Scalable Sampling Algorithms via Langevin Dynamics
Abstract
Langevin algorithms are popular Markov chain Monte Carlo (MCMC) methods for large-scale sampling from distributions with probability density \(\pi(x)\propto e^{-f(x)}\). In this talk, we present three Langevin sampling algorithms under strong convexity and smoothness assumptions on \(f\).
We first study the decentralized proximal stochastic gradient Langevin dynamics (DE-PSGLD) algorithm for sampling from a convex body \(\mathcal{K}\subset\mathbb{R}^d\) when the data is distributed among \(N\) computational agents. Constraints are handled through a shared proximal regularization based on the Moreau–Yosida envelope, allowing unconstrained updates to approximate the constrained target distribution.
Next, we study unconstrained sampling in a decentralized setting, where agents perform Bayesian learning collaboratively without sharing individual data. Existing decentralized SGLD algorithms introduce network-induced bias that persists even when using full batches. To address this bias, we propose the generalized EXTRA stochastic gradient Langevin dynamics.
Finally, we propose the \(P\)-th order Langevin dynamics for any \(P\geq 3\), with improved dependence on dimension and sampling accuracy. For all three algorithms, we obtain convergence guarantees and iteration complexities in the 2-Wasserstein distance. We also validate our algorithms through numerical experiments in Bayesian regression, classification, and deep learning.