On the Provable Generalization of Recurrent Neural Networks

TL;DR

This paper provides theoretical guarantees for the generalization of over-parameterized RNNs, demonstrating learnability of complex functions without normalization constraints and analyzing the impact of input sequence structure.

Contribution

It introduces new generalization bounds for RNNs trained with random initialization, extending learnability results to non-additive and more complex function classes.

Findings

Learnability of certain functions without normalized input conditions

Almost-polynomial scaling of iterations and samples with input length

Extension to non-additive functions of input sequences

Abstract

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 $L$.