In this paper, we focus on non-asymptotic bounds related to the Euler scheme of an ergodic diffusion with a possibly multiplicative diffusion term (non-constant diffusion coefficient). More precisely, the objective of this paper is to control the distance of the standard Euler scheme with decreasing step ({usually called Unadjusted Langevin Algorithm in the Monte Carlo literature}) to the invariant distribution of such an ergodic diffusion. In an appropriate Lyapunov setting and under {uniform} ellipticity assumptions on the diffusion coefficient, we establish (or improve) such bounds for Total Variation and $L^1$-Wasserstein distances in both multiplicative and additive and frameworks. These bounds rely on weak error expansions using {Stochastic Analysis} adapted to decreasing step setting.
Delocalization of bias in unadjusted Hamiltonian Monte Carlo and underdamped Langevin
Department of Mathematics, University of California, Los Angeles, CA 90095, USA（加州大学洛杉矶分校数学系）；Courant Institute, New York University, NY 10012, USA（纽约大学Courant研究所）
The efficiency of a Markov sampler based on the underdamped Langevin diffusion is studied for high dimensional targets with convex and smooth potentials. We consider a classical second-order integrator which requires only one gradient computation per iteration. Contrary to previous works on similar samplers, a dimension-free contraction of Wasserstein distances and convergence rate for the total variance distance are proven for the discrete time chain itself. Non-asymptotic Wasserstein and total variation efficiency bounds and concentration inequalities are obtained for both the Metropolis adj…
Stochastic approximation methods play a central role in maximum likelihood estimation problems involving intractable likelihood functions, such as marginal likelihoods arising in problems with missing or incomplete data, and in parametric empirical Bayesian estimation. Combined with Markov chain Monte Carlo algorithms, these stochastic optimisation methods have been successfully applied to a wide range of problems in science and industry. However, this strategy scales poorly to large problems because of methodological and theoretical difficulties related to using high-dimensional Markov chain …
In this paper, we provide new insights on the Unadjusted Langevin Algorithm. We show that this method can be formulated as a first order optimization algorithm of an objective functional defined on the Wasserstein space of order $2$. Using this interpretation and techniques borrowed from convex optimization, we give a non-asymptotic analysis of this method to sample from logconcave smooth target distribution on $\mathbb{R}^d$. Based on this interpretation, we propose two new methods for sampling from a non-smooth target distribution, which we analyze as well. Besides, these new algorithms are …
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…
A new (unadjusted) Langevin Monte Carlo (LMC) algorithm with improved rates in total variation and in Wasserstein distance is presented. All these are obtained in the context of sampling from a target distribution $π$ that has a density $\hatπ$ on $\mathbb{R}^d$ known up to a normalizing constant. Moreover, $-\log \hatπ$ is assumed to have a locally Lipschitz gradient and its third derivative is locally Hölder continuous with exponent $β\in (0,1]$. Non-asymptotic bounds are obtained for the convergence to stationarity of the new sampling method with convergence rate $1+ β/2$ in Wasserstein dis…
A randomized time integrator is suggested for unadjusted Hamiltonian Monte Carlo (uHMC) which involves a very minor modification to the usual Verlet time integrator, and hence, is easy to implement. For target distributions of the form $μ(dx) \propto e^{-U(x)} dx$ where $U: \mathbb{R}^d \to \mathbb{R}_{\ge 0}$ is $K$-strongly convex but only $L$-gradient Lipschitz, and initial distributions $ν$ with finite second moment, coupling proofs reveal that an $\varepsilon$-accurate approximation of the target distribution in $L^2$-Wasserstein distance $\boldsymbol{\mathcal{W}}^2$ can be achieved by th…
We study the problem of sampling from a distribution $\target$ using the Langevin Monte Carlo algorithm and provide rate of convergences for this algorithm in terms of Wasserstein distance of order $2$. Our result holds as long as the continuous diffusion process associated with the algorithm converges exponentially fast to the target distribution along with some technical assumptions. While such an exponential convergence holds for example in the log-concave measure case, it also holds for the more general case of asymptoticaly log-concave measures. Our results thus extends the known rates of…
Nanyang Technological University（南洋理工大学）；Imperial College London（伦敦帝国理工学院）；The Alan Turing Institute（艾伦·图灵研究所）；The University of Warwick（华威大学）；The University of Edinburgh（爱丁堡大学）；National Technical University of Athens（雅典国立技术大学）
