Papers by P. Koiran and E.D. Sontag

This paper provides lower and upper bounds for the VC dimension of recurrent networks. Several types of activation functions are discussed, including threshold, polynomial, piecewise-polynomial and sigmoidal functions. The bounds depend on two independent parameters: the number w of weights in the network, and the length k of the input sequence. Ignoring multiplicative constants, the main results say roughly the following: 1. For architectures whose activation is any fixed nonlinear polynomial, the VC dimension is proportional to wk. 2. For architectures whose activation is any fixed piecewise polynomial, the VC dimension is between wk and w**2k. 3. For architectures with threshold activations, the VC dimension is between wlog(k/w) and the smallest of wklog(wk) and w**2+wlog(wk). 4. For the standard sigmoid tanh(x), the VC dimension is between wk and w**4 k**2.

This paper shows that neural networks which use continuous activation functions have VC dimension at least as large as the square of the number of weights w. This result settles the open question of whether whether the well-known O(w log w) bound, known for hard-threshold nets, also held for more general sigmoidal nets. Implications for the number of samples needed for valid generalization are discussed.