Advanced Seminar in Analysis: Prof. Sayan Banerjee: Long-term behaviour of optimal transport-based sampling algorithms
Time: Tuesday, 10 am c.t.Organizer: Institute of Applied Analysis
Location: Ulm University, Helmholtzstraße 22, 1.42
As part of the advanced seminar at the Institute of Applied Analysis, Prof. Sayan Banerjee will give a talk on the
Title: Long-time behaviour of optimal transport-based sampling algorithms.
Tuesday, 13 October at 10.00 am c.t.
Helmholtzstrasse 22, Room 1.42
Abstract:
Stein variational gradient descent (SVGD) is a deterministic interacting-particle method for sampling from a target distribution. It is based on evolving the empirical distribution of $N$ particles via a kernelised projection of the Wasserstein gradient flow of relative entropy. Although its mean-field formulation is well understood, obtaining quantitative guarantees for a finite number of particles—particularly over long time horizons—presents several challenges.
In this talk, I will describe recent progress on these issues. I will first present an entropy-dissipation approach that yields near-i.i.d. $N^{-1/2}$ finite-particle convergence rates in kernel Stein discrepancy, as well as convergence in Wasserstein distance and long-time propagation of chaos (POC) for time-averaged particle laws. I will then discuss a cutoff principle that combines finite-time propagation-of-chaos estimates with quantitative convergence of the particle and mean-field dynamics to the target. This yields uniform-in-time POC in kernel Stein discrepancy and Wasserstein metrics, but the associated rates are logarithmic in $N$.
Finally, I will show that for noisy SVGD—obtained by adding a small Brownian diffusion to the particle dynamics—the resulting Fisher information dissipation can be utilised to obtain polynomial-in-$N$ uniform-in-time POC rates.