# Implicit Bias of Gradient Descent for Logistic Regression at the Edge of Stability

arXiv: https://arxiv.org/abs/2305.11788 · HTML: https://mlsanity.com/paper/2305.11788/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 41 references that are in the map.
- First posted on arXiv 19 May 2023.

## Builds on (most cited first, 29 of 29)

- [Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour](https://mlsanity.com/paper/1706.02677/) (2017)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [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)
- [The large learning rate phase of deep learning: the catapult mechanism](https://mlsanity.com/paper/2003.02218/) (2020)
- [Gradient descent aligns the layers of deep linear networks](https://mlsanity.com/paper/1810.02032/) (2018)
- [Self-Stabilization: The Implicit Bias of Gradient Descent at the Edge of Stability](https://mlsanity.com/paper/2209.15594/) (2022)
- [Understanding Gradient Descent on Edge of Stability in Deep Learning](https://mlsanity.com/paper/2205.09745/) (2022)
- [Convergence of Gradient Descent on Separable Data](https://mlsanity.com/paper/1803.01905/) (2018)
- [Understanding the unstable convergence of gradient descent](https://mlsanity.com/paper/2204.01050/) (2022)
- [Understanding the Generalization Benefit of Normalization Layers: Sharpness Reduction](https://mlsanity.com/paper/2206.07085/) (2022)
- [Risk and parameter convergence of logistic regression](https://mlsanity.com/paper/1803.07300/) (2018)
- [A Walk with SGD](https://mlsanity.com/paper/1802.08770/) (2018)
- [SGD with Large Step Sizes Learns Sparse Features](https://mlsanity.com/paper/2210.05337/) (2022)
- [Stochastic Gradient Descent on Separable Data: Exact Convergence with a Fixed Learning Rate](https://mlsanity.com/paper/1806.01796/) (2018)
- [Analyzing Sharpness along GD Trajectory: Progressive Sharpening and Edge of Stability](https://mlsanity.com/paper/2207.12678/) (2022)
- [Understanding Edge-of-Stability Training Dynamics with a Minimalist Example](https://mlsanity.com/paper/2210.03294/) (2022)
- [Lexicographic and Depth-Sensitive Margins in Homogeneous and Non-Homogeneous Deep Models](https://mlsanity.com/paper/1905.07325/) (2019)
- [The Dynamics of Sharpness-Aware Minimization: Bouncing Across Ravines and Drifting Towards Wide Minima](https://mlsanity.com/paper/2210.01513/) (2022)
- [Learning threshold neurons via the "edge of stability"](https://mlsanity.com/paper/2212.07469/) (2022)
- [Large Learning Rate Tames Homogeneity: Convergence and Balancing Effect](https://mlsanity.com/paper/2110.03677/) (2021)
- [Beyond the Quadratic Approximation: the Multiscale Structure of Neural Network Loss Landscapes](https://mlsanity.com/paper/2204.11326/) (2022)
- [Fast Margin Maximization via Dual Acceleration](https://mlsanity.com/paper/2107.00595/) (2021)
- [Stochasticity of Deterministic Gradient Descent: Large Learning Rate for Multiscale Objective Function](https://mlsanity.com/paper/2002.06189/) (2020)

