Underdamped Langevin dynamics (ULD) is a widely-used sampler for Gibbs distributions $π\propto e^{-V}$, and is often empirically effective in high dimensions. However, existing non-asymptotic convergence guarantees for discretized ULD typically scale polynomially with the ambient dimension $d$, leading to vacuous bounds when $d$ is large. The main known dimension-free result concerns the randomized midpoint discretization in Wasserstein-2 distance (Liu et al.,2023), while dimension-independent guarantees for ULD discretizations in KL divergence have remained open. We close this gap by proving the first dimension-free KL divergence bounds for discretized ULD. Our analysis refines the KL local error framework (Altschuler et al., 2025) to a dimension-free setting and yields bounds that depend on $\mathrm{tr}(\mathbf{H})$, where $\mathbf{H}$ upper bounds the Hessian of $V$, rather than on $d$. As a consequence, we obtain improved iteration complexity for underdamped Langevin Monte Carlo relative to overdamped Langevin methods in regimes where $\mathrm{tr}(\mathbf{H})\ll d$.
Recent works have derived non-asymptotic upper bounds for convergence of underdamped Langevin MCMC. We revisit these bound and consider introducing scaling terms in the underlying underdamped Langevin equation. In particular, we provide conditions under which an appropriate scaling allows to improve the error bounds in terms of the condition number of the underlying density of interest.
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 …
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Duke University（杜克大学）
The Underdamped Langevin Monte Carlo (ULMC) is a popular Markov chain Monte Carlo sampling method. It requires the computation of the full gradient of the log-density at each iteration, an expensive operation if the dimension of the problem is high. We propose a sampling method called Random Coordinate ULMC (RC-ULMC), which selects a single coordinate at each iteration to be updated and leaves the other coordinates untouched. We investigate the computational complexity of RC-ULMC and compare it with the classical ULMC for strongly log-concave probability distributions. We show that RC-ULMC is …
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…
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…
Underdamped Langevin Monte Carlo (ULMC) is an algorithm used to sample from unnormalized densities by leveraging the momentum of a particle moving in a potential well. We provide a novel analysis of ULMC, motivated by two central questions: (1) Can we obtain improved sampling guarantees beyond strong log-concavity? (2) Can we achieve acceleration for sampling? For (1), prior results for ULMC only hold under a log-Sobolev inequality together with a restrictive Hessian smoothness condition. Here, we relax these assumptions by removing the Hessian smoothness condition and by considering distribut…
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…
We formulate gradient-based Markov chain Monte Carlo (MCMC) sampling as optimization on the space of probability measures, with Kullback-Leibler (KL) divergence as the objective functional. We show that an underdamped form of the Langevin algorithm performs accelerated gradient descent in this metric. To characterize the convergence of the algorithm, we construct a Lyapunov functional and exploit hypocoercivity of the underdamped Langevin algorithm. As an application, we show that accelerated rates can be obtained for a class of nonconvex functions with the Langevin algorithm.
