# The Implicit Bias of Minima Stability in Multivariate Shallow ReLU Networks

arXiv: https://arxiv.org/abs/2306.17499 · HTML: https://mlsanity.com/paper/2306.17499/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 57 references that are in the map.
- First posted on arXiv 30 Jun 2023.

## Builds on (most cited first, 30 of 33)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [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)
- [Don't Decay the Learning Rate, Increase the Batch Size](https://mlsanity.com/paper/1711.00489/) (2017)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018)
- [Implicit Bias of Gradient Descent on Linear Convolutional Networks](https://mlsanity.com/paper/1806.00468/) (2018)
- [On the Origin of Implicit Regularization in Stochastic Gradient Descent](https://mlsanity.com/paper/2101.12176/) (2021)
- [Gradient descent aligns the layers of deep linear networks](https://mlsanity.com/paper/1810.02032/) (2018)
- [Towards Explaining the Regularization Effect of Initial Large Learning Rate in Training Neural Networks](https://mlsanity.com/paper/1907.04595/) (2019)
- [The Power of Interpolation: Understanding the Effectiveness of SGD in Modern Over-parametrized Learning](https://mlsanity.com/paper/1712.06559/) (2017)
- [Implicit Gradient Regularization](https://mlsanity.com/paper/2009.11162/) (2020)
- [Implicit Regularization of Discrete Gradient Dynamics in Linear Neural Networks](https://mlsanity.com/paper/1904.13262/) (2019)
- [Implicit Regularization in Deep Learning May Not Be Explainable by Norms](https://mlsanity.com/paper/2005.06398/) (2020)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [Convergence of Gradient Descent on Separable Data](https://mlsanity.com/paper/1803.01905/) (2018)
- [On the Implicit Bias of Initialization Shape: Beyond Infinitesimal Mirror Descent](https://mlsanity.com/paper/2102.09769/) (2021)
- [Stochastic Gradient Descent on Separable Data: Exact Convergence with a Fixed Learning Rate](https://mlsanity.com/paper/1806.01796/) (2018)
- [Neural Network with Unbounded Activation Functions is Universal Approximator](https://mlsanity.com/paper/1505.03654/) (2015)
- [On Linear Stability of SGD and Input-Smoothness of Neural Networks](https://mlsanity.com/paper/2105.13462/) (2021)
- [Gradient descent follows the regularization path for general losses](https://mlsanity.com/paper/2006.11226/) (2020)
- [Lexicographic and Depth-Sensitive Margins in Homogeneous and Non-Homogeneous Deep Models](https://mlsanity.com/paper/1905.07325/) (2019)

## Built on by (most cited first, 15 of 15)

- [Taking a Big Step: Large Learning Rates in Denoising Score Matching Prevent Memorization](https://mlsanity.com/paper/2502.03435/) (2025)
- [Stable Minima Cannot Overfit in Univariate ReLU Networks: Generalization by Large Step Sizes](https://mlsanity.com/paper/2406.06838/) (2024)
- [How do Minimum-Norm Shallow Denoisers Look in Function Space?](https://mlsanity.com/paper/2311.06748/) (2023)
- [Analyzing Neural Network-Based Generative Diffusion Models through Convex Optimization](https://mlsanity.com/paper/2402.01965/) (2024)
- [Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Region](https://mlsanity.com/paper/2509.25351/) (2025)
- [Where Do Large Learning Rates Lead Us?](https://mlsanity.com/paper/2410.22113/) (2024)
- [When Are Bias-Free ReLU Networks Effectively Linear Networks?](https://mlsanity.com/paper/2406.12615/) (2024)
- [Generalization Below the Edge of Stability: The Role of Data Geometry](https://mlsanity.com/paper/2510.18120/) (2025)
- [On the Stability of Nonlinear Dynamics in GD and SGD: Beyond Quadratic Potentials](https://mlsanity.com/paper/2602.14789/) (2026)
- [Structured Transformations for Stable and Interpretable Neural Computation](https://mlsanity.com/paper/2508.00127/) (2025)
- [Sharpness of Minima in Deep Matrix Factorization](https://mlsanity.com/paper/2509.25783/) (2025)
- [Does Sparse Connectivity Improve Generalization? Convolutional Networks Below the Edge of Stability](https://mlsanity.com/paper/2603.04807/) (2026)
- [Slowing Down Forgetting in Continual Learning](https://mlsanity.com/paper/2411.06916/) (2024)
- [Variational Deep Learning via Implicit Regularization](https://mlsanity.com/paper/2505.20235/) (2025)
- [Residual-Stream Burden Shapes Representation Learning in Diffusion Transformers](https://mlsanity.com/paper/2609.33895/) (2026)

## Abstract (arXiv)

We study the type of solutions to which stochastic gradient descent converges when used to train a single hidden-layer multivariate ReLU network with the quadratic loss. Our results are based on a dynamical stability analysis. In the univariate case, it was shown that linearly stable minima correspond to network functions (predictors), whose second derivative has a bounded weighted L1 norm. Notably, the bound gets smaller as the step size increases, implying that training with a large step size leads to `smoother' predictors. Here we generalize this result to the multivariate case, showing that a similar result applies to the Laplacian of the predictor. We demonstrate the tightness of our bound on the MNIST dataset, and show that it accurately captures the behavior of the solutions as a function of the step size. Additionally, we prove a depth separation result on the approximation power of ReLU networks corresponding to stable minima of the loss. Specifically, although shallow ReLU networks are universal approximators, we prove that stable shallow networks are not. Namely, there is a function that cannot be well-approximated by stable single hidden-layer ReLU networks trained with a non-vanishing step size. This is while the same function can be realized as a stable two hidden-layer ReLU network. Finally, we prove that if a function is sufficiently smooth (in a Sobolev sense) then it can be approximated arbitrarily well using shallow ReLU networks that correspond to stable solutions of gradient descent.