Towards Understanding the Importance of Shortcut Connections in Residual Networks

TL;DR

This paper investigates why residual networks train efficiently, showing that gradient descent with proper normalization can find global optima despite non-convexity, especially with specific initializations.

Contribution

It provides a theoretical analysis demonstrating convergence guarantees for training a two-layer residual network under certain conditions.

Findings

Gradient descent with normalization avoids spurious local optima.

Proper initialization ensures polynomial-time convergence.

Numerical experiments support the theoretical results.

Abstract

Residual Network (ResNet) is undoubtedly a milestone in deep learning. ResNet is equipped with shortcut connections between layers, and exhibits efficient training using simple first order algorithms. Despite of the great empirical success, the reason behind is far from being well understood. In this paper, we study a two-layer non-overlapping convolutional ResNet. Training such a network requires solving a non-convex optimization problem with a spurious local optimum. We show, however, that gradient descent combined with proper normalization, avoids being trapped by the spurious local optimum, and converges to a global optimum in polynomial time, when the weight of the first layer is initialized at 0, and that of the second layer is initialized arbitrarily in a ball. Numerical experiments are provided to support our theory.