# On the ISS Property of the Gradient Flow for Single Hidden-Layer Neural Networks with Linear Activations

arXiv: https://arxiv.org/abs/2305.09904 · HTML: https://mlsanity.com/paper/2305.09904/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 31 references that are in the map.
- First posted on arXiv 17 May 2023.

## Builds on (most cited first, 23 of 23)

- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Adversarial Examples Are Not Bugs, They Are Features](https://mlsanity.com/paper/1905.02175/) (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 Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Deep Learning without Poor Local Minima](https://mlsanity.com/paper/1605.07110/) (2016)
- [To understand deep learning we need to understand kernel learning](https://mlsanity.com/paper/1802.01396/) (2018)
- [Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced](https://mlsanity.com/paper/1806.00900/) (2018)
- [Rademacher Complexity for Adversarially Robust Generalization](https://mlsanity.com/paper/1810.11914/) (2018)
- [Precise Tradeoffs in Adversarial Training for Linear Regression](https://mlsanity.com/paper/2002.10477/) (2020)
- [Benign Overfitting without Linearity: Neural Network Classifiers Trained by Gradient Descent for Noisy Linear Data](https://mlsanity.com/paper/2202.05928/) (2022)
- [How benign is benign overfitting?](https://mlsanity.com/paper/2007.04028/) (2020)
- [The Curious Case of Adversarially Robust Models: More Data Can Help, Double Descend, or Hurt Generalization](https://mlsanity.com/paper/2002.11080/) (2020)
- [Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions](https://mlsanity.com/paper/1605.00405/) (2016)
- [Overparameterized Linear Regression under Adversarial Attacks](https://mlsanity.com/paper/2204.06274/) (2022)
- [Training Linear Neural Networks: Non-Local Convergence and Complexity Results](https://mlsanity.com/paper/2002.09852/) (2020)
- [A Geometric Approach of Gradient Descent Algorithms in Linear Neural Networks](https://mlsanity.com/paper/1811.03568/) (2018)
- [Remarks on input to state stability of perturbed gradient flows, motivated by model-free feedback control learning](https://mlsanity.com/paper/2108.02632/) (2021)
- [Extending the step-size restriction for gradient descent to avoid strict saddle points](https://mlsanity.com/paper/1908.01753/) (2019)

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

- [Ensuring Both Positivity and Stability Using Sector-Bounded Nonlinearity for Systems with Neural Network Controllers](https://mlsanity.com/paper/2406.12744/) (2024)

## Abstract (arXiv)

Recent research in neural networks and machine learning suggests that using many more parameters than strictly required by the initial complexity of a regression problem can result in more accurate or faster-converging models -- contrary to classical statistical belief. This phenomenon, sometimes known as ``benign overfitting'', raises questions regarding in what other ways might overparameterization affect the properties of a learning problem. In this work, we investigate the effects of overfitting on the robustness of gradient-descent training when subject to uncertainty on the gradient estimation. This uncertainty arises naturally if the gradient is estimated from noisy data or directly measured. Our object of study is a linear neural network with a single, arbitrarily wide, hidden layer and an arbitrary number of inputs and outputs. In this paper we solve the problem for the case where the input and output of our neural-network are one-dimensional, deriving sufficient conditions for robustness of our system based on necessary and sufficient conditions for convergence in the undisturbed case. We then show that the general overparametrized formulation introduces a set of spurious equilibria which lay outside the set where the loss function is minimized, and discuss directions of future work that might extend our current results for more general formulations.