Almost Linear VC Dimension Bounds for Piecewise Polynomial Networks

VitalyMaiorov Department of Mathematics Technion, Haifa 32000 Israel Ron Meir Department of Electrical Engineering Technion, Haifa 32000 Israel rmeir@dumbo.technion.ac.il Abstract We compute upper and lower bounds on the VC dimension of feedforward networks of units with piecewise polynomial activation functions.We show that if the number of layers is fixed, then the VC dimension grows as W log W, where W is the number of parameters in the network. The VC dimension is an important measure of the complexity of a class of binaryvalued functions,since it characterizes the amount of data required for learning in the PAC setting (see [BEHW89, Vap82]). In this paper, we establish upper and lower bounds on the VC dimension of a specific class of multi-layered feedforward neural networks. Let F be the class of binary-valued functions computed by a feedforward neural network with W weights and k computational (non-input) units, each with a piecewise polynomial activation function. O(W2), which would lead one to conclude that the bounds Almost Linear VC Dimension Bounds for Piecewise Polynomial Networks 191 are in fact tight up to a constant.