Optimization Landscape and Expressivity of Deep CNNs

TL;DR

This paper investigates the loss landscape and expressivity of deep CNNs, revealing how width and depth influence optimization and representational power, with wide networks having favorable loss surfaces and conditions for zero training error.

Contribution

It provides theoretical conditions for global minima in wide CNNs and shows that width and depth critically affect the loss landscape and expressiveness.

Findings

Wide CNNs produce linearly independent features at wide layers.

Almost all critical points in wide CNNs are global minima with zero training error.

Depth enhances representational power, while width smooths the loss landscape.

Abstract

We analyze the loss landscape and expressiveness of practical deep convolutional neural networks (CNNs) with shared weights and max pooling layers. We show that such CNNs produce linearly independent features at a "wide" layer which has more neurons than the number of training samples. This condition holds e.g. for the VGG network. Furthermore, we provide for such wide CNNs necessary and sufficient conditions for global minima with zero training error. For the case where the wide layer is followed by a fully connected layer we show that almost every critical point of the empirical loss is a global minimum with zero training error. Our analysis suggests that both depth and width are very important in deep learning. While depth brings more representational power and allows the network to learn high level features, width smoothes the optimization landscape of the loss function in the sense that a sufficiently wide network has a well-behaved loss surface with almost no bad local minima.