We study sampling as optimization in the space of measures. We focus on gradient flow-based optimization with the Langevin dynamics as a case study. We investigate the source of the bias of the unadjusted Langevin algorithm (ULA) in discrete time, and consider how to remove or reduce the bias. We point out the difficulty is that the heat flow is exactly solvable, but neither its forward nor backward method is implementable in general, except for Gaussian data. We propose the symmetrized Langevin algorithm (SLA), which should have a smaller bias than ULA, at the price of implementing a proximal gradient step in space. We show SLA is in fact consistent for Gaussian target measure, whereas ULA is not. We also illustrate various algorithms explicitly for Gaussian target measure, including gradient descent, proximal gradient, and Forward-Backward, and show they are all consistent.
University of California, San Diego（加利福尼亚大学圣地亚哥分校）；Google Research（谷歌研究院）；The Hong Kong University of Science and Technology（香港科技大学）
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 La…
Approximate Thompson sampling with Langevin Monte Carlo broadens its reach from Gaussian posterior sampling to encompass more general smooth posteriors. However, it still encounters scalability issues in high-dimensional problems when demanding high accuracy. To address this, we propose an approximate Thompson sampling strategy, utilizing underdamped Langevin Monte Carlo, where the latter is the go-to workhorse for simulations of high-dimensional posteriors. Based on the standard smoothness and log-concavity conditions, we study the accelerated posterior concentration and sampling using a spec…
We proposed a new technique to accelerate sampling methods for solving difficult optimization problems. Our method investigates the intrinsic connection between posterior distribution sampling and optimization with Langevin dynamics, and then we propose an interacting particle scheme that approximates a Reweighted Interacting Langevin Diffusion system (RILD). The underlying system is designed by adding a multiplicative source term into the classical Langevin operator, leading to a higher convergence rate and a more concentrated invariant measure. We analyze the convergence rate of our algorith…
UC Berkeley（加州大学伯克利分校）；International Computer Science Institute（国际计算机科学研究所）；Queensland University of Technology（昆士兰科技大学）；University of Queensland（昆士兰大学）
For sampling from a log-concave density, we study implicit integrators resulting from $θ$-method discretization of the overdamped Langevin diffusion stochastic differential equation. Theoretical and algorithmic properties of the resulting sampling methods for $ θ\in [0,1] $ and a range of step sizes are established. Our results generalize and extend prior works in several directions. In particular, for $θ\ge1/2$, we prove geometric ergodicity and stability of the resulting methods for all step sizes. We show that obtaining subsequent samples amounts to solving a strongly-convex optimization pr…
Langevin algorithms are gradient descent methods with additive noise. They have been used for decades in Markov chain Monte Carlo (MCMC) sampling, optimization, and learning. Their convergence properties for unconstrained non-convex optimization and learning problems have been studied widely in the last few years. Other work has examined projected Langevin algorithms for sampling from log-concave distributions restricted to convex compact sets. For learning and optimization, log-concave distributions correspond to convex losses. In this paper, we analyze the case of non-convex losses with comp…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We provide a new convergence analysis of stochastic gradient Langevin dynamics (SGLD) for sampling from a class of distributions that can be non-log-concave. At the core of our approach is a novel conductance analysis of SGLD using an auxiliary time-reversible Markov Chain. Under certain conditions on the target distribution, we prove that $\tilde O(d^4ε^{-2})$ stochastic gradient evaluations suffice to guarantee $ε$-sampling error in terms of the total variation distance, where $d$ is the problem dimension. This improves existing results on the convergence rate of SGLD (Raginsky et al., 2017;…
A fundamental problem in Bayesian inference and statistical machine learning is to efficiently sample from multimodal distributions. Due to metastability, multimodal distributions are difficult to sample using standard Markov chain Monte Carlo methods. We propose a new sampling algorithm based on a birth-death mechanism to accelerate the mixing of Langevin diffusion. Our algorithm is motivated by its mean field partial differential equation (PDE), which is a Fokker-Planck equation supplemented by a nonlocal birth-death term. This PDE can be viewed as a gradient flow of the Kullback-Leibler div…
We propose a new discretization of the mirror-Langevin diffusion and give a crisp proof of its convergence. Our analysis uses relative convexity/smoothness and self-concordance, ideas which originated in convex optimization, together with a new result in optimal transport that generalizes the displacement convexity of the entropy. Unlike prior works, our result both (1) requires much weaker assumptions on the mirror map and the target distribution, and (2) has vanishing bias as the step size tends to zero. In particular, for the task of sampling from a log-concave distribution supported on a c…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Microsoft Research（微软研究院）
