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 R p . The present work focuses on this framework and studies the behavior of the Monte Carlo algorithm 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 optimal in terms of its dependence on the condition number. We then use this result for obtaining improved guarantees of sampling using the kinetic Langevin Monte Carlo method, when the quality of sampling is measured by the Wasserstein distance. We also consider the situation where the Hessian of the log-density of the target distribution is Lipschitzcontinuous. In this case, we introduce a new discretization of the kinetic Langevin diffusion and prove that this leads to a substantial improvement of the upper bound on the sampling error measured in Wasserstein distance.

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A statistical model is said to be un-normalised when its likelihood function involves an intractable normalising constant. Two popular methods for parameter inference for these models are MC-MLE (Monte Carlo maximum likelihood estimation), and NCE (noise contrastive estimation); both methods rely on simulating artificial data-points to approximate the normalising constant. While the asymptotics of MC-MLE have been established under general hypotheses (Geyer, 1994), this is not so for NCE. We establish consistency and asymptotic normality of NCE estimators under mild assumptions. We compare NCE and MC-MLE under several asymptotic regimes. In particular, we show that, when m → ∞ while n is fixed (m and n being respectively the number of artificial data-points, and actual data-points), the two estimators are asymptotically equivalent. Conversely, we prove that, when the artificial data-points are IID, and when n → ∞ while m/n converges to a positive constant, the asymptotic variance of a NCE estimator is always smaller than the asymptotic variance of the corresponding MC-MLE estimator. We illustrate the variance reduction brought by NCE through a numerical study.

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 R p , but not necessarily strongly log-concave. Bounds on the computational complexity are obtained under two types of smoothness assumption: the potential has a Lipschitz-continuous gradient and the potential has a Lipschitz-continuous Hessian matrix. The error of sampling is measured by Wasserstein-q distances and the bounded-Lipschitz distance. We advocate for the use of a new dimension-adapted scaling in the definition of the computational complexity, when Wasserstein-q distances are considered. The obtained results show that the number of iterations to achieve a scaled-error smaller than a prescribed value depends only polynomially in the dimension.

Hamiltonian Monte Carlo (HMC) is a widely used sampler, known for its efficiency on high dimensional distributions. Yet HMC remains quite sensitive to the choice of integration time. Randomizing the length of Hamiltonian trajectories (RHMC) has been suggested to smooth the Auto-Correlation Functions (ACF), ensuring robustness of tuning. We present the Langevin diffusion as an alternative to control these ACFs by inducing randomness in Hamiltonian trajectories through a continuous refreshment of the velocities. We connect and compare the two processes in terms of quantitative mixing rates for the 2-Wasserstein and L 2 distances. The Langevin diffusion is presented as a limit of RHMC achieving the fastest mixing rate for strongly log-concave targets. We introduce a robust alternative to HMC built upon these dynamics, named Metropolis Adjusted Langevin Trajectories (MALT). Studying the scaling limit of MALT, we obtain optimal tuning guidelines similar to HMC, and recover the same scaling with respect to the dimension without additional assumptions. We illustrate numerically the efficiency of MALT compared to HMC and RHMC.