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Analysis, Geometry and Probability Seminar: Fall 2026

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September 9th: Rafiq Islam (ISU)

Title: 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.

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September 16th: No speaker. Miller Family Endowed Mathematics Lecture Series.

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