish a polynomial-time complexity Õ(dε −1 ) to obtain ε total variation distance to the target density, better than all existing results under the same assumptions. Our method is based on the proximal bundle method and an alternating sampling framework. This framework requires the so-called restricted Gaussian oracle, which can be viewed as a sampling counterpart of the proximal mapping in convex optimization. One key contribution of this work is a fast algorithm that realizes the restricted Gaussian oracle for any convex non-smooth potential with bounded Lipschitz constant.

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“…It is possible to remove the bias by applying the Metropolis filter (accept-reject step) in each iteration; this has a geometric interpretation as projection in total variation (TV) distance [5]. With the Metropolis filter, it is possible to prove the algorithm still converges exponentially fast in discrete time, and obtain an iteration complexity of O(log 1 δ ) to reach error δ in TV distance with warm start and under various conditions such as strong logconcavity, isoperimetry, or distant dissipativity [7,21,37,41]. However, if we want convergence in KL divergence-which is stronger-then Metropolis filter does not work because it makes the distributions singular (have point masses).…”

Section: Introductionmentioning

confidence: 99%

Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetry

Wibisono¹

2019

Preprint

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We study the Proximal Langevin Algorithm (PLA) for sampling from a probability distribution ν = e −f on R n under isoperimetry. We prove a convergence guarantee for PLA in Kullback-Leibler (KL) divergence when ν satisfies log-Sobolev inequality (LSI) and f has bounded second and third derivatives. This improves on the result for the Unadjusted Langevin Algorithm (ULA), and matches the fastest known rate for sampling under LSI (without Metropolis filter) with a better dependence on the LSI constant. We also prove convergence guarantees for PLA in Rényi divergence of order q > 1 when the biased limit satisfies either LSI or Poincaré inequality.

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An Efficient Sampling Algorithm for Non-smooth Composite Potentials

Cited by 7 publications

References 59 publications

When is the Convergence Time of Langevin Algorithms Dimension Independent? A Composite Optimization Viewpoint

When is the Convergence Time of Langevin Algorithms Dimension Independent? A Composite Optimization Viewpoint

A Proximal Algorithm for Sampling from Non-smooth Potentials

Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetry

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