Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) is a momentum version of stochastic gradient descent with properly injected Gaussian noise to find a global minimum. In this paper, non-asymptotic convergence analysis of SGHMC is given in the context of non-convex optimization, where subsampling techniques are used over an i.i.d dataset for gradient updates. Our results complement those of [RRT17] and improve on those of [GGZ18].
Non-asymptotic convergence rate Langevin dynamics based algorithms for approximate sampling log-concave distributions are intensively studied in recent years. For example, overdamped Langevin dynamics are discussed in @cite_11 , @cite_1 , @cite_4 , @cite_28 , @cite_12 and others. Recently, @cite_16 treats the case of non-i.i.d. data streams with a certain mixing property. Underdamped Langevin dynamics are examined in @cite_20 , @cite_21 , @cite_23 , etc. Further analysis on HMC are discussed on @cite_25 , @cite_10 . Subsampling methods are applied to speed up HMC for large datasets, see @cite_29 , @cite_0 .
@cite_4 @cite_28 @cite_10 @cite_29 @cite_21 @cite_1 @cite_0 @cite_23 @cite_16 @cite_20 @cite_25 @cite_12 @cite_11
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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 @math error (in 2-Wasserstein distance) in @math steps. This is a significant improvement over the best known rate for overdamped Langevin MCMC, which is @math 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 outperform overdamped Langevin MCMC methods in a number of application areas. We provide quantitative rates that support this empirical wisdom. In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings. Sampling logconcave functions arising in statistics and machine learning has been a subject of intensive study. Recent developments include analyses for Langevin dynamics and Hamiltonian Monte Carlo (HMC). While both approaches have dimension-independent bounds for the underlying @math processes under sufficiently strong smoothness conditions, the resulting discrete algorithms have complexity and number of function evaluations growing with the dimension. Motivated by this problem, in this paper, we give a general algorithm for solving multivariate ordinary differential equations whose solution is close to the span of a known basis of functions (e.g., polynomials or piecewise polynomials). The resulting algorithm has polylogarithmic depth and essentially tight runtime - it is nearly linear in the size of the representation of the solution. We apply this to the sampling problem to obtain a nearly linear implementation of HMC for a broad class of smooth, strongly logconcave densities, with the number of iterations (parallel depth) and gradient evaluations being @math in the dimension (rather than polynomial as in previous work). This class includes the widely-used loss function for logistic regression with incoherent weight matrices and has been subject of much study recently. We also give a faster algorithm with @math for the more general and standard class of strongly convex functions with Lipschitz gradient. These results are based on (1) an improved contraction bound for the exact HMC process and (2) logarithmic bounds on the degree of polynomials that approximate solutions of the differential equations arising in implementing HMC. We study the problem of sampling from a distribution where the negative logarithm of the target density is @math -smooth everywhere and @math -strongly convex outside a ball of radius @math , but potentially non-convex inside this ball. We study both overdamped and underdamped Langevin MCMC and prove upper bounds on the time required to obtain a sample from a distribution that is within @math of the target distribution in @math -Wasserstein distance. For the first-order method (overdamped Langevin MCMC), the time complexity is @math , where @math is the dimension of the underlying space. For the second-order method (underdamped Langevin MCMC), the time complexity is @math for some explicit positive constant @math . Surprisingly, the convergence rate is only polynomial in the dimension @math and the target accuracy @math . It is however exponential in the problem parameter @math , which is a measure of non-logconcavity of the target distribution. Sampling distribution over high-dimensional state-space is a problem which has recently attracted a lot of research efforts; applications include Bayesian non-parametrics, Bayesian inverse problems and aggregation of estimators. All these problems boil down to sample a target distribution @math having a density the Lebesgue measure on @math , known up to a normalisation factor @math where @math is continuously differentiable and smooth. In this paper, we study a sampling technique based on the Euler discretization of the Langevin stochastic differential equation. Contrary to the Metropolis Adjusted Langevin Algorithm (MALA), we do not apply a Metropolis-Hastings correction. We obtain for both constant and decreasing step sizes in the Euler discretization, non-asymptotic bounds for the convergence to stationarity in both total variation and Wasserstein distances. A particular attention is paid on the dependence on the dimension of the state space, to demonstrate the applicability of this method in the high dimensional setting, at least when @math is convex. These bounds are based on recently obtained estimates of the convergence of the Langevin diffusion to stationarity using Poincar 'e and log-Sobolev inequalities. These bounds improve and extend the results of (Dalalyan, 2014). We also investigate the convergence of an appropriately weighted empirical measure and we report sharp bounds for the mean square error and exponential deviation inequality for Lipschitz functions. A limited Monte Carlo experiment is carried out to support our findings. 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 noisy evaluations of the gradient. We obtain several quantitative bounds on the mixing properties of the Hamiltonian Monte Carlo (HMC) algorithm for a strongly log-concave target distribution @math on @math , showing that HMC mixes quickly in this setting. One of our main results is a dimension-free bound on the mixing of an "ideal" HMC chain, which is used to show that the usual leapfrog implementation of HMC can sample from @math using only @math gradient evaluations. This dependence on dimension is sharp, and our results significantly extend and improve previous quantitative bounds on the mixing of HMC. 