# Rethinking generalization of classifiers in separable classes scenarios and over-parameterized regimes

arXiv: https://arxiv.org/abs/2410.16868 · HTML: https://mlsanity.com/paper/2410.16868/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 34 of its 55 references that are in the map.
- First posted on arXiv 22 Oct 2024.

## Builds on (most cited first, 30 of 34)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [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)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Large Batch Optimization for Deep Learning: Training BERT in 76 minutes](https://mlsanity.com/paper/1904.00962/) (2019)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Fantastic Generalization Measures and Where to Find Them](https://mlsanity.com/paper/1912.02178/) (2019)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [Stronger generalization bounds for deep nets via a compression approach](https://mlsanity.com/paper/1802.05296/) (2018)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Learning Overparameterized Neural Networks via Stochastic Gradient Descent on Structured Data](https://mlsanity.com/paper/1808.01204/) (2018)
- [High-dimensional dynamics of generalization error in neural networks](https://mlsanity.com/paper/1710.03667/) (2017)
- [Uniform convergence may be unable to explain generalization in deep learning](https://mlsanity.com/paper/1902.04742/) (2019)
- [Fisher-Rao Metric, Geometry, and Complexity of Neural Networks](https://mlsanity.com/paper/1711.01530/) (2017)
- [On the Implicit Bias in Deep-Learning Algorithms](https://mlsanity.com/paper/2208.12591/) (2022)
- [Generalization Bounds of SGLD for Non-convex Learning: Two Theoretical Viewpoints](https://mlsanity.com/paper/1707.05947/) (2017)
- [SGD Learns Over-parameterized Networks that Provably Generalize on Linearly Separable Data](https://mlsanity.com/paper/1710.10174/) (2017)
- [Data-dependent Sample Complexity of Deep Neural Networks via Lipschitz Augmentation](https://mlsanity.com/paper/1905.03684/) (2019)

## Abstract (arXiv)

We investigate the learning dynamics of classifiers in scenarios where classes are separable or classifiers are over-parameterized. In both cases, Empirical Risk Minimization (ERM) results in zero training error. However, there are many global minima with a training error of zero, some of which generalize well and some of which do not. We show that in separable classes scenarios the proportion of "bad" global minima diminishes exponentially with the number of training data n. Our analysis provides bounds and learning curves dependent solely on the density distribution of the true error for the given classifier function set, irrespective of the set's size or complexity (e.g., number of parameters). This observation may shed light on the unexpectedly good generalization of over-parameterized Neural Networks. For the over-parameterized scenario, we propose a model for the density distribution of the true error, yielding learning curves that align with experiments on MNIST and CIFAR-10.