# Better Depth-Width Trade-offs for Neural Networks through the lens of Dynamical Systems

arXiv: https://arxiv.org/abs/2003.00777 · HTML: https://mlsanity.com/paper/2003.00777/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Mar 2020.

## Built on by (most cited first, 9 of 9)

- [Statistically Meaningful Approximation: a Case Study on Approximating Turing Machines with Transformers](https://mlsanity.com/paper/2107.13163/) (2021)
- [A Primer on Bayesian Neural Networks: Review and Debates](https://mlsanity.com/paper/2309.16314/) (2023)
- [The Connection Between Approximation, Depth Separation and Learnability in Neural Networks](https://mlsanity.com/paper/2102.00434/) (2021)
- [Bifurcations and loss jumps in RNN training](https://mlsanity.com/paper/2310.17561/) (2023)
- [On Scrambling Phenomena for Randomly Initialized Recurrent Networks](https://mlsanity.com/paper/2210.05212/) (2022)
- [Depth Separation in Norm-Bounded Infinite-Width Neural Networks](https://mlsanity.com/paper/2402.08808/) (2024)
- [Expressivity of Neural Networks via Chaotic Itineraries beyond Sharkovsky's Theorem](https://mlsanity.com/paper/2110.10295/) (2021)
- [Understanding the Trade-offs in Accuracy and Uncertainty Quantification: Architecture and Inference Choices in Bayesian Neural Networks](https://mlsanity.com/paper/2503.11808/) (2025)
- [Constants of Motion: The Antidote to Chaos in Optimization and Game Dynamics](https://mlsanity.com/paper/2109.03974/) (2021)

## Abstract (arXiv)

The expressivity of neural networks as a function of their depth, width and type of activation units has been an important question in deep learning theory. Recently, depth separation results for ReLU networks were obtained via a new connection with dynamical systems, using a generalized notion of fixed points of a continuous map f, called periodic points. In this work, we strengthen the connection with dynamical systems and we improve the existing width lower bounds along several aspects. Our first main result is period-specific width lower bounds that hold under the stronger notion of L1-approximation error, instead of the weaker classification error. Our second contribution is that we provide sharper width lower bounds, still yielding meaningful exponential depth-width separations, in regimes where previous results wouldn't apply. A byproduct of our results is that there exists a universal constant characterizing the depth-width trade-offs, as long as f has odd periods. Technically, our results follow by unveiling a tighter connection between the following three quantities of a given function: its period, its Lipschitz constant and the growth rate of the number of oscillations arising under compositions of the function f with itself.