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On PAC learning of functions with smoothness properties using feedforward sigmoidal networks

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2 Author(s)
Rao, N.S.V. ; Center for Eng. Syst. Adv. Res., Oak Ridge Nat. Lab., TN, USA ; Protopopescu, V.A.

We consider the problem of learning functions based on finite samples by using feedforward sigmoidal networks. The unknown function f is chosen from a family that has either bounded modulus of smoothness and/or bounded capacity. The sample is given by (X1, f(X1)), (X2, f(X2)), ...(Xn, f(Xn)). Where X1, X2, ..., Xn, are independently and identically distributed according to an unknown distribution PX. General results guarantee the existence of a neural network, fw*, that best approximates f in terms of expected error. However, since both f and PX are unknown, computing fw* is impossible in general. We propose to compute probability and approximately correct (PAC) approximations to fw*, based on alternative estimators, namely: 1) the nearest neighbor rule, 2) local averaging, and 3) Nadaraya-Watson estimators, all computed using the Haar system. We show that given a sufficiently large sample, each of these estimators guarantees a performance as close as desired to that of fw*. The practical importance of this result sterns from the fact that, unlike neural networks, the three estimators above are linear-time computable in terms of the sample size

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Proceedings of the IEEE  (Volume:84 ,  Issue: 10 )