Theoretical Analysis of the Advantage of Deepening Neural Networks

TL;DR

This paper introduces two new criteria to analyze the expressivity of deep neural networks, demonstrating that increasing layers enhances expressivity more effectively than increasing units per layer.

Contribution

It proposes novel criteria for evaluating neural network expressivity independently of learning efficiency, providing theoretical insights into the benefits of deepening networks.

Findings

Increasing layers improves expressivity more than increasing units.

The criteria evaluate approximation accuracy and linear region properties.

Deep networks' expressivity is crucial for their performance.

Abstract

We propose two new criteria to understand the advantage of deepening neural networks. It is important to know the expressivity of functions computable by deep neural networks in order to understand the advantage of deepening neural networks. Unless deep neural networks have enough expressivity, they cannot have good performance even though learning is successful. In this situation, the proposed criteria contribute to understanding the advantage of deepening neural networks since they can evaluate the expressivity independently from the efficiency of learning. The first criterion shows the approximation accuracy of deep neural networks to the target function. This criterion has the background that the goal of deep learning is approximating the target function by deep neural networks. The second criterion shows the property of linear regions of functions computable by deep neural networks. This criterion has the background that deep neural networks whose activation functions are piecewise linear are also piecewise linear. Furthermore, by the two criteria, we show that to increase layers is more effective than to increase units at each layer on improving the expressivity of deep neural networks.