Convergence Rates for Gaussian Mixtures of Experts

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 \emph{algebraic independence} of the expert functions. Drawing on optimal transport theory, we establish a connection between the algebraic independence and a certain class of partial differential equations (PDEs). Exploiting this connection allows us to derive convergence rates and minimax lower bounds for parameter estimation.