On the Provable Generalization of Recurrent Neural Networks

Recurrent Neural Network (RNN) is a fundamental structure in deep learning. Recently, some works study the training process of over-parameterized neural networks, and show that over-parameterized networks can learn functions in some notable concept classes with a provable generalization error bound. In this paper, we analyze the training and generalization for RNNs with random initialization, and provide the following improvements over recent works: (1) For a RNN with input sequence x = (X-1, X-2,..., X-L), previous works study to learn functions that are summation of f(ss(T)(l) X-l) and require normalized conditions that ||X-l|| <= epsilon with some very small. depending on the complexity of f. In this paper, using detailed analysis about the neural tangent kernel matrix, we prove a generalization error bound to learn such functions without normalized conditions and show that some notable concept classes are learnable with the numbers of iterations and samples scaling almost-polynomially in the input length L. (2) Moreover, we prove a novel result to learn N-variables functions of input sequence with the form f(ss(T) [X-l1,..., X-lN]), which do not belong to the "additive" concept class, i,e., the summation of function f(X-l). And we show that when either N or l(0) = max(l(1),.., l(N)) - min(l(1),.., l(N)) is small, f(ss(T) [X-l1,..., X-lN]) will be learnable with the number iterations and samples scaling almost-polynomially in the input length L.