# Convergence of optimizers implies eigenvalues filtering at equilibrium

arXiv: https://arxiv.org/abs/2510.09034 · HTML: https://mlsanity.com/paper/2510.09034/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 52 references that are in the map.
- First posted on arXiv 10 Oct 2025.

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

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [Self-Stabilization: The Implicit Bias of Gradient Descent at the Edge of Stability](https://mlsanity.com/paper/2209.15594/) (2022)
- [Towards Understanding Sharpness-Aware Minimization](https://mlsanity.com/paper/2206.06232/) (2022)
- [Understanding Gradient Descent on Edge of Stability in Deep Learning](https://mlsanity.com/paper/2205.09745/) (2022)
- [Adaptive Gradient Methods at the Edge of Stability](https://mlsanity.com/paper/2207.14484/) (2022)
- [A Modern Look at the Relationship between Sharpness and Generalization](https://mlsanity.com/paper/2302.07011/) (2023)
- [Understanding the unstable convergence of gradient descent](https://mlsanity.com/paper/2204.01050/) (2022)
- [Accelerated Gradient Descent Escapes Saddle Points Faster than Gradient Descent](https://mlsanity.com/paper/1711.10456/) (2017)
- [On the Maximum Hessian Eigenvalue and Generalization](https://mlsanity.com/paper/2206.10654/) (2022)
- [Implicit Bias of Gradient Descent for Logistic Regression at the Edge of Stability](https://mlsanity.com/paper/2305.11788/) (2023)
- [Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency](https://mlsanity.com/paper/2402.15926/) (2024)
- [Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions](https://mlsanity.com/paper/1605.00405/) (2016)
- [Edge of Stochastic Stability: Revisiting the Edge of Stability for SGD](https://mlsanity.com/paper/2412.20553/) (2024)
- [The Crucial Role of Normalization in Sharpness-Aware Minimization](https://mlsanity.com/paper/2305.15287/) (2023)
- [SAM operates far from home: eigenvalue regularization as a dynamical phenomenon](https://mlsanity.com/paper/2302.08692/) (2023)
- [Sharpness-Aware Minimization Efficiently Selects Flatter Minima Late in Training](https://mlsanity.com/paper/2410.10373/) (2024)
- [High dimensional analysis reveals conservative sharpening and a stochastic edge of stability](https://mlsanity.com/paper/2404.19261/) (2024)
- [Deep linear networks for regression are implicitly regularized towards flat minima](https://mlsanity.com/paper/2405.13456/) (2024)
- [Taking a Big Step: Large Learning Rates in Denoising Score Matching Prevent Memorization](https://mlsanity.com/paper/2502.03435/) (2025)
- [Stability Analysis of Sharpness-Aware Minimization](https://mlsanity.com/paper/2301.06308/) (2023)
- [Heavy-ball Algorithms Always Escape Saddle Points](https://mlsanity.com/paper/1907.09697/) (2019)
- [Stable Minima Cannot Overfit in Univariate ReLU Networks: Generalization by Large Step Sizes](https://mlsanity.com/paper/2406.06838/) (2024)
- [Stabilizing Sharpness-aware Minimization Through A Simple Renormalization Strategy](https://mlsanity.com/paper/2401.07250/) (2024)
- [Inertial Newton Algorithms Avoiding Strict Saddle Points](https://mlsanity.com/paper/2111.04596/) (2021)

## Abstract (arXiv)

Ample empirical evidence in deep neural network training suggests that a variety of optimizers tend to find nearly global optima. In this article, we adopt the reversed perspective that convergence to an arbitrary point is assumed rather than proven, focusing on the consequences of this assumption. From this viewpoint, in line with recent advances on the edge-of-stability phenomenon, we argue that different optimizers effectively act as eigenvalue filters determined by their hyperparameters. Specifically, the standard gradient descent method inherently avoids the sharpest minima, whereas Sharpness-Aware Minimization (SAM) algorithms go even further by actively favoring wider basins. Inspired by these insights, we propose two novel algorithms that exhibit enhanced eigenvalue filtering, effectively promoting wider minima. Our theoretical analysis leverages a generalized Hadamard--Perron stable manifold theorem and applies to general semialgebraic C2 functions, without requiring additional non-degeneracy conditions or global Lipschitz bound assumptions. We support our conclusions with numerical experiments on feed-forward neural networks.