Sampling with Markov chain Monte Carlo methods often amounts to discretizing some continuous-time dynamics with numerical integration. In this paper, we establish the convergence rate of sampling algorithms obtained by discretizing smooth Itô diffusions exhibiting fast Wasserstein-$2$ contraction, based on local deviation properties of the integration scheme. In particular, we study a sampling algorithm constructed by discretizing the overdamped Langevin diffusion with the method of stochastic Runge-Kutta. For strongly convex potentials that are smooth up to a certain order, its iterates conve…
Non-Reversible Langevin Algorithms for Constrained Sampling
University of California, Irvine（加利福尼亚大学欧文分校）；Florida State University（佛罗里达州立大学）；Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
We propose a new algorithm---Stochastic Proximal Langevin Algorithm (SPLA)---for sampling from a log concave distribution. Our method is a generalization of the Langevin algorithm to potentials expressed as the sum of one stochastic smooth term and multiple stochastic nonsmooth terms. In each iteration, our splitting technique only requires access to a stochastic gradient of the smooth term and a stochastic proximal operator for each of the nonsmooth terms. We establish nonasymptotic sublinear and linear convergence rates under convexity and strong convexity of the smooth term, respectively, e…
Microsoft Research（微软研究院）；Weizmann Institute（魏茨曼科学研究所）；Université Paris-Dauphine（巴黎第九大学）
We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected Stochastic Gradient Descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main result shows in particular that when the target distribution is uniform, LMC mixes in $\tilde{O}(n^7)$ steps (where $n$ is the dimension). We also provide preliminary experimental evidence that LMC performs at least as well as hit-and-run,…
Motivated by the problem of sampling from ill-conditioned log-concave distributions, we give a clean non-asymptotic convergence analysis of mirror-Langevin diffusions as introduced in Zhang et al. (2020). As a special case of this framework, we propose a class of diffusions called Newton-Langevin diffusions and prove that they converge to stationarity exponentially fast with a rate which not only is dimension-free, but also has no dependence on the target distribution. We give an application of this result to the problem of sampling from the uniform distribution on a convex body using a strate…
Controlled Langevin Dynamics for Sampling of Feedforward Neural Networks Trained with Minibatches
Department of Physics, Università degli Studi di Milano（米兰大学物理系）；Department of Computing Sciences（计算科学系）；Bocconi Institute for Data Science（博科尼数据科学研究所）；Analytics (BIDSA), Bocconi University（分析（BIDSA），博科尼大学）
Laboratory for Information Inference Systems (LIONS)（信息推理系统实验室（LIONS））；EPFL（洛桑联邦理工学院）
We consider the problem of sampling from constrained distributions, which has posed significant challenges to both non-asymptotic analysis and algorithmic design. We propose a unified framework, which is inspired by the classical mirror descent, to derive novel first-order sampling schemes. We prove that, for a general target distribution with strongly convex potential, our framework implies the existence of a first-order algorithm achieving $\tilde{O}(ε^{-2}d)$ convergence, suggesting that the state-of-the-art $\tilde{O}(ε^{-6}d^5)$ can be vastly improved. With the important Latent Dirichlet …
Discretized Langevin diffusions are efficient Monte Carlo methods for sampling from high dimensional target densities that are log-Lipschitz-smooth and (strongly) log-concave. In particular, the Euclidean Langevin Monte Carlo sampling algorithm has received much attention lately, leading to a detailed understanding of its non-asymptotic convergence properties and of the role that smoothness and log-concavity play in the convergence rate. Distributions that do not possess these regularity properties can be addressed by considering a Riemannian Langevin diffusion with a metric capturing the loca…
Massachusetts Institute of Technology（麻省理工学院）；Carnegie Mellon University（卡内基梅隆大学）
In this paper, we study the problem of sampling from distributions of the form p(x) \propto e^{-βf(x)} for some function f whose values and gradients we can query. This mode of access to f is natural in the scenarios in which such problems arise, for instance sampling from posteriors in parametric Bayesian models. Classical results show that a natural random walk, Langevin diffusion, mixes rapidly when f is convex. Unfortunately, even in simple examples, the applications listed above will entail working with functions f that are nonconvex -- for which sampling from p may in general require an …
The University of Edinburgh（爱丁堡大学）；The Alan Turing Institute（艾伦·图灵研究所）；National Technical University of Athens（雅典国立技术大学）
In this article we consider sampling from log concave distributions in Hamiltonian setting, without assuming that the objective gradient is globally Lipschitz. We propose two algorithms based on monotone polygonal (tamed) Euler schemes, to sample from a target measure, and provide non-asymptotic 2-Wasserstein distance bounds between the law of the process of each algorithm and the target measure. Finally, we apply these results to bound the excess risk optimization error of the associated optimization problem.
