# Understanding Square Loss in Training Overparametrized Neural Network Classifiers

arXiv: https://arxiv.org/abs/2112.03657 · HTML: https://mlsanity.com/paper/2112.03657/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 62 references that are in the map.
- First posted on arXiv 7 Dec 2021.

## Builds on (most cited first, 30 of 40)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Supervised Contrastive Learning](https://mlsanity.com/paper/2004.11362/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks](https://mlsanity.com/paper/2003.01690/) (2020)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [Gradient Starvation: A Learning Proclivity in Neural Networks](https://mlsanity.com/paper/2011.09468/) (2020)
- [Learning Overparameterized Neural Networks via Stochastic Gradient Descent on Structured Data](https://mlsanity.com/paper/1808.01204/) (2018)
- [Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural Networks](https://mlsanity.com/paper/1905.13210/) (2019)
- [On the Inductive Bias of Neural Tangent Kernels](https://mlsanity.com/paper/1905.12173/) (2019)
- [MMA Training: Direct Input Space Margin Maximization through Adversarial Training](https://mlsanity.com/paper/1812.02637/) (2018)
- [Evaluating model calibration in classification](https://mlsanity.com/paper/1902.06977/) (2019)
- [Exploring Deep Neural Networks via Layer-Peeled Model: Minority Collapse in Imbalanced Training](https://mlsanity.com/paper/2101.12699/) (2021)
- [Neural Collapse Under MSE Loss: Proximity to and Dynamics on the Central Path](https://mlsanity.com/paper/2106.02073/) (2021)
- [Learning Diverse and Discriminative Representations via the Principle of Maximal Coding Rate Reduction](https://mlsanity.com/paper/2006.08558/) (2020)
- [Evaluation of Neural Architectures Trained with Square Loss vs Cross-Entropy in Classification Tasks](https://mlsanity.com/paper/2006.07322/) (2020)
- [Parametric Noise Injection: Trainable Randomness to Improve Deep Neural Network Robustness against Adversarial Attack](https://mlsanity.com/paper/1811.09310/) (2018)
- [Polylogarithmic width suffices for gradient descent to achieve arbitrarily small test error with shallow ReLU networks](https://mlsanity.com/paper/1909.12292/) (2019)
- [Large Margin Deep Networks for Classification](https://mlsanity.com/paper/1803.05598/) (2018)
- [The Neural Tangent Kernel in High Dimensions: Triple Descent and a Multi-Scale Theory of Generalization](https://mlsanity.com/paper/2008.06786/) (2020)
- [Classification vs regression in overparameterized regimes: Does the loss function matter?](https://mlsanity.com/paper/2005.08054/) (2020)

## Built on by (most cited first, 14 of 14)

- [The Implicit Bias of Benign Overfitting](https://mlsanity.com/paper/2201.11489/) (2022)
- [Selftok: Discrete Visual Tokens of Autoregression, by Diffusion, and for Reasoning](https://mlsanity.com/paper/2505.07538/) (2025)
- [Inducing Neural Collapse in Deep Long-tailed Learning](https://mlsanity.com/paper/2302.12453/) (2023)
- [From Tempered to Benign Overfitting in ReLU Neural Networks](https://mlsanity.com/paper/2305.15141/) (2023)
- [Classification with Deep Neural Networks and Logistic Loss](https://mlsanity.com/paper/2307.16792/) (2023)
- [Explore and Exploit the Diverse Knowledge in Model Zoo for Domain Generalization](https://mlsanity.com/paper/2306.02595/) (2023)
- [Exact Count of Boundary Pieces of ReLU Classifiers: Towards the Proper Complexity Measure for Classification](https://mlsanity.com/paper/2306.08805/) (2023)
- [GraphCleaner: Detecting Mislabelled Samples in Popular Graph Learning Benchmarks](https://mlsanity.com/paper/2306.00015/) (2023)
- [Random Smoothing Regularization in Kernel Gradient Descent Learning](https://mlsanity.com/paper/2305.03531/) (2023)
- [Generalization of Scaled Deep ResNets in the Mean-Field Regime](https://mlsanity.com/paper/2403.09889/) (2024)
- [The Optimality of Kernel Classifiers in Sobolev Space](https://mlsanity.com/paper/2402.01148/) (2024)
- [Probabilistic Neural Networks (PNNs) with t-Distributed Outputs: Adaptive Prediction Intervals Beyond Gaussian Assumptions](https://mlsanity.com/paper/2503.12354/) (2025)
- [Revisiting and Advancing Adversarial Training Through A Simple Baseline](https://mlsanity.com/paper/2306.07613/) (2023)
- [Super-fast Rates of Convergence for Neural Network Classifiers under the Hard Margin Condition](https://mlsanity.com/paper/2505.08262/) (2025)

## Abstract (arXiv)

Deep learning has achieved many breakthroughs in modern classification tasks. Numerous architectures have been proposed for different data structures but when it comes to the loss function, the cross-entropy loss is the predominant choice. Recently, several alternative losses have seen revived interests for deep classifiers. In particular, empirical evidence seems to promote square loss but a theoretical justification is still lacking. In this work, we contribute to the theoretical understanding of square loss in classification by systematically investigating how it performs for overparametrized neural networks in the neural tangent kernel (NTK) regime. Interesting properties regarding the generalization error, robustness, and calibration error are revealed. We consider two cases, according to whether classes are separable or not. In the general non-separable case, fast convergence rate is established for both misclassification rate and calibration error. When classes are separable, the misclassification rate improves to be exponentially fast. Further, the resulting margin is proven to be lower bounded away from zero, providing theoretical guarantees for robustness. We expect our findings to hold beyond the NTK regime and translate to practical settings. To this end, we conduct extensive empirical studies on practical neural networks, demonstrating the effectiveness of square loss in both synthetic low-dimensional data and real image data. Comparing to cross-entropy, square loss has comparable generalization error but noticeable advantages in robustness and model calibration.