We study the problem of sampling from a distribution $p^*(x) \propto \exp\left(-U(x)\right)$, where the function $U$ is $L$-smooth everywhere and $m$-strongly convex outside a ball of radius $R$, but potentially nonconvex inside this ball. We study both overdamped and underdamped Langevin MCMC and establish upper bounds on the number of steps required to obtain a sample from a distribution that is within $ε$ of $p^*$ in $1$-Wasserstein distance. For the first-order method (overdamped Langevin MCMC), the iteration complexity is $\tilde{\mathcal{O}}\left(e^{cLR^2}d/ε^2\right)$, where $d$ is the …
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Department of Mathematics（数学系）；Madison, WI 53706（威斯康星州麦迪逊市 53706（注：该地址非机构，此处仅按要求提取原文中明确出现的机构相关内容））
Sampling from a log-concave distribution function is one core problem that has wide applications in Bayesian statistics and machine learning. While most gradient free methods have slow convergence rate, the Langevin Monte Carlo (LMC) that provides fast convergence requires the computation of gradients. In practice one uses finite-differencing approximations as surrogates, and the method is expensive in high-dimensions. A natural strategy to reduce computational cost in each iteration is to utilize random gradient approximations, such as random coordinate descent (RCD) or simultaneous perturbat…
Two popular classes of methods for approximate inference are Markov chain Monte Carlo (MCMC) and variational inference. MCMC tends to be accurate if run for a long enough time, while variational inference tends to give better approximations at shorter time horizons. However, the amount of time needed for MCMC to exceed the performance of variational methods can be quite high, motivating more fine-grained tradeoffs. This paper derives a distribution over variational parameters, designed to minimize a bound on the divergence between the resulting marginal distribution and the target, and gives a…
We provide convergence guarantees in Wasserstein distance for a variety of variance-reduction methods: SAGA Langevin diffusion, SVRG Langevin diffusion and control-variate underdamped Langevin diffusion. We analyze these methods under a uniform set of assumptions on the log-posterior distribution, assuming it to be smooth, strongly convex and Hessian Lipschitz. This is achieved by a new proof technique combining ideas from finite-sum optimization and the analysis of sampling methods. Our sharp theoretical bounds allow us to identify regimes of interest where each method performs better than th…
Langevin diffusion is a commonly used tool for sampling from a given distribution. In this work, we establish that when the target density $p^*$ is such that $\log p^*$ is $L$ smooth and $m$ strongly convex, discrete Langevin diffusion produces a distribution $p$ with $KL(p||p^*)\leq ε$ in $\tilde{O}(\frac{d}ε)$ steps, where $d$ is the dimension of the sample space. We also study the convergence rate when the strong-convexity assumption is absent. By considering the Langevin diffusion as a gradient flow in the space of probability distributions, we obtain an elegant analysis that applies to th…
Ergodicity of the underdamped mean-field Langevin dynamics
Université Paris-Saclay（巴黎萨克雷大学）；Université Paris-Dauphine, PSL（巴黎多芬大学，PSL大学）；The Chinese University of Hong Kong（香港中文大学）；Vienna University of Technology（维也纳工业大学）
Monte Carlo sampling techniques have broad applications in machine learning, Bayesian posterior inference, and parameter estimation. Often the target distribution takes the form of a product distribution over a dataset with a large number of entries. For sampling schemes utilizing gradient information it is cheaper for the derivative to be approximated using a random small subset of the data, introducing extra noise into the system. We present a new discretization scheme for underdamped Langevin dynamics when utilizing a stochastic (noisy) gradient. This scheme is shown to bias computed averag…
Error estimates between SGD with momentum and underdamped Langevin diffusion
Laboratoire de Mathematiques Blaise Pascal, CNRS-UMR 6620, Universite Clermont-Auvergne（克莱蒙奥弗涅大学）；University of Macau（澳门大学）；UM Zhuhai Research Institute（澳门大学珠海研究院）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Beijing International Center for Mathematical Research (BICMR), Peking University（北京大学北京国际数学研究中心）
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…
Variance Reduction for Stochastic Gradient Generalized Non-reversible Langevin Monte Carlo Algorithms
FinTech Thrust, Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州）金融科技方向）；School of Mathematical Science, Fudan University（复旦大学数学学院）；Department of Mathematics, Florida State University（佛罗里达州立大学数学系）
Improved Analysis for Hessian-free High-resolution Monte Carlo Sampling
School of Mathematics and Statistics, Donghua University（东华大学数学与统计学院）；Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）；Department of Mathematics, Florida State University（佛罗里达州立大学数学系）
We propose efficient Langevin Monte Carlo algorithms for sampling distributions with nonsmooth convex composite potentials, which is the sum of a continuously differentiable function and a possibly nonsmooth function. We devise such algorithms leveraging recent advances in convex analysis and optimization methods involving Bregman divergences, namely the Bregman--Moreau envelopes and the Bregman proximity operators, and in the Langevin Monte Carlo algorithms reminiscent of mirror descent. The proposed algorithms extend existing Langevin Monte Carlo algorithms in two aspects -- the ability to s…
