Neural Nets with Superlinear VC-Dimension

It has been known for quite a while that the Vapnik-Chervonenkis dimension (VC-dimension) of a feedforward neural net with linear threshold gates is at most O(w . log w), where w is the total number of weights in the neural net. We show in this paper that this bound is in fact asymptotically optimal. More precisely, we exhibit for any depth d 2 3 a large class of feedforward neural nets of depth d with w weights that have VC-dimension Q(w . log w). This lower bound holds even if the inputs are restricted to Boolean values. The proof of this result relies on a new method that allows us to encode more "program-bits" in the weights of a neural net than previously thought possible.