We study Hamiltonian Monte Carlo (HMC) for sampling from a strongly logconcave density proportional to @math where @math is @math -strongly convex and @math -smooth (the condition number is @math ). We show that the relaxation time (inverse of the spectral gap) of ideal HMC is @math , improving on the previous best bound of @math ; we complement this with an example where the relaxation time is @math . When implemented using a nearly optimal ODE solver, HMC returns an @math -approximate point in @math -Wasserstein distance using @math gradient evaluations per step and @math total time.
Various versions of Langevin dynamics have been studied in many recent papers, see @cite_15 @cite_14 @cite_10 @cite_20 @cite_19 @cite_6 @cite_3 @cite_7 @cite_13 @cite_17 @cite_9 @cite_8 . The convergence rate of HMC is also studied recently in @cite_18 @cite_1 @cite_2 @cite_0 @cite_12 @cite_11 . The first bound for our setting was obtained by Mangoubi and Smith @cite_2 , who gave an @math bound on the convergence rate of ideal HMC.
@cite_13 @cite_18 @cite_14 @cite_11 @cite_7 @cite_8 @cite_9 @cite_1 @cite_6 @cite_3 @cite_0 @cite_19 @cite_2 @cite_15 @cite_10 @cite_12 @cite_20 @cite_17
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We introduce a new probabilistic approach to quantify convergence to equilibrium for (kinetic) Langevin processes. In contrast to previous analytic approaches that focus on the associated kinetic Fokker-Planck equation, our approach is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. It yields contractions in a particular Wasserstein distance, and it provides rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. In particular, we are able to recover kinetic behavior in terms of explicit lower bounds for the contraction rate. For example, for a rescaled double-well potential with local minima at distance @math , we obtain a lower bound for the contraction rate of order @math provided the friction coefficient is of order @math . 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. 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. 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. Stochastic Gradient Langevin Dynamics (SGLD) is a combination of a Robbins-Monro type algorithm with Langevin dynamics. In this paper, the SGLD method with fixed step size @math is considered in order to sample from a logconcave target distribution @math , known up to a normalisation factor. We assume that unbiased estimates of the gradient from possibly dependent observations are available. It is shown that, for all @math , the Wasserstein- @math distance of the @math th iterate of the SGLD algorithm from @math is dominated by @math with appropriate constants @math and @math , whereas for the case of i.i.d. data the upper bound is improved to @math . 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. Consider a graph, G, for which the vertices can have two modes, 0 or 1. Suppose that a particle moves around on G according to a discrete time Markov chain with the following rules. With (strictly positive) probabilities pm, pc and pr it moves to a randomly chosen neighbour, changes the mode of the vertex it is at or just stands still, respectively. We call such a random process a (pm, pc, pr)-lamplighter process on G. Assume that the process starts with the particle in a fixed position and with all vertices having mode 0. The convergence rate to stationarity in terms of the total variation norm is studied for the special cases with G = KN, the complete graph with N vertices, and when G = mod N. In the former case we prove that as N --> [infinity], ((2pc + pm) 4pcpm)N log N is a threshold for the convergence rate. In the latter case we show that the convergence rate is asymptotically determined by the cover time CN in that the total variation norm after aN2 steps is given by P(CN > aN2). The limit of this probability can in turn be calculated by considering a Brownian motion with two absorbing barriers. In particular, this means that there is no threshold for this case. This paper considers local convergence and rate of convergence results for algorithms for minimizing the composite functionF(x)=f(x)+h(c(x)) wheref andc are smooth buth(c) may be nonsmooth. Local convergence at a second order rate is established for the generalized Gauss—Newton method whenh is convex and globally Lipschitz and the minimizer is strongly unique. Local convergence at a second order rate is established for a generalized Newton method when the minimizer satisfies nondegeneracy, strict complementarity and second order sufficiency conditions. Assuming the minimizer satisfies these conditions, necessary and sufficient conditions for a superlinear rate of convergence for curvature approximating methods are established. Necessary and sufficient conditions for a two-step superlinear rate of convergence are also established when only reduced curvature information is available. All these local convergence and rate of convergence results are directly applicable to nonlinearing programming problems. The convergence of a class of Metropolis-type Markov-chain annealing algorithms for global optimization of a smooth function @math on @math is established. No prior information is assumed as to what bounded region contains a global minimum. The analysis contained herein is based on writing the Metropolis-type algorithm in the form of a recursive stochastic algorithm @math , where @math is a standard white Gaussian sequence, @math are random variables, and @math , @math for k large. Convergence results for @math are then applied from our previous work [SIAM Journal on Control and Optimization, 29 (1991), pp. 999–1018]. Since the analysis of @math is based on the asymptotic behavior of the related Langevin-type Markov diffusion annealing algorithm @math , where @math is a standard Wiener process and $c(t) = ... Abstract At the heart of this article will be the study of a branching Brownian motion (BBM) with killing , where individual particles move as Brownian motions with drift − ρ , perform dyadic branching at rate β and are killed on hitting the origin. Firstly, by considering properties of the right-most particle and the extinction probability, we will provide a probabilistic proof of the classical result that the ‘one-sided’ FKPP travelling-wave equation of speed − ρ with solutions f : [ 0 , ∞ ) → [ 0 , 1 ] satisfying f ( 0 ) = 1 and f ( ∞ ) = 0 has a unique solution with a particular asymptotic when ρ 2 β , and no solutions otherwise. Our analysis is in the spirit of the standard BBM studies of [S.C. Harris, Travelling-waves for the FKPP equation via probabilistic arguments, Proc. Roy. Soc. Edinburgh Sect. A 129 (3) (1999) 503–517] and [A.E. Kyprianou, Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis, Ann. Inst. H. Poincare Probab. Statist. 