A Gradient Flow Perspective on Minimum MMD Estimation

2026-07-04

AI Summary

This work addresses the lack of convergence guarantees for Maximum Mean Discrepancy (MMD) estimation in non-convex settings. By adopting the perspective of MMD gradient flows, the authors propose a Preconditioned Gradient Descent (PGD) algorithm that performs parameter optimization in the space of probability measures. They establish, for the first time, global asymptotic convergence of PGD under non-convexity by introducing gradient domination and projected residual conditions, thereby bridging nonparametric gradient flows with parametric optimization. Experimental results demonstrate that PGD significantly outperforms standard gradient descent in both parameter estimation and composite hypothesis testing tasks, corroborating both the theoretical rigor and practical efficacy of the proposed method.

Abstract

Minimum maximum mean discrepancy (MMD) estimation has emerged as a robust and likelihood-free alternative to maximum likelihood estimation for parameter estimation. Yet, despite its practical success, the associated optimization problem remains poorly understood, with theoretical guarantees for existing algorithms hinging on convexity assumptions that rarely hold in practice. We address this gap by proposing a preconditioned gradient descent (PGD) scheme, establishing its asymptotic \emph{global} convergence under explicit gradient-dominance and projection-residual conditions. Our approach is inspired by recent progress on MMD gradient flows, a nonparametric descent scheme on the space of probability measures. We provide extensive empirical evidence that our PGD scheme outperforms standard gradient descent across a range of challenging parameter estimation and composite hypothesis testing problems.

Problem

• Minimum MMD estimation
• optimization problem
• non-convexity
• theoretical guarantees
• parameter estimation

Innovation

• preconditioned gradient descent
• maximum mean discrepancy
• gradient flow
• global convergence
• likelihood-free estimation