Sampling from a high-dimensional distribution is a fundamental task in statistics, engineering, and the sciences. A canonical approach is the Langevin Algorithm, i.e., the Markov chain for the discretized Langevin Diffusion. This is the sampling analog of Gradient Descent. Despite being studied for several decades in multiple communities, tight mixing bounds for this algorithm remain unresolved even in the seemingly simple setting of log-concave distributions over a bounded domain. This paper completely characterizes the mixing time of the Langevin Algorithm to its stationary distribution in t…
Athena/Archimedes Research Centre（雅典娜/阿基米德研究中心）；The University of Edinburgh（爱丁堡大学）；Univ. Grenoble Alpes（格勒诺布尔大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；Grenoble INP（格勒诺布尔工业大学）；LIG（格勒诺布尔信息与信号处理实验室）
Discretizations of Langevin diffusions provide a powerful method for sampling and Bayesian inference. However, such discretizations require evaluation of the gradient of the potential function. In several real-world scenarios, obtaining gradient evaluations might either be computationally expensive, or simply impossible. In this work, we propose and analyze stochastic zeroth-order sampling algorithms for discretizing overdamped and underdamped Langevin diffusions. Our approach is based on estimating the gradients, based on Gaussian Stein's identities, widely used in the stochastic optimization…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
Federated sampling algorithms have recently gained great popularity in the community of machine learning and statistics. This paper studies variants of such algorithms called Error Feedback Langevin algorithms (ELF). In particular, we analyze the combinations of EF21 and EF21-P with the federated Langevin Monte-Carlo. We propose three algorithms: P-ELF, D-ELF, and B-ELF that use, respectively, primal, dual, and bidirectional compressors. We analyze the proposed methods under Log-Sobolev inequality and provide non-asymptotic convergence guarantees.
Operator-Level Quantum Acceleration of Non-Logconcave Sampling
Simons Institute for the Theory of Computing, University of California, Berkeley, USA（Simons理论计算研究所，加州大学伯克利分校）；Department of Mathematics, University of California, Berkeley, USA（加州大学伯克利分校数学系）；Department of Mathematics, University of Michigan, Ann Arbor, USA（密歇根大学安娜堡分校数学系）；Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, USA（伯克利国家实验室应用数学与计算研究部）
University of Washington（华盛顿大学）；Stanford University（斯坦福大学）；Microsoft Research（微软研究院）
We consider sampling from composite densities on $\mathbb{R}^d$ of the form $dπ(x) \propto \exp(-f(x) - g(x))dx$ for well-conditioned $f$ and convex (but possibly non-smooth) $g$, a family generalizing restrictions to a convex set, through the abstraction of a restricted Gaussian oracle. For $f$ with condition number $κ$, our algorithm runs in $O \left(κ^2 d \log^2\tfrac{κd}ε\right)$ iterations, each querying a gradient of $f$ and a restricted Gaussian oracle, to achieve total variation distance $ε$. The restricted Gaussian oracle, which draws samples from a distribution whose negative log-lik…
