# Learning Functions: When Is Deep Better Than Shallow

arXiv: https://arxiv.org/abs/1603.00988 · HTML: https://mlsanity.com/paper/1603.00988/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Mar 2016.

## Built on by (most cited first, 19 of 19)

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- [Universal Approximation Property of Hamiltonian Deep Neural Networks](https://mlsanity.com/paper/2303.12147/) (2023)
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- [Cooperative Deep Q-learning Framework for Environments Providing Image Feedback](https://mlsanity.com/paper/2110.15305/) (2021)
- [Deep neural network approximation of composite functions without the curse of dimensionality](https://mlsanity.com/paper/2304.05790/) (2023)
- [Freely Long-Thinking Transformer (FraiLT)](https://mlsanity.com/paper/2401.11626/) (2024)
- [BEACONS: Bounded-Error, Algebraically-Composable Neural Solvers for Partial Differential Equations](https://mlsanity.com/paper/2602.14853/) (2026)

## Abstract (arXiv)

While the universal approximation property holds both for hierarchical and shallow networks, we prove that deep (hierarchical) networks can approximate the class of compositional functions with the same accuracy as shallow networks but with exponentially lower number of training parameters as well as VC-dimension. This theorem settles an old conjecture by Bengio on the role of depth in networks. We then define a general class of scalable, shift-invariant algorithms to show a simple and natural set of requirements that justify deep convolutional networks.