Depth-Width Trade-offs for Neural Networks via Topological Entropy

TL;DR

This paper establishes a novel connection between neural network expressivity and topological entropy, providing bounds on network complexity and insights into depth-width trade-offs in deep learning models.

Contribution

It introduces a new theoretical framework linking topological entropy to neural network expressivity and derives bounds on network parameters based on this connection.

Findings

Topological entropy of ReLU networks is bounded by O(l log m)

Network size has an exponential lower bound related to the entropy of the target function

Discusses relationships between entropy, oscillations, periods, and Lipschitz constants

Abstract

One of the central problems in the study of deep learning theory is to understand how the structure properties, such as depth, width and the number of nodes, affect the expressivity of deep neural networks. In this work, we show a new connection between the expressivity of deep neural networks and topological entropy from dynamical system, which can be used to characterize depth-width trade-offs of neural networks. We provide an upper bound on the topological entropy of neural networks with continuous semi-algebraic units by the structure parameters. Specifically, the topological entropy of ReLU network with $l$ layers and $m$ nodes per layer is upper bounded by $O(l\log m)$. Besides, if the neural network is a good approximation of some function $f$, then the size of the neural network has an exponential lower bound with respect to the topological entropy of $f$. Moreover, we discuss the relationship between topological entropy, the number of oscillations, periods and Lipschitz constant.