  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 $f(\beta^T_lX_l)$ and require
normalized conditions that $||X_l||\leq\epsilon$ with some very small
$\epsilon$ 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 $f(\beta^T[X_{l_1},...,X_{l_N}])$, 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, $f(\beta^T[X_{l_1},...,X_{l_N}])$ will be learnable with the number
iterations and samples scaling almost-polynomially in the input length $L$.

Machine Learning;Machine Learning