Generalization and Risk Bounds for Recurrent Neural Networks

TL;DR

A new generalization error bound for vanilla RNNs is established, a unified framework to calculate the Rademacher complexity is provided, and a sharp estimation error bound for RNN-based estimators obtained through empirical risk minimization in multi-class classification problems when the loss function satisfies a Bernstein condition is derived.

Abstract

Recurrent Neural Networks (RNNs) have achieved great success in the prediction of sequential data. However, their theoretical studies are still lagging behind because of their complex interconnected structures. In this paper, we establish a new generalization error bound for vanilla RNNs, and provide a unified framework to calculate the Rademacher complexity that can be applied to a variety of loss functions. When the ramp loss is used, we show that our bound is tighter than the existing bounds based on the same assumptions on the Frobenius and spectral norms of the weight matrices and a few mild conditions. Our numerical results show that our new generalization bound is the tightest among all existing bounds in three public datasets. Our bound improves the second tightest one by an average percentage of 13.80% and 3.01% when the $\tanh$ and ReLU activation functions are used, respectively. Moreover, we derive a sharp estimation error bound for RNN-based estimators obtained through empirical risk minimization (ERM) in multi-class classification problems when the loss function satisfies a Bernstein condition.

Theorems & Definitions (15)

• Definition 1
• Lemma 1
• Theorem 1
• Corollary 1
• Theorem 2
• Lemma 2
• Theorem 3
• Theorem 4
• Lemma A.1
• Lemma A.2