# Deep Learning Generalization, Extrapolation, and Over-parameterization

arXiv: https://arxiv.org/abs/2203.10366 · HTML: https://mlsanity.com/paper/2203.10366/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 21 references that are in the map.
- First posted on arXiv 19 Mar 2022.

## Builds on (most cited first, 17 of 17)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Certified Adversarial Robustness via Randomized Smoothing](https://mlsanity.com/paper/1902.02918/) (2019)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Robustness May Be at Odds with Accuracy](https://mlsanity.com/paper/1805.12152/) (2018)
- [The Limitations of Deep Learning in Adversarial Settings](https://mlsanity.com/paper/1511.07528/) (2015)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Just Interpolate: Kernel "Ridgeless" Regression Can Generalize](https://mlsanity.com/paper/1808.00387/) (2018)
- [To understand deep learning we need to understand kernel learning](https://mlsanity.com/paper/1802.01396/) (2018)
- [Towards Understanding the Role of Over-Parametrization in Generalization of Neural Networks](https://mlsanity.com/paper/1805.12076/) (2018)
- [Are adversarial examples inevitable?](https://mlsanity.com/paper/1809.02104/) (2018)
- [Predicting the Generalization Gap in Deep Networks with Margin Distributions](https://mlsanity.com/paper/1810.00113/) (2018)
- [Large Margin Deep Networks for Classification](https://mlsanity.com/paper/1803.05598/) (2018)
- [How do infinite width bounded norm networks look in function space?](https://mlsanity.com/paper/1902.05040/) (2019)
- [Overfitting or perfect fitting? Risk bounds for classification and regression rules that interpolate](https://mlsanity.com/paper/1806.05161/) (2018)
- [On the Expressive Power of Deep Polynomial Neural Networks](https://mlsanity.com/paper/1905.12207/) (2019)

## Built on by (most cited first, 1 of 1)

- [Escaping Local Minima Provably in Non-convex Matrix Sensing: A Deterministic Framework via Simulated Lifting](https://mlsanity.com/paper/2602.05887/) (2026)

## Abstract (arXiv)

We study the generalization of over-parameterized deep networks (for image classification) in relation to the convex hull of their training sets. Despite their great success, generalization of deep networks is considered a mystery. These models have orders of magnitude more parameters than their training samples, and they can achieve perfect accuracy on their training sets, even when training images are randomly labeled, or the contents of images are replaced with random noise. The training loss function of these models has infinite number of near zero minimizers, where only a small subset of those minimizers generalize well. Overall, it is not clear why models need to be over-parameterized, why we should use a very specific training regime to train them, and why their classifications are so susceptible to imperceivable adversarial perturbations (phenomenon known as adversarial vulnerability) citepapernot2016limitations,shafahi2018adversarial,tsipras2018robustness. Some recent studies have made advances in answering these questions, however, they only consider interpolation. We show that interpolation is not adequate to understand generalization of deep networks and we should broaden our perspective.