We study the temperature control problem for Langevin diffusions in the context of non-convex optimization. The classical optimal control of such a problem is of the bang-bang type, which is overly sensitive to errors. A remedy is to allow the diffusions to explore other temperature values and hence smooth out the bang-bang control. We accomplish this by a stochastic relaxed control formulation incorporating randomization of the temperature control and regularizing its entropy. We derive a state-dependent, truncated exponential distribution, which can be used to sample temperatures in a Langevin algorithm, in terms of the solution to an HJB partial differential equation. We carry out a numerical experiment on a one-dimensional baseline example, in which the HJB equation can be easily solved, to compare the performance of the algorithm with three other available algorithms in search of a global optimum.
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…
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…
University of Toronto（多伦多大学）；Vector Institute（矢量研究所）；Borealis AI（Borealis人工智能公司）；LG Toronto AI Lab（LG多伦多人工智能实验室）
We propose a novel approach to analyze generalization error for discretizations of Langevin diffusion, such as the stochastic gradient Langevin dynamics (SGLD). For an $ε$ tolerance of expected generalization error, it is known that a first order discretization can reach this target if we run $Ω(ε^{-1} \log (ε^{-1}) )$ iterations with $Ω(ε^{-1})$ samples. In this article, we show that with additional smoothness assumptions, even first order methods can achieve arbitrarily runtime complexity. More precisely, for each $N>0$, we provide a sufficient smoothness condition on the loss function such …
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）
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…
Cornell University（康奈尔大学）；University of Connecticut（康涅狄格大学）
It is well known that adding any skew symmetric matrix to the gradient of Langevin dynamics algorithm results in a non-reversible diffusion with improved convergence rate. This paper presents a gradient algorithm to adaptively optimize the choice of the skew symmetric matrix. The resulting algorithm involves a non-reversible diffusion algorithm cross coupled with a stochastic gradient algorithm that adapts the skew symmetric matrix. The algorithm uses the same data as the classical Langevin algorithm. A weak convergence proof is given for the optimality of the choice of the skew symmetric matr…
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…
What is the information leakage of an iterative randomized learning algorithm about its training data, when the internal state of the algorithm is \emph{private}? How much is the contribution of each specific training epoch to the information leakage through the released model? We study this problem for noisy gradient descent algorithms, and model the \emph{dynamics} of Rényi differential privacy loss throughout the training process. Our analysis traces a provably \emph{tight} bound on the Rényi divergence between the pair of probability distributions over parameters of models trained on neigh…
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 …
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…
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…
Stochastic Gradient Langevin Dynamics (SGLD) is a powerful algorithm for optimizing a non-convex objective, where a controlled and properly scaled Gaussian noise is added to the stochastic gradients to steer the iterates towards a global minimum. SGLD is based on the overdamped Langevin diffusion which is reversible in time. By adding an anti-symmetric matrix to the drift term of the overdamped Langevin diffusion, one gets a non-reversible diffusion that converges to the same stationary distribution with a faster convergence rate. In this paper, we study the non reversible Stochastic Gradient …
We prove, under mild conditions, that the stochastic gradient Langevin dynamics converges to a limiting law as time tends to infinity, even in the case where the driving data sequence is dependent.
We propose a Markov chain Monte Carlo (MCMC) algorithm based on third-order Langevin dynamics for sampling from distributions with log-concave and smooth densities. The higher-order dynamics allow for more flexible discretization schemes, and we develop a specific method that combines splitting with more accurate integration. For a broad class of $d$-dimensional distributions arising from generalized linear models, we prove that the resulting third-order algorithm produces samples from a distribution that is at most $\varepsilon > 0$ in Wasserstein distance from the target distribution in $O\l…
