On the Approximation Rate of Hierarchical Mixtures-of-Experts for Generalized Linear Models

We investigate a class of hierarchical mixtures-of-experts (HME) models where generalized linear models with nonlinear mean functions of the form ψ(α + x^Tβ) are mixed. Here ψ(·) is the inverse link function. It is shown that mixtures of such mean functions can approximate a class of smooth functions of the form ψ(h(x)), where h(·) ε W^∞_2;k (a Sobolev class over [0, 1]^s, as the number of experts m in the network increases. An upper bound of the approximation rate is given as O(m^−2/s) in L_p norm. This rate can be achieved within the family of HME structures with no more than s-layers, where s is the dimension of the predictor x.