# Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss

arXiv: https://arxiv.org/abs/2002.04486 · HTML: https://mlsanity.com/paper/2002.04486/

## Key facts

- Cited by 268 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Feb 2020.

## Built on by (most cited first, 30 of 268)

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- [On the Implicit Bias in Deep-Learning Algorithms](https://mlsanity.com/paper/2208.12591/) (2022)
- [Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity](https://mlsanity.com/paper/2106.15933/) (2021)
- [Gradient Descent on Two-layer Nets: Margin Maximization and Simplicity Bias](https://mlsanity.com/paper/2110.13905/) (2021)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs](https://mlsanity.com/paper/2206.00939/) (2022)
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- [Label-Imbalanced and Group-Sensitive Classification under Overparameterization](https://mlsanity.com/paper/2103.01550/) (2021)
- [Generalization on the Unseen, Logic Reasoning and Degree Curriculum](https://mlsanity.com/paper/2301.13105/) (2023)
- [On the Implicit Bias of Initialization Shape: Beyond Infinitesimal Mirror Descent](https://mlsanity.com/paper/2102.09769/) (2021)
- [Classifying high-dimensional Gaussian mixtures: Where kernel methods fail and neural networks succeed](https://mlsanity.com/paper/2102.11742/) (2021)
- [SGD with Large Step Sizes Learns Sparse Features](https://mlsanity.com/paper/2210.05337/) (2022)
- [Depthwise Hyperparameter Transfer in Residual Networks: Dynamics and Scaling Limit](https://mlsanity.com/paper/2309.16620/) (2023)
- [Neural Networks Efficiently Learn Low-Dimensional Representations with SGD](https://mlsanity.com/paper/2209.14863/) (2022)
- [Max-Margin Token Selection in Attention Mechanism](https://mlsanity.com/paper/2306.13596/) (2023)
- [Saddle-to-Saddle Dynamics in Diagonal Linear Networks](https://mlsanity.com/paper/2304.00488/) (2023)
- [Implicit Regularization Towards Rank Minimization in ReLU Networks](https://mlsanity.com/paper/2201.12760/) (2022)
- [Do Input Gradients Highlight Discriminative Features?](https://mlsanity.com/paper/2102.12781/) (2021)

## Abstract (arXiv)

Neural networks trained to minimize the logistic (a.k.a. cross-entropy) loss with gradient-based methods are observed to perform well in many supervised classification tasks. Towards understanding this phenomenon, we analyze the training and generalization behavior of infinitely wide two-layer neural networks with homogeneous activations. We show that the limits of the gradient flow on exponentially tailed losses can be fully characterized as a max-margin classifier in a certain non-Hilbertian space of functions. In presence of hidden low-dimensional structures, the resulting margin is independent of the ambiant dimension, which leads to strong generalization bounds. In contrast, training only the output layer implicitly solves a kernel support vector machine, which a priori does not enjoy such an adaptivity. Our analysis of training is non-quantitative in terms of running time but we prove computational guarantees in simplified settings by showing equivalences with online mirror descent. Finally, numerical experiments suggest that our analysis describes well the practical behavior of two-layer neural networks with ReLU activation and confirm the statistical benefits of this implicit bias.