# Piecewise Strong Convexity of Neural Networks

arXiv: https://arxiv.org/abs/1810.12805 · HTML: https://mlsanity.com/paper/1810.12805/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Oct 2018.

## Built on by (most cited first, 13 of 13)

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- [On generalization bounds for deep networks based on loss surface implicit regularization](https://mlsanity.com/paper/2201.04545/) (2022)
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- [On The Relationship Between Continual Learning and Long-Tailed Recognition](https://mlsanity.com/paper/2306.13275/) (2023)
- [Data value estimation on private gradients](https://mlsanity.com/paper/2412.17008/) (2024)
- [M-HOF-Opt: Multi-Objective Hierarchical Output Feedback Optimization via Multiplier Induced Loss Landscape Scheduling](https://mlsanity.com/paper/2403.13728/) (2024)
- [WTNN: Weibull-Tailored Neural Networks for survival analysis](https://mlsanity.com/paper/2512.09163/) (2025)
- [Obliviate: Efficient Unlearning in Recommender Systems](https://mlsanity.com/paper/2607.22665/) (2026)

## Abstract (arXiv)

We study the loss surface of a feed-forward neural network with ReLU non-linearities, regularized with weight decay. We show that the regularized loss function is piecewise strongly convex on an important open set which contains, under some conditions, all of its global minimizers. This is used to prove that local minima of the regularized loss function in this set are isolated, and that every differentiable critical point in this set is a local minimum, partially addressing an open problem given at the Conference on Learning Theory (COLT) 2015; our result is also applied to linear neural networks to show that with weight decay regularization, there are no non-zero critical points in a norm ball obtaining training error below a given threshold. We also include an experimental section where we validate our theoretical work and show that the regularized loss function is almost always piecewise strongly convex when restricted to stochastic gradient descent trajectories for three standard image classification problems.