40 (1) (2004) 53–72] and includes an intuitive application of a change of measure inducing a spine decomposition that, as a by product, gives the new result that the asymptotic speed of the right-most particle in the killed BBM is 2 β − ρ on the survival set. Secondly, we introduce and discuss the convergence of an additive martingale for the killed BBM, W λ , that appears of fundamental importance as well as facilitating some new results on the almost-sure exponential growth rate of the number of particles of speed λ ∈ ( 0 , 2 β − ρ ) . Finally, we prove a new result for the asymptotic behaviour of the probability of finding the right-most particle with speed λ > 2 β − ρ . This result combined with Chauvin and Rouault's [B. Chauvin, A. Rouault, KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees, Probab. Theory Related Fields 80 (2) (1988) 299–314] arguments for standard BBM readily yields an analogous Yaglom-type conditional limit theorem for the killed BBM and reveals W λ as the limiting Radon–Nikodým derivative when conditioning the right-most particle to travel at speed λ into the distant future. 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 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 study the maximum of a Gaussian field on @math ( @math ) whose correlations decay logarithmically with the distance. Kahane Kah85 introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas DRSV12a DRSV12b extended Kahane's construction to the critical case and established the KPZ formula at criticality. Moreover, they made in DRSV12a several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in DRSV12a : we establish the convergence in law of the maximum and show that the limit law is the Gumbel distribution convoluted by the limit of the derivative martingale. Stochastic Gradient Langevin Dynamics (SGLD) is a popular variant of Stochastic Gradient Descent, where properly scaled isotropic Gaussian noise is added to an unbiased estimate of the gradient at each iteration. This modest change allows SGLD to escape local minima and suffices to guarantee asymptotic convergence to global minimizers for sufficiently regular non-convex objectives (Gelfand and Mitter, 1991). The present work provides a nonasymptotic analysis in the context of non-convex learning problems, giving finite-time guarantees for SGLD to find approximate minimizers of both empirical and population risks. As in the asymptotic setting, our analysis relates the discrete-time SGLD Markov chain to a continuous-time diffusion process. A new tool that drives the results is the use of weighted transportation cost inequalities to quantify the rate of convergence of SGLD to a stationary distribution in the Euclidean @math -Wasserstein distance. Hamiltonian Monte Carlo (HMC) sampling methods provide a mechanism for defining distant proposals with high acceptance probabilities in a Metropolis-Hastings framework, enabling more efficient exploration of the state space than standard random-walk proposals. The popularity of such methods has grown significantly in recent years. However, a limitation of HMC methods is the required gradient computation for simulation of the Hamiltonian dynamical system--such computation is infeasible in problems involving a large sample size or streaming data. Instead, we must rely on a noisy gradient estimate computed from a subset of the data. In this paper, we explore the properties of such a stochastic gradient HMC approach. Surprisingly, the natural implementation of the stochastic approximation can be arbitrarily bad. To address this problem we introduce a variant that uses second-order Langevin dynamics with a friction term that counteracts the effects of the noisy gradient, maintaining the desired target distribution as the invariant distribution. Results on simulated data validate our theory. We also provide an application of our methods to a classification task using neural networks and to online Bayesian matrix factorization. Exact generalized Langevin equations are derived for arbitrarily nonlinear systems interacting with specially chosen heat baths. An example is displayed in which the Langevin equation is nonlinear but approximately Markovian. A general, inexact, efficient proximal quasi-Newton algorithm for composite optimization problems has been proposed by Scheinberg and Tang (Math Program 160:495–529, 2016) and a sublinear global convergence rate has been established. In this paper, we analyze the global convergence rate of this method, in the both exact and inexact settings, in the case when the objective function is strongly convex. We also investigate a practical variant of this method by establishing a simple stopping criterion for the subproblem optimization. Furthermore, we consider an accelerated variant, based on FISTA of Beck and Teboulle (SIAM 2:183–202, 2009), to the proximal quasi-Newton algorithm. (SIAM 22:1042–1064, 2012) considered a similar accelerated method, where the convergence rate analysis relies on very strong impractical assumptions on Hessian estimates. We present a modified analysis while relaxing these assumptions and perform a numerical comparison of the accelerated proximal quasi-Newton algorithm and the regular one. Our analysis and computational results show that acceleration may not bring any benefit in the quasi-Newton setting. The stochastic dynamical behavior of a well-stirred mixture of N molecular species that chemically interact through M reaction channels is accurately described by the chemical master equation. It is shown here that, whenever two explicit dynamical conditions are satisfied, the microphysical premise from which the chemical master equation is derived leads directly to an approximate time-evolution equation of the Langevin type. This chemical Langevin equation is the same as one studied earlier by Kurtz, in contradistinction to some other earlier proposed forms that assume a deterministic macroscopic evolution law. The novel aspect of the present analysis is that it shows that the accuracy of the equation depends on the satisfaction of certain specific conditions that can change from moment to moment, rather than on a static system size parameter. The derivation affords a new perspective on the origin and magnitude of noise in a chemically reacting system. It also clarifies the connection between the stochas... 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. In this paper we employ a parallel version of a randomized (block) coordinate descent method for minimizing the sum of a partially separable smooth convex function and a fully separable nonsmooth convex function. Under the assumption of Lipschitz continuity of the gradient of the smooth function, this method has a sublinear convergence rate. Linear convergence rate of the method is obtained for the newly introduced class of generalized error bound functions. We prove that the new class of generalized error bound functions encompasses both global local error bound functions and smooth strongly convex functions. We also show that the theoretical estimates on the convergence rate depend on the number of blocks chosen randomly and a natural measure of separability of the smooth component of the objective function.