Depth-Width Trade-offs for ReLU Networks via Sharkovsky's Theorem

Vaggos Chatziafratis, Sai Ganesh Nagarajan, Ioannis Panageas, Xiao Wang

TL;DR: In this work, we point to a new connection between DNNs expressivity and Sharkovsky’s Theorem from dynamical systems, that enables us to characterize the depth-width trade-offs of ReLU networks

Abstract:

Understanding the representational power of Deep Neural Networks (DNNs) and how their structural properties (e.g., depth, width, type of activation unit) affect the functions they can compute, has been an important yet challenging question in deep learning and approximation theory. In a seminal paper, Telgarsky high- lighted the benefits of depth by presenting a family of functions (based on sim- ple triangular waves) for which DNNs achieve zero classification error, whereas shallow networks with fewer than exponentially many nodes incur constant error. Even though Telgarsky’s work reveals the limitations of shallow neural networks, it doesn’t inform us on why these functions are difficult to represent and in fact he states it as a tantalizing open question to characterize those functions that cannot be well-approximated by smaller depths.
In this work, we point to a new connection between DNNs expressivity and Sharkovsky’s Theorem from dynamical systems, that enables us to characterize the depth-width trade-offs of ReLU networks for representing functions based on the presence of a generalized notion of fixed points, called periodic points (a fixed point is a point of period 1). Motivated by our observation that the triangle waves used in Telgarsky’s work contain points of period 3 – a period that is special in that it implies chaotic behaviour based on the celebrated result by Li-Yorke – we proceed to give general lower bounds for the width needed to represent periodic functions as a function of the depth. Technically, the crux of our approach is based on an eigenvalue analysis of the dynamical systems associated with such functions.

Code: https://docs.google.com/document/d/1qr-sROZ7q93OhigF6CoPde5NQ901wI17wmnBvbZRT9s/edit?fbclid=IwAR1HwkNZ1g2QgMmTGRZ0ktCYNgeKKk91tvRNLb59QJwU3dRmuGCJbTNMwj0

Keywords: Depth-Width trade-offs, ReLU networks, chaos theory, Sharkovsky Theorem, dynamical systems