Georgia Institute of Technology（佐治亚理工学院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Institute of Artificial Intelligence and Robotics for Society（深圳人工智能与机器人研究院）
Nesterov's Accelerated Gradient (NAG) for optimization has better performance than its continuous time limit (noiseless kinetic Langevin) when a finite step-size is employed \citep{shi2021understanding}. This work explores the sampling counterpart of this phenonemon and proposes a diffusion process, whose discretizations can yield accelerated gradient-based MCMC methods. More precisely, we reformulate the optimizer of NAG for strongly convex functions (NAG-SC) as a Hessian-Free High-Resolution ODE, change its high-resolution coefficient to a hyperparameter, inject appropriate noise, and discre…
LTCI, Télécom Paris, Institut Polytechnique de Paris（巴黎理工学院巴黎电信学院LTCI实验室）；Department of Statistics, University of Oxford（牛津大学统计系）；Department of Mathematics, Florida State University（佛罗里达州立大学数学系）；Department of Management Science and Information Systems, Rutgers Business School（罗格斯大学商学院管理科学与信息系统系）
Stochastic gradient descent with momentum (SGDm) is one of the most popular optimization algorithms in deep learning. While there is a rich theory of SGDm for convex problems, the theory is considerably less developed in the context of deep learning where the problem is non-convex and the gradient noise might exhibit a heavy-tailed behavior, as empirically observed in recent studies. In this study, we consider a \emph{continuous-time} variant of SGDm, known as the underdamped Langevin dynamics (ULD), and investigate its asymptotic properties under heavy-tailed perturbations. Supported by recen…
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…
Various Markov chain Monte Carlo (MCMC) methods are studied to improve upon random walk Metropolis sampling, for simulation from complex distributions. Examples include Metropolis-adjusted Langevin algorithms, Hamiltonian Monte Carlo, and other recent algorithms related to underdamped Langevin dynamics. We propose a broad class of irreversible sampling algorithms, called Hamiltonian assisted Metropolis sampling (HAMS), and develop two specific algorithms with appropriate tuning and preconditioning strategies. Our HAMS algorithms are designed to achieve two distinctive properties, while using a…
Various differentially private algorithms instantiate the exponential mechanism, and require sampling from the distribution $\exp(-f)$ for a suitable function $f$. When the domain of the distribution is high-dimensional, this sampling can be computationally challenging. Using heuristic sampling schemes such as Gibbs sampling does not necessarily lead to provable privacy. When $f$ is convex, techniques from log-concave sampling lead to polynomial-time algorithms, albeit with large polynomials. Langevin dynamics-based algorithms offer much faster alternatives under some distance measures such as…
We study the Proximal Langevin Algorithm (PLA) for sampling from a probability distribution $ν= e^{-f}$ on $\mathbb{R}^n$ 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 >…
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…
The Chinese University of Hong Kong（香港中文大学）；Rutgers University（罗格斯大学）；Florida State University（佛罗里达州立大学）
Stochastic gradient Hamiltonian Monte Carlo (SGHMC) is a variant of stochastic gradient with momentum where a controlled and properly scaled Gaussian noise is added to the stochastic gradients to steer the iterates towards a global minimum. Many works reported its empirical success in practice for solving stochastic non-convex optimization problems, in particular it has been observed to outperform overdamped Langevin Monte Carlo-based methods such as stochastic gradient Langevin dynamics (SGLD) in many applications. Although asymptotic global convergence properties of SGHMC are well known, its…
We analyse the privacy leakage of noisy stochastic gradient descent by modeling Rényi divergence dynamics with Langevin diffusions. Inspired by recent work on non-stochastic algorithms, we derive similar desirable properties in the stochastic setting. In particular, we prove that the privacy loss converges exponentially fast for smooth and strongly convex objectives under constant step size, which is a significant improvement over previous DP-SGD analyses. We also extend our analysis to arbitrary sequences of varying step sizes and derive new utility bounds. Last, we propose an implementation …
The stochastic gradient Langevin Dynamics is one of the most fundamental algorithms to solve sampling problems and non-convex optimization appearing in several machine learning applications. Especially, its variance reduced versions have nowadays gained particular attention. In this paper, we study two variants of this kind, namely, the Stochastic Variance Reduced Gradient Langevin Dynamics and the Stochastic Recursive Gradient Langevin Dynamics. We prove their convergence to the objective distribution in terms of KL-divergence under the sole assumptions of smoothness and Log-Sobolev inequalit…
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…
