Convergence Rates for Mixture-of-Experts

In this paper we study the mixture-of-experts (ME) model with experts in an one-exponential family with mean $\phi(h_k)$, where $h_k$ is a $k^{th}$ order polynomial and $\phi(\cdot)$ is the inverse link function. We derive sharp approximation rates with respect to the Kullback-Leibler divergence and convergence rate of the maximum likelihood estimator to densities in an one-parameter exponential family with mean $\phi(h)$ where $h\in\WK$, a Sobolev class with $\alpha$ derivatives. We found that the convergence rate of the maximum likelihood estimator to the true density is $O_p(m^{-2[\alpha\wedge(k+1)]/s}+ (mJ_k+v_m) n^{-1}\log n)$, where $n$ is the number of observations, $s$ is the number of covariates, $J_k$ is the number of parameters of the polynomial $h_k$, $m$ the number of experts and $v_m$ is the number of parameters on the weight functions. Further, if the maximum likelihood estimator is uniquely identified we can remove the "$\log n$" term of the convergence rates. We close the paper discussing model specification and the effects on approximation and estimation errors and conclude that the best error bound is achieved using a balance between $k$ and $m$. We also explain how the results in this paper can be extended to more general approximation and target classes of densities.