# Support Vectors and Gradient Dynamics of Single-Neuron ReLU Networks

arXiv: https://arxiv.org/abs/2202.05510 · HTML: https://mlsanity.com/paper/2202.05510/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 40 references that are in the map.
- First posted on arXiv 11 Feb 2022.

## Builds on (most cited first, 27 of 27)

- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Fantastic Generalization Measures and Where to Find Them](https://mlsanity.com/paper/1912.02178/) (2019)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [Stronger generalization bounds for deep nets via a compression approach](https://mlsanity.com/paper/1802.05296/) (2018)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018)
- [Two models of double descent for weak features](https://mlsanity.com/paper/1903.07571/) (2019)
- [Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced](https://mlsanity.com/paper/1806.00900/) (2018)
- [Complexity of Linear Regions in Deep Networks](https://mlsanity.com/paper/1901.09021/) (2019)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [Towards Resolving the Implicit Bias of Gradient Descent for Matrix Factorization: Greedy Low-Rank Learning](https://mlsanity.com/paper/2012.09839/) (2020)
- [Implicit Regularization in Deep Learning May Not Be Explainable by Norms](https://mlsanity.com/paper/2005.06398/) (2020)
- [Data-dependent Sample Complexity of Deep Neural Networks via Lipschitz Augmentation](https://mlsanity.com/paper/1905.03684/) (2019)
- [On the Implicit Bias of Initialization Shape: Beyond Infinitesimal Mirror Descent](https://mlsanity.com/paper/2102.09769/) (2021)
- [Deterministic PAC-Bayesian generalization bounds for deep networks via generalizing noise-resilience](https://mlsanity.com/paper/1905.13344/) (2019)
- [Implicit Regularization in ReLU Networks with the Square Loss](https://mlsanity.com/paper/2012.05156/) (2020)
- [Learning a Single Neuron with Bias Using Gradient Descent](https://mlsanity.com/paper/2106.01101/) (2021)
- [Traversing the Local Polytopes of ReLU Neural Networks: A Unified Approach for Network Verification](https://mlsanity.com/paper/2111.08922/) (2021)
- [Unsupervised Representation Learning via Neural Activation Coding](https://mlsanity.com/paper/2112.04014/) (2021)
- [Regularization by Misclassification in ReLU Neural Networks](https://mlsanity.com/paper/2111.02154/) (2021)

## Abstract (arXiv)

Understanding implicit bias of gradient descent for generalization capability of ReLU networks has been an important research topic in machine learning research. Unfortunately, even for a single ReLU neuron trained with the square loss, it was recently shown impossible to characterize the implicit regularization in terms of a norm of model parameters (Vardi & Shamir, 2021). In order to close the gap toward understanding intriguing generalization behavior of ReLU networks, here we examine the gradient flow dynamics in the parameter space when training single-neuron ReLU networks. Specifically, we discover an implicit bias in terms of support vectors, which plays a key role in why and how ReLU networks generalize well. Moreover, we analyze gradient flows with respect to the magnitude of the norm of initialization, and show that the norm of the learned weight strictly increases through the gradient flow. Lastly, we prove the global convergence of single ReLU neuron for d = 2 case.