# Consistency for Large Neural Networks: Regression and Classification

arXiv: https://arxiv.org/abs/2409.14123 · HTML: https://mlsanity.com/paper/2409.14123/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 58 references that are in the map.
- First posted on arXiv 21 Sep 2024.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [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)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [Overfitting in adversarially robust deep learning](https://mlsanity.com/paper/2002.11569/) (2020)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [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)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Consistency of random forests](https://mlsanity.com/paper/1405.2881/) (2014)
- [The Interpolation Phase Transition in Neural Networks: Memorization and Generalization under Lazy Training](https://mlsanity.com/paper/2007.12826/) (2020)
- [Double Descent Demystified: Identifying, Interpreting & Ablating the Sources of a Deep Learning Puzzle](https://mlsanity.com/paper/2303.14151/) (2023)
- [A U-turn on Double Descent: Rethinking Parameter Counting in Statistical Learning](https://mlsanity.com/paper/2310.18988/) (2023)
- [Large Scale Prediction with Decision Trees](https://mlsanity.com/paper/2104.13881/) (2021)
- [How do noise tails impact on deep ReLU networks?](https://mlsanity.com/paper/2203.10418/) (2022)
- [Quantile regression with deep ReLU Networks: Estimators and minimax rates](https://mlsanity.com/paper/2010.08236/) (2020)
- [Nonparametric regression using over-parameterized shallow ReLU neural networks](https://mlsanity.com/paper/2306.08321/) (2023)
- [Approximation bounds for norm constrained neural networks with applications to regression and GANs](https://mlsanity.com/paper/2201.09418/) (2022)
- [Convergence rates of deep ReLU networks for multiclass classification](https://mlsanity.com/paper/2108.00969/) (2021)
- [Dropout Rademacher Complexity of Deep Neural Networks](https://mlsanity.com/paper/1402.3811/) (2014)
- [Generalization Performance of Empirical Risk Minimization on Over-parameterized Deep ReLU Nets](https://mlsanity.com/paper/2111.14039/) (2021)
- [Nonasymptotic theory for two-layer neural networks: Beyond the bias-variance trade-off](https://mlsanity.com/paper/2106.04795/) (2021)

## Abstract (arXiv)

Although overparameterized models have achieved remarkable practical success, their theoretical properties, particularly their generalization behavior, remain incompletely understood. The well known double descents phenomenon suggests that the test error curve of neural networks decreases monotonically as model size grows and eventually converges to a non-zero constant. This work aims to explain the theoretical mechanism underlying this tail behavior and study the statistical consistency of deep overparameterized neural networks in many different learning tasks including regression and classification. Firstly, we prove that as the number of parameters increases, the approximation error decreases monotonically, while explicit or implicit regularization (e.g., weight decay) keeps the generalization error existing but bounded. Consequently, the overall error curve eventually converges to a constant determined by the bounded generalization error and the optimization error. Secondly, we prove that deep overparameterized neural networks are statistical consistency across multiple learning tasks if regularization technique is used. Our theoretical findings coincide with numerical experiments and provide a perspective for understanding the generalization behavior of overparameterized neural networks.