Past events

Abstract:

Probabilistic representations provide a particle viewpoint to studying evolution equations.  For continuous-time dynamics, we review how Dynkin’s formula and Itô calculus lead to probabilistic representations of solutions to PDEs in terms of Markov processes with both continuous and jump-type potentials. In particular we derive the  probabilistic representation of the one-dimensional linear Dirac equation and show that it arises as a scaling limit of the discrete-time quantum walk via a Feynman-type formula, derived using a Poissonization trick.

Abstract:

Mean-field games (MFGs) provide elegant approximations to stochastic differential games with a large number of infinitesimal players and have become increasingly important in financial applications. In this talk, we introduce the mean-field actor-critic flow (MFAC), a reinforcement learning (RL)-based algorithm designed to solve MFGs, and establish its validity through both theoretical analysis and numerical experiments.

Unlike traditional RL approaches, our method employs partial differential equations (PDEs) to model the continuous-time training dynamics of the actor, critic, and population distribution. Inspired by optimal transport, the distributional dynamics evolve along a novel geodesic flow, and we prove convergence under a single time scale. Furthermore, borrowing ideas from generative modeling, we parameterize the distribution via score functions, the sampling from which is supported by Langevin Monte Carlo. Through numerical experiments, we demonstrate that MFAC effectively solves MFGs, achieving state-of-the-art performance.

This is joint work with Ruimeng Hu (UCSB) and Mo Zhou (UCLA).

Abstract:

Concentration of measure refers to the phenomenon that functions of independent random variables are sharply concentrated around their expected values. In this talk, we introduce the Hanson–Wright inequality, a class of concentration inequalities for quadratic forms of sub-Gaussian random vectors. We begin by reviewing classical concentration results, including Gaussian concentration and Bernstein’s inequality, derived via the Chernoff bound. We then present the Hanson–Wright inequality and outline its proof using Gaussian replacement and decoupling techniques. Finally, we discuss several applications of this inequality in machine learning and high-dimensional data analysis.