In this paper, we are concerned with a non-asymptotic analysis of sampling algorithms used in nonconvex optimization. In particular, we obtain non-asymptotic estimates in Wasserstein-1 and Wasserstein-2 distances for a popular class of algorithms called Stochastic Gradient Langevin Dynamics (SGLD). In addition, the aforementioned Wasserstein-2 convergence result can be applied to establish a non-asymptotic error bound for the expected excess risk. Crucially, these results are obtained under a local Lipschitz condition and a local dissipativity condition where we remove the uniform dependence i…
Langevin diffusion processes and their discretizations are often used for sampling from a target density. The most convenient framework for assessing the quality of such a sampling scheme corresponds to smooth and strongly log-concave densities defined on $\mathbb R^p$. The present work focuses on this framework and studies the behavior of Monte Carlo algorithms based on discretizations of the kinetic Langevin diffusion. We first prove the geometric mixing property of the kinetic Langevin diffusion with a mixing rate that is, in the overdamped regime, optimal in terms of its dependence on the …
Algorithms based on discretizing Langevin diffusion are popular tools for sampling from high-dimensional distributions. We develop novel connections between such Monte Carlo algorithms, the theory of Wasserstein gradient flow, and the operator splitting approach to solving PDEs. In particular, we show that a proximal version of the Unadjusted Langevin Algorithm corresponds to a scheme that alternates between solving the gradient flows of two specific functionals on the space of probability measures. Using this perspective, we derive some new non-asymptotic results on the convergence properties…
In this paper, we consider the underdamped Langevin diffusion (ULD) and propose a numerical approximation using its associated ordinary differential equation (ODE). When used as a Markov Chain Monte Carlo (MCMC) algorithm, we show that the ODE approximation achieves a $2$-Wasserstein error of $\varepsilon$ in $\mathcal{O}\big(d^{\frac{1}{3}}/\varepsilon^{\frac{2}{3}}\big)$ steps under the standard smoothness and strong convexity assumptions on the target distribution. This matches the complexity of the randomized midpoint method proposed by Shen and Lee [NeurIPS 2019] which was shown to be ord…
ENSAE, CREST, IP Paris（国立统计与经济行政学校，经济研究与应用统计中心，巴黎综合理工学院）；University of Warwick（华威大学）
In this paper, we provide non-asymptotic upper bounds on the error of sampling from a target density using three schemes of discretized Langevin diffusions. The first scheme is the Langevin Monte Carlo (LMC) algorithm, the Euler discretization of the Langevin diffusion. The second and the third schemes are, respectively, the kinetic Langevin Monte Carlo (KLMC) for differentiable potentials and the kinetic Langevin Monte Carlo for twice-differentiable potentials (KLMC2). The main focus is on the target densities that are smooth and log-concave on $\mathbb R^p$, but not necessarily strongly log-…
In this paper, we revisit the recently established theoretical guarantees for the convergence of the Langevin Monte Carlo algorithm of sampling from a smooth and (strongly) log-concave density. We improve the existing results when the convergence is measured in the Wasserstein distance and provide further insights on the very tight relations between, on the one hand, the Langevin Monte Carlo for sampling and, on the other hand, the gradient descent for optimization. Finally, we also establish guarantees for the convergence of a version of the Langevin Monte Carlo algorithm that is based on noi…
Georgia Institute of Technology（佐治亚理工学院）；Shenzhen Institute of Artificial Intelligence and Robotics for Society（深圳人工智能与机器人研究院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
This article considers the popular MCMC method of unadjusted Langevin Monte Carlo (LMC) and provides a non-asymptotic analysis of its sampling error in 2-Wasserstein distance. The proof is based on a refinement of mean-square analysis in Li et al. (2019), and this refined framework automates the analysis of a large class of sampling algorithms based on discretizations of contractive SDEs. Using this framework, we establish an $\tilde{O}(\sqrt{d}/ε)$ mixing time bound for LMC, without warm start, under the common log-smooth and log-strongly-convex conditions, plus a growth condition on the 3rd-…
A new methodology is presented for the construction of control variates to reduce the variance of additive functionals of Markov Chain Monte Carlo (MCMC) samplers. Our control variates are definedthrough the minimization of the asymptotic variance of the Langevin diffusion over a family of functions, which can be seen as a quadratic risk minimization procedure. The use of these control variates is theoretically justified. We show that the asymptotic variances of some well-known MCMC algorithms, including the Random Walk Metropolis and the (Metropolis) Unadjusted/Adjusted Langevin Algorithm, ar…
We propose discrete Langevin proposal (DLP), a simple and scalable gradient-based proposal for sampling complex high-dimensional discrete distributions. In contrast to Gibbs sampling-based methods, DLP is able to update all coordinates in parallel in a single step and the magnitude of changes is controlled by a stepsize. This allows a cheap and efficient exploration in the space of high-dimensional and strongly correlated variables. We prove the efficiency of DLP by showing that the asymptotic bias of its stationary distribution is zero for log-quadratic distributions, and is small for distrib…
