Lénaic Chizat : "Quantitative Convergence of Kernelized Wasserstein Gradient Flows"
Abstract:
Several machine-learning algorithms can be described, in the mean-field limit, as transport/continuity PDEs whose velocity field is regularized by a smoothing kernel. This is the case for Wasserstein gradient flows of Kernel Mean Discrepancies (KMD), which arise in the large-width limit of shallow neural-network training, and for Stein Variational Gradient Flow (SVGF), a sampling method based on interacting particles.
Understanding the performance of these algorithms therefore leads to questions about the long-time behavior of nonlocal PDEs. I will present an approach to quantitative local convergence for these flows, that yields sharp polynomial rates for Riesz-type kernels. The key difficulty is that the energy dissipation controls a weaker norm than the energy. The proof compensates for this loss through Sobolev interpolation and propagation of regularity.