On the Convergence Rate of Training Recurrent Neural Networks

TL;DR

This paper provides theoretical analysis demonstrating that stochastic gradient descent can efficiently train recurrent neural networks with ReLU activations, showing their ability to memorize data and avoid common training issues.

Contribution

It extends convergence analysis to multi-layer RNNs, offering new tools for understanding ReLU networks and their training dynamics.

Findings

SGD achieves linear convergence in training RNNs with sufficiently many neurons.

ReLU activations prevent exponential gradient explosion or vanishing.

Theoretical evidence of RNNs' ability to memorize data.

Abstract

How can local-search methods such as stochastic gradient descent (SGD) avoid bad local minima in training multi-layer neural networks? Why can they fit random labels even given non-convex and non-smooth architectures? Most existing theory only covers networks with one hidden layer, so can we go deeper? In this paper, we focus on recurrent neural networks (RNNs) which are multi-layer networks widely used in natural language processing. They are harder to analyze than feedforward neural networks, because the $\textit{same}$ recurrent unit is repeatedly applied across the entire time horizon of length $L$, which is analogous to feedforward networks of depth $L$. We show when the number of neurons is sufficiently large, meaning polynomial in the training data size and in $L$, then SGD is capable of minimizing the regression loss in the linear convergence rate. This gives theoretical evidence of how RNNs can memorize data. More importantly, in this paper we build general toolkits to analyze multi-layer networks with ReLU activations. For instance, we prove why ReLU activations can prevent exponential gradient explosion or vanishing, and build a perturbation theory to analyze first-order approximation of multi-layer networks.