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

Department of Statistics, Northwestern University, Evanston, IL 60208, U.S.A.

Martin A. Tanner

Department of Statistics, Northwestern University, Evanston, IL 60208, U.S.A.

We investigate a class of hierarchical mixtures-of-experts (HME) models where generalized linear models with nonlinear mean functions of the form ( + x^T ß) are mixed. Here (·) is the inverse link function. It is shown that mixtures of such mean functions can approximate a class of smooth functions of the form (h(x)), where h(·) 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.

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