On the Optimization Landscape of Maximum Mean Discrepancy

TL;DR

This paper analyzes the optimization landscape of Maximum Mean Discrepancy (MMD) in generative models, showing that in certain cases the landscape is benign and allows for global optimization via gradient methods.

Contribution

It provides the first theoretical analysis demonstrating conditions under which MMD-based generative models can be globally optimized despite non-convexity.

Findings

MMD landscape is benign for low-rank Gaussian distributions.

Gradient methods can globally minimize MMD in these cases.

Analysis extends to mixtures of Gaussians.

Abstract

Generative models have been successfully used for generating realistic signals. Because the likelihood function is typically intractable in most of these models, the common practice is to use "implicit" models that avoid likelihood calculation. However, it is hard to obtain theoretical guarantees for such models. In particular, it is not understood when they can globally optimize their non-convex objectives. Here we provide such an analysis for the case of Maximum Mean Discrepancy (MMD) learning of generative models. We prove several optimality results, including for a Gaussian distribution with low rank covariance (where likelihood is inapplicable) and a mixture of Gaussians. Our analysis shows that that the MMD optimization landscape is benign in these cases, and therefore gradient based methods will globally minimize the MMD objective.