# Non-asymptotic implicit bias of logistic regression at early-stage gradient descent dynamics

arXiv: https://arxiv.org/abs/2608.04382 · HTML: https://mlsanity.com/paper/2608.04382/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 38 references that are in the map.
- First posted on arXiv 5 Aug 2026.

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

- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [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)
- [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)
- [Train longer, generalize better: closing the generalization gap in large batch training of neural networks](https://mlsanity.com/paper/1705.08741/) (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)
- [Polylogarithmic width suffices for gradient descent to achieve arbitrarily small test error with shallow ReLU networks](https://mlsanity.com/paper/1909.12292/) (2019)
- [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)
- [How do infinite width bounded norm networks look in function space?](https://mlsanity.com/paper/1902.05040/) (2019)
- [Benign Overfitting in Two-layer Convolutional Neural Networks](https://mlsanity.com/paper/2202.06526/) (2022)
- [Understanding the unstable convergence of gradient descent](https://mlsanity.com/paper/2204.01050/) (2022)
- [Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs](https://mlsanity.com/paper/2206.00939/) (2022)
- [Gradient descent follows the regularization path for general losses](https://mlsanity.com/paper/2006.11226/) (2020)
- [Max-Margin Token Selection in Attention Mechanism](https://mlsanity.com/paper/2306.13596/) (2023)
- [Implicit Bias of AdamW: ℓ_∞ Norm Constrained Optimization](https://mlsanity.com/paper/2404.04454/) (2024)
- [Learning Gradient Descent: Better Generalization and Longer Horizons](https://mlsanity.com/paper/1703.03633/) (2017)
- [Implicit Bias of Spectral Descent and Muon on Multiclass Separable Data](https://mlsanity.com/paper/2502.04664/) (2025)
- [SGD Finds then Tunes Features in Two-Layer Neural Networks with near-Optimal Sample Complexity: A Case Study in the XOR problem](https://mlsanity.com/paper/2309.15111/) (2023)
- [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)
- [Gradient Methods Never Overfit On Separable Data](https://mlsanity.com/paper/2007.00028/) (2020)
- [Stability vs Implicit Bias of Gradient Methods on Separable Data and Beyond](https://mlsanity.com/paper/2202.13441/) (2022)
- [Flavors of Margin: Implicit Bias of Steepest Descent in Homogeneous Neural Networks](https://mlsanity.com/paper/2410.22069/) (2024)
- [Gradient Descent on Logistic Regression with Non-Separable Data and Large Step Sizes](https://mlsanity.com/paper/2406.05033/) (2024)
- [The Rich and the Simple: On the Implicit Bias of Adam and SGD](https://mlsanity.com/paper/2505.24022/) (2025)
- [The Implicit Bias of Gradient Descent on Separable Multiclass Data](https://mlsanity.com/paper/2411.01350/) (2024)
- [Implicit Bias of Mirror Flow for Shallow Neural Networks in Univariate Regression](https://mlsanity.com/paper/2410.03988/) (2024)

## Abstract (arXiv)

Gradient descent has been of particular interest in modern machine learning beyond sole focus on optimization. Implicit bias emerging from optimization, though not being encoded by the learning objective, often prevents from overfitting to spurious patterns. A typical instance is the max-margin implicit bias of a linear classifier, widely established for exponentially tailed loss functions. Even after having a given dataset separated, the parameter vector continues to evolve towards the max-margin direction asymptotically along the gradient descent dynamics. This phenomenon corroborates a frequent empirical observation of "train longer, generalize better." However, the max-margin convergence is an asymptotic phenomenon, and what is worse, this asymptotic convergence rate is significantly slower than pure convex optimization. Even so, the parameter vector along gradient descent dynamics commonly correlates with the max-margin direction positively (though not exactly) within considerably fewer iterations than the asymptotic rate. By shedding another light on this classical problem, this work aims to understand the mechanism of this early-stage alignment phenomenon. Our theoretical results demonstrate that the parameter vector weakly aligns with the max-margin direction within O(exp(exp(-δ))) iterations, where δ>0 is the permissible alignment error, which is shown to be tight. By tracking the radial and tangential flows, our proof operates on the alignment dynamics directly with dataset geometry and gets rid of the asymptotic expansion, which is a key insight to establishing faster weak alignment.