# Zero Generalization Error Theorem for Random Interpolators via Algebraic Geometry

arXiv: https://arxiv.org/abs/2512.06347 · HTML: https://mlsanity.com/paper/2512.06347/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 41 references that are in the map.
- First posted on arXiv 6 Dec 2025.

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

- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [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)
- [Confident Learning: Estimating Uncertainty in Dataset Labels](https://mlsanity.com/paper/1911.00068/) (2019)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [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)
- [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)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Deep learning generalizes because the parameter-function map is biased towards simple functions](https://mlsanity.com/paper/1805.08522/) (2018)
- [Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation](https://mlsanity.com/paper/2105.14368/) (2021)
- [Spurious Local Minima are Common in Two-Layer ReLU Neural Networks](https://mlsanity.com/paper/1712.08968/) (2017)
- [On the Implicit Bias in Deep-Learning Algorithms](https://mlsanity.com/paper/2208.12591/) (2022)
- [A Unifying View on Implicit Bias in Training Linear Neural Networks](https://mlsanity.com/paper/2010.02501/) (2020)
- [Scikit-dimension: a Python package for intrinsic dimension estimation](https://mlsanity.com/paper/2109.02596/) (2021)
- [An Analytical Formula of Population Gradient for two-layered ReLU network and its Applications in Convergence and Critical Point Analysis](https://mlsanity.com/paper/1703.00560/) (2017)
- [Is SGD a Bayesian sampler? Well, almost](https://mlsanity.com/paper/2006.15191/) (2020)
- [Semi-flat minima and saddle points by embedding neural networks to overparameterization](https://mlsanity.com/paper/1906.04868/) (2019)
- [On the Implicit Bias of Adam](https://mlsanity.com/paper/2309.00079/) (2023)
- [Over-Parameterization Exponentially Slows Down Gradient Descent for Learning a Single Neuron](https://mlsanity.com/paper/2302.10034/) (2023)
- [How Uniform Random Weights Induce Non-uniform Bias: Typical Interpolating Neural Networks Generalize with Narrow Teachers](https://mlsanity.com/paper/2402.06323/) (2024)
- [Can Implicit Bias Explain Generalization? Stochastic Convex Optimization as a Case Study](https://mlsanity.com/paper/2003.06152/) (2020)
- [Exact Gap between Generalization Error and Uniform Convergence in Random Feature Models](https://mlsanity.com/paper/2103.04554/) (2021)
- [Beyond Implicit Bias: The Insignificance of SGD Noise in Online Learning](https://mlsanity.com/paper/2306.08590/) (2023)
- [Investigating Generalization by Controlling Normalized Margin](https://mlsanity.com/paper/2205.03940/) (2022)

## Abstract (arXiv)

We theoretically demonstrate that the generalization error of interpolators for machine learning models under teacher-student settings becomes 0 once the number of training samples exceeds a certain threshold. Understanding the high generalization ability of large-scale models such as deep neural networks (DNNs) remains one of the central open problems in machine learning theory. While recent theoretical studies have attributed this phenomenon to the implicit bias of stochastic gradient descent (SGD) toward well-generalizing solutions, empirical evidences indicate that it primarily stems from properties of the model itself. Specifically, even randomly sampled interpolators, which are parameters that achieve zero training error, have been observed to generalize effectively. In this study, under a teacher-student framework, we prove that the generalization error of randomly sampled interpolators becomes exactly zero once the number of training samples exceeds a threshold determined by the geometric structure of the interpolator set in parameter space. As a proof technique, we leverage tools from algebraic geometry to mathematically characterize this geometric structure.