arXiv subjects

Linshan Liu

Publications and source records attributed to Linshan Liu.

3 recordsLinked to original sources

arXiv2026

On propagation of chaos for the Fisher-Rao gradient flow in entropic mean-field optimization

We consider a class of optimization problems on the space of probability measures motivated by the mean-field approach to studying neural networks. Such problems can be solved by constructing continuous-time gradient flows that converge to the minimizer of the energy function under consideration, and then implementing discrete-time algorithms that approximate the flow. In this work, we focus on the Fisher-Rao gradient flow and we construct an interacting particle system that approximates the flow as its mean-field limit. We discuss the connection between the energy function, the gradient flow and the particle system and explain different approaches to smoothing out the energy function with an appropriate kernel in a way that allows for the particle system to be well-defined. We provide a rigorous proof of the existence and uniqueness of thus obtained kernelized flows, as well as a propagation of chaos result that provides a theoretical justification for using the corresponding kernelized particle systems as approximation algorithms in entropic mean-field optimization.

math.OC↗

arXiv2023

$L^2$-Wasserstein contraction for Euler schemes of elliptic diffusions and interacting particle systems

We show the $L^2$-Wasserstein contraction for the transition kernel of a discretised diffusion process, under a contractivity at infinity condition on the drift and a sufficiently high diffusivity requirement. This extends recent results that, under similar assumptions on the drift but without the diffusivity restrictions, showed the $L^1$-Wasserstein contraction, or $L^p$-Wasserstein bounds for $p > 1$ that were, however, not true contractions. We explain how showing the true $L^2$-Wasserstein contraction is crucial for obtaining the local Poincaré inequality for the transition kernel of the Euler scheme of a diffusion. Moreover, we discuss other consequences of our contraction results, such as concentration inequalities and convergence rates in KL-divergence and total variation. We also study the corresponding $L^2$-Wasserstein contraction for discretisations of interacting diffusions. As a particular application, this allows us to analyse the behaviour of particle systems that can be used to approximate a class of McKean-Vlasov SDEs that were recently studied in the mean-field optimization literature.

math.PR↗

arXiv2023

Polyak-Łojasiewicz inequality on the space of measures and convergence of mean-field birth-death processes

The Polyak-Lojasiewicz inequality (PLI) in $\mathbb{R}^d$ is a natural condition for proving convergence of gradient descent algorithms. In the present paper, we study an analogue of PLI on the space of probability measures $\mathcal{P}(\mathbb{R}^d)$ and show that it is a natural condition for showing exponential convergence of a class of birth-death processes related to certain mean-field optimization problems. We verify PLI for a broad class of such problems for energy functions regularised by the KL-divergence.

math.OC↗