Georgia Institute of Technology（佐治亚理工学院）；Yale University（耶鲁大学）
We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution $ν= e^{-f}$ on $\mathbb{R}^n$. We prove a convergence guarantee in Kullback-Leibler (KL) divergence assuming $ν$ satisfies a log-Sobolev inequality and the Hessian of $f$ is bounded. Notably, we do not assume convexity or bounds on higher derivatives. We also prove convergence guarantees in Rényi divergence of order $q > 1$ assuming the limit of ULA satisfies either the log-Sobolev or Poincaré inequality. We also prove a bound on the bias of the limiting distribution of ULA assuming third-order smooth…
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））
The recent statistical finite element method (statFEM) provides a coherent statistical framework to synthesise finite element models with observed data. Through embedding uncertainty inside of the governing equations, finite element solutions are updated to give a posterior distribution which quantifies all sources of uncertainty associated with the model. However to incorporate all sources of uncertainty, one must integrate over the uncertainty associated with the model parameters, the known forward problem of uncertainty quantification. In this paper, we make use of Langevin dynamics to solv…
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…
Discretization of continuous-time diffusion processes is a widely recognized method for sampling. However, it seems to be a considerable restriction when the potentials are often required to be smooth (gradient Lipschitz). This paper studies the problem of sampling through Euler discretization, where the potential function is assumed to be a mixture of weakly smooth distributions and satisfies weakly dissipative. We establish the convergence in Kullback-Leibler (KL) divergence with the number of iterations to reach $ε$-neighborhood of a target distribution in only polynomial dependence on the …
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…
Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；Fudan University（复旦大学）；Beijing Institute of Technology（北京理工大学）；Florida State University（佛罗里达州立大学）
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.
We study the underdamped Langevin diffusion when the log of the target distribution is smooth and strongly concave. We present a MCMC algorithm based on its discretization and show that it achieves $\varepsilon$ error (in 2-Wasserstein distance) in $\mathcal{O}(\sqrt{d}/\varepsilon)$ steps. This is a significant improvement over the best known rate for overdamped Langevin MCMC, which is $\mathcal{O}(d/\varepsilon^2)$ steps under the same smoothness/concavity assumptions. The underdamped Langevin MCMC scheme can be viewed as a version of Hamiltonian Monte Carlo (HMC) which has been observed to …
Modern imaging methods rely strongly on Bayesian inference techniques to solve challenging imaging problems. Currently, the predominant Bayesian computation approach is convex optimisation, which scales very efficiently to high dimensional image models and delivers accurate point estimation results. However, in order to perform more complex analyses, for example image uncertainty quantification or model selection, it is necessary to use more computationally intensive Bayesian computation techniques such as Markov chain Monte Carlo methods. This paper presents a new and highly efficient Markov …
University of Toronto（多伦多大学）；Vector Institute（向量研究所）
We study sampling from a target distribution $ν_* = e^{-f}$ using the unadjusted Langevin Monte Carlo (LMC) algorithm when the potential $f$ satisfies a strong dissipativity condition and it is first-order smooth with a Lipschitz gradient. We prove that, initialized with a Gaussian random vector that has sufficiently small variance, iterating the LMC algorithm for $\widetilde{\mathcal{O}}(λ^2 dε^{-1})$ steps is sufficient to reach $ε$-neighborhood of the target in both Chi-squared and Renyi divergence, where $λ$ is the logarithmic Sobolev constant of $ν_*$. Our results do not require warm-star…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）
We study sampling from a target distribution ${ν_* = e^{-f}}$ using the unadjusted Langevin Monte Carlo (LMC) algorithm. For any potential function $f$ whose tails behave like ${\|x\|^α}$ for ${α\in [1,2]}$, and has $β$-Hölder continuous gradient, we prove that ${\widetilde{\mathcal{O}} \Big(d^{\frac{1}β+\frac{1+β}β(\frac{2}α - \boldsymbol{1}_{\{α\neq 1\}})} ε^{-\frac{1}β}\Big)}$ steps are sufficient to reach the $ε$-neighborhood of a $d$-dimensional target distribution $ν_*$ in KL-divergence. This convergence rate, in terms of $ε$ dependency, is not directly influenced by the tail growth rate…
NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problems
Shanghai Jiao Tong University（上海交通大学）；University of Birmingham（伯明翰大学）；Indian Institute of Technology (IIT) Kharagpur（印度理工学院卡拉格布尔分校）；University of Cambridge（剑桥大学）
