Seminar Announcement
Speaker: Siddharth Mitra, Yale University
Date: Friday, 22 December 2023
Time: 2:15 PM
Venue: Lecture Hall 01
On the Convergence of Mutual Information for (Strongly) Log-concave Sampling
Siddharth Mitra
Yale University.
22-12-23

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Abstract

We study the question of obtaining independent samples along two canonical Markov chains for continuous spaces -- the Langevin Diffusion (LD) and the Unadjusted Langevin Algorithm (ULA). Mixing time guarantees for these processes tell us when we get a (single approximate) sample from the target distribution, but motivated by obtaining multiple independent samples from the target distribution, we seek to answer the rate at which the mutual information decreases along these processes. The mutual information is a fundamental quantity lying at the intersection of statistics and information theory, and is zero iff two random variables are independent. We show that the mutual information decreases exponentially fast along both the LD and the ULA for strongly log-concave targets, and at a linear rate for log-concave targets along the LD. We begin by introducing the sampling problem and discuss the mixing time guarantees of the LD and ULA. We then talk about the question of obtaining independent samples along these Markov chains and discuss our main results and approaches. Joint work with Jiaming Liang and Andre Wibisono.