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 function. We show that an underdamped form of the Langevin algorithm perform 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 present a unified framework to analyze the global convergence of Langevin dynamics based algorithms for nonconvex finite-sum optimization with @math component functions. At the core of our analysis is a direct analysis of the ergodicity of the numerical approximations to Langevin dynamics, which leads to faster convergence rates. Specifically, we show that gradient Langevin dynamics (GLD) and stochastic gradient Langevin dynamics (SGLD) converge to the almost minimizer within @math and @math stochastic gradient evaluations respectively, where @math is the problem dimension, and @math is the spectral gap of the Markov chain generated by GLD. Both of the results improve upon the best known gradient complexity results. Furthermore, for the first time we prove the global convergence guarantee for variance reduced stochastic gradient Langevin dynamics (VR-SGLD) to the almost minimizer after @math stochastic gradient evaluations, which outperforms the gradient complexities of GLD and SGLD in a wide regime. Our theoretical analyses shed some light on using Langevin dynamics based algorithms for nonconvex optimization with provable guarantees. Sampling from various kinds of distributions is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators, test procedures or confidence intervals. In many situations, the exact sampling from a given distribution is impossible or computationally expensive and, therefore, one needs to resort to approximate sampling strategies. However, there is no well-developed theory providing meaningful nonasymptotic guarantees for the approximate sampling procedures, especially in the high-dimensional problems. This paper makes some progress in this direction by considering the problem of sampling from a distribution having a smooth and log-concave density defined on Rp, for some integer p > 0. We establish nonasymptotic bounds for the error of approximating the true distribution by the one obtained by the Langevin Monte Carlo method and its variants. We illustrate the effectiveness of the established guarantees with various experiments. Underlying our analysis are insights from the theory of continuous-time diffusion processes, which may be of interest beyond the framework of distributions with log-concave densities considered in the present work. Recent advances in Bayesian learning with large-scale data have witnessed emergence of stochastic gradient MCMC algorithms (SG-MCMC), such as stochastic gradient Langevin dynamics (SGLD), stochastic gradient Hamiltonian MCMC (SGHMC), and the stochastic gradient thermostat. While finite-time convergence properties of the SGLD with a 1st-order Euler integrator have recently been studied, corresponding theory for general SG-MCMCs has not been explored. In this paper we consider general SG-MCMCs with high-order integrators, and develop theory to analyze finite-time convergence properties and their asymptotic invariant measures. Our theoretical results show faster convergence rates and more accurate invariant measures for SG-MCMCs with higher-order integrators. For example, with the proposed efficient 2nd-order symmetric splitting integrator, the mean square error (MSE) of the posterior average for the SGHMC achieves an optimal convergence rate of L-4 5 at L iterations, compared to L-2 3 for the SGHMC and SGLD with 1st-order Euler integrators. Furthermore, convergence results of decreasing-step-size SG-MCMCs are also developed, with the same convergence rates as their fixed-step-size counterparts for a specific decreasing sequence. Experiments on both synthetic and real datasets verify our theory, and show advantages of the proposed method in two large-scale real applications. Recently, Stochastic Gradient Markov Chain Monte Carlo (SG-MCMC) methods have been proposed for scaling up Monte Carlo computations to large data problems. Whilst these approaches have proven useful in many applications, vanilla SG-MCMC might suffer from poor mixing rates when random variables exhibit strong couplings under the target densities or big scale differences. In this study, we propose a novel SG-MCMC method that takes the local geometry into account by using ideas from Quasi-Newton optimization methods. These second order methods directly approximate the inverse Hessian by using a limited history of samples and their gradients. Our method uses dense approximations of the inverse Hessian while keeping the time and memory complexities linear with the dimension of the problem. We provide a formal theoretical analysis where we show that the proposed method is asymptotically unbiased and consistent with the posterior expectations. We illustrate the effectiveness of the approach on both synthetic and real datasets. Our experiments on two challenging applications show that our method achieves fast convergence rates similar to Riemannian approaches while at the same time having low computational requirements similar to diagonal preconditioning approaches. We introduce efficient margin-based algorithms for selective sampling and filtering in binary classification tasks. Experiments on real-world textual data reveal that our algorithms perform significantly better than popular and similarly efficient competitors. Using the so-called Mammen-Tsybakov low noise condition to parametrize the instance distribution, and assuming linear label noise, we show bounds on the convergence rate to the Bayes risk of a weaker adaptive variant of our selective sampler. Our analysis reveals that, excluding logarithmic factors, the average risk of this adaptive sampler converges to the Bayes risk at rate N ?(1+?)(2+?) 2(3+?) where N denotes the number of queried labels, and ?>0 is the exponent in the low noise condition. For all @math this convergence rate is asymptotically faster than the rate N ?(1+?) (2+?) achieved by the fully supervised version of the base selective sampler, which queries all labels. Moreover, for ??? (hard margin condition) the gap between the semi- and fully-supervised rates becomes exponential.