## Built on by (most cited first, 30 of 42)

- [Heavy-Tailed Class Imbalance and Why Adam Outperforms Gradient Descent on Language Models](https://mlsanity.com/paper/2402.19449/) (2024)
- [The Implicit Regularization of Dynamical Stability in Stochastic Gradient Descent](https://mlsanity.com/paper/2305.17490/) (2023)
- [Implicit Bias of Spectral Descent and Muon on Multiclass Separable Data](https://mlsanity.com/paper/2502.04664/) (2025)
- [Does SGD really happen in tiny subspaces?](https://mlsanity.com/paper/2405.16002/) (2024)
- [The Implicit Bias of Adam on Separable Data](https://mlsanity.com/paper/2406.10650/) (2024)
- [Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency](https://mlsanity.com/paper/2402.15926/) (2024)
- [Trajectory Alignment: Understanding the Edge of Stability Phenomenon via Bifurcation Theory](https://mlsanity.com/paper/2307.04204/) (2023)
- [Universal Sharpness Dynamics in Neural Network Training: Fixed Point Analysis, Edge of Stability, and Route to Chaos](https://mlsanity.com/paper/2311.02076/) (2023)
- [From Stability to Chaos: Analyzing Gradient Descent Dynamics in Quadratic Regression](https://mlsanity.com/paper/2310.01687/) (2023)
- [Large Stepsize Gradient Descent for Non-Homogeneous Two-Layer Networks: Margin Improvement and Fast Optimization](https://mlsanity.com/paper/2406.08654/) (2024)
- [Learning Associative Memories with Gradient Descent](https://mlsanity.com/paper/2402.18724/) (2024)
- [Gradient Descent on Logistic Regression with Non-Separable Data and Large Step Sizes](https://mlsanity.com/paper/2406.05033/) (2024)
- [Benefits of Early Stopping in Gradient Descent for Overparameterized Logistic Regression](https://mlsanity.com/paper/2502.13283/) (2025)
- [Large Stepsizes Accelerate Gradient Descent for Regularized Logistic Regression](https://mlsanity.com/paper/2506.02336/) (2025)
- [Benign Oscillation of Stochastic Gradient Descent with Large Learning Rates](https://mlsanity.com/paper/2310.17074/) (2023)
- [Minimax Optimal Convergence of Gradient Descent in Logistic Regression via Large and Adaptive Stepsizes](https://mlsanity.com/paper/2504.04105/) (2025)
- [Understanding Sharpness Dynamics in NN Training with a Minimalist Example: The Effects of Dataset Difficulty, Depth, Stochasticity, and More](https://mlsanity.com/paper/2506.06940/) (2025)
- [Gradient Descent on Logistic Regression: Do Large Step-Sizes Work with Data on the Sphere?](https://mlsanity.com/paper/2507.11228/) (2025)
- [Achieving Margin Maximization Exponentially Fast via Progressive Norm Rescaling](https://mlsanity.com/paper/2311.14387/) (2023)
- [Convergence Rates for Gradient Descent on the Edge of Stability in Overparametrised Least Squares](https://mlsanity.com/paper/2510.17506/) (2025)
- [Gradient Descent with Polyak's Momentum Finds Flatter Minima via Large Catapults](https://mlsanity.com/paper/2311.15051/) (2023)
- [Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective](https://mlsanity.com/paper/2601.06597/) (2026)
- [Implicit Bias of Mirror Flow in Homogeneous Neural Networks: Sparse and Dense Feature Learning](https://mlsanity.com/paper/2605.19458/) (2026)
- [Edge Flow: A Tractable and Predictive Continuous-Time Model for Gradient Descent at the Edge of Stability](https://mlsanity.com/paper/2606.18080/) (2026)
- [Variational Learning Finds Flatter Solutions at the Edge of Stability](https://mlsanity.com/paper/2506.12903/) (2025)
- [Constant Stepsize Local GD for Logistic Regression: Acceleration by Instability](https://mlsanity.com/paper/2506.13974/) (2025)
- [Diagonalizing the Softmax: Hadamard Initialization for Tractable Cross-Entropy Dynamics](https://mlsanity.com/paper/2512.04006/) (2025)
- [Training on the Edge of Stability Is Caused by Layerwise Jacobian Alignment](https://mlsanity.com/paper/2406.00127/) (2024)
- [Any-stepsize Gradient Descent for Separable Data under Fenchel-Young Losses](https://mlsanity.com/paper/2502.04889/) (2025)
- [Tight Bounds for Logistic Regression with Large Stepsize Gradient Descent in Low Dimension](https://mlsanity.com/paper/2602.12471/) (2026)

## Abstract (arXiv)

Recent research has observed that in machine learning optimization, gradient descent (GD) often operates at the edge of stability (EoS) [Cohen, et al., 2021], where the stepsizes are set to be large, resulting in non-monotonic losses induced by the GD iterates. This paper studies the convergence and implicit bias of constant-stepsize GD for logistic regression on linearly separable data in the EoS regime. Despite the presence of local oscillations, we prove that the logistic loss can be minimized by GD with any constant stepsize over a long time scale. Furthermore, we prove that with any constant stepsize, the GD iterates tend to infinity when projected to a max-margin direction (the hard-margin SVM direction) and converge to a fixed vector that minimizes a strongly convex potential when projected to the orthogonal complement of the max-margin direction. In contrast, we also show that in the EoS regime, GD iterates may diverge catastrophically under the exponential loss, highlighting the superiority of the logistic loss. These theoretical findings are in line with numerical simulations and complement existing theories on the convergence and implicit bias of GD for logistic regression, which are only applicable when the stepsizes are sufficiently small.