Convergence Rates for Gaussian Mixtures of Experts

Abstract

We provide a theoretical treatment of over-specified Gaussian mixtures of experts with covariate-free gating networks. We establish the convergence rates of the maximum likelihood estimation (MLE) for these models. Our proof technique is based on a novel notion of algebraic independence of the expert functions. Drawing on optimal transport, we establish a connection between the algebraic independence of the expert functions and a certain class of partial differential equations (PDEs) with respect to the parameters. Exploiting this connection allows us to derive convergence rates for parameter estimation.