We show the Randomized Hamiltonian Monte Carlo (RHMC) algorithm has accelerated mixing time guarantees for sampling from log-concave probability distributions. RHMC proceeds by repeatedly simulating the continuous-time Hamiltonian dynamics for some random integration times, and resetting the velocity to be an independent Gaussian random variable between each simulation. We show that when the target distribution is log-concave and satisfies an α-Talagrand inequality (for example, if the target distribution is α-strongly log-concave), if we use a random integration time from either the triangular or the exponential distribution with mean Θ(α−1/2), then RHMC converges exponentially fast in KL divergence, and the total integration time to reach error ε in KL divergence scales as O(α−1/2log(ε−1)). We also show that when the target distribution is log-concave, if we use a sequence of random integration times from the triangular distribution with exponentially increasing means, then the total integration time to reach error ε in KL divergence scales as O(ε−1/2). Our analysis relies on a bound on the average KL divergence along Hamiltonian dynamics, which is inspired by an analogous result on accelerated optimization methods based on Hamiltonian dynamics.
We propose Decentralized Proximal Stochastic Gradient Langevin Dynamics (DE-PSGLD), a decentralized Markov chain Monte Carlo (MCMC) algorithm for sampling from a log-concave probability distribution constrained to a convex domain. Constraints are enforced through a shared proximal regularization based on the Moreau-Yosida envelope, enabling unconstrained updates while preserving consistency with the target constrained posterior. We establish non-asymptotic convergence guarantees in the 2-Wasserstein distance for both individual agent iterates and their network averages. Our analysis shows that DE-PSGLD converges to a regularized Gibbs distribution and quantifies the bias introduced by the proximal approximation. We evaluate DE-PSGLD for different sampling problems on synthetic and real datasets. As the first decentralized approach for constrained domains, our algorithm exhibits fast posterior concentration and high predictive accuracy.
We give a simple, unified, and nearly tight bound for sampling arbitrary logconcave distributions from a warm start using the In-and-Out algorithm along with exponential lifting. The main new ingredient in the analysis is an improved bound on the Poincaré constant of a lifted distribution. As a consequence, the resulting convergence rate is nearly tight for both constrained settings (e.g., Gaussian restricted to a convex body) and well-conditioned settings (e.g., strongly logconcave and smooth densities).
We provide new estimates in Wasserstein distance for the asymptotic bias of the unadjusted Langevin algorithm, in the classical setting of log-smooth strongly log-concave measures. Our bound implies a Wasserstein mixing time of order κd/ε, where κ is the condition number, d is the dimension, and ε is the target precision: this improves by a factor of d/ε over the previous state-of-the-art results.