# Cross-Entropy Loss Functions: Theoretical Analysis and Applications

arXiv: https://arxiv.org/abs/2304.07288 · HTML: https://mlsanity.com/paper/2304.07288/

## Key facts

- Cited by 210 papers in the map (data through 14 Oct 2026).
- Builds on 95 of its 160 references that are in the map.
- First posted on arXiv 14 Apr 2023.

## Builds on (most cited first, 30 of 95)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Theoretically Principled Trade-off between Robustness and Accuracy](https://mlsanity.com/paper/1901.08573/) (2019)
- [Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks](https://mlsanity.com/paper/2003.01690/) (2020)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [Generalized Cross Entropy Loss for Training Deep Neural Networks with Noisy Labels](https://mlsanity.com/paper/1805.07836/) (2018)
- [Adversarial Machine Learning at Scale](https://mlsanity.com/paper/1611.01236/) (2016)
- [Robustness May Be at Odds with Accuracy](https://mlsanity.com/paper/1805.12152/) (2018)
- [Ensemble Adversarial Training: Attacks and Defenses](https://mlsanity.com/paper/1705.07204/) (2017)
- [Fast is better than free: Revisiting adversarial training](https://mlsanity.com/paper/2001.03994/) (2020)
- [Evasion Attacks against Machine Learning at Test Time](https://mlsanity.com/paper/1708.06131/) (2017)
- [Adversarial Training for Free!](https://mlsanity.com/paper/1904.12843/) (2019)
- [Overfitting in adversarially robust deep learning](https://mlsanity.com/paper/2002.11569/) (2020)
- [Robust Loss Functions under Label Noise for Deep Neural Networks](https://mlsanity.com/paper/1712.09482/) (2017)
- [Unlabeled Data Improves Adversarial Robustness](https://mlsanity.com/paper/1905.13736/) (2019)
- [Adversarially Robust Generalization Requires More Data](https://mlsanity.com/paper/1804.11285/) (2018)
- [Feature Denoising for Improving Adversarial Robustness](https://mlsanity.com/paper/1812.03411/) (2018)
- [Uncovering the Limits of Adversarial Training against Norm-Bounded Adversarial Examples](https://mlsanity.com/paper/2010.03593/) (2020)
- [Attacks Which Do Not Kill Training Make Adversarial Learning Stronger](https://mlsanity.com/paper/2002.11242/) (2020)
- [Adversarial Logit Pairing](https://mlsanity.com/paper/1803.06373/) (2018)
- [Understanding and Improving Fast Adversarial Training](https://mlsanity.com/paper/2007.02617/) (2020)
- [Improving Adversarial Robustness via Promoting Ensemble Diversity](https://mlsanity.com/paper/1901.08846/) (2019)
- [Fixing Data Augmentation to Improve Adversarial Robustness](https://mlsanity.com/paper/2103.01946/) (2021)
- [Bag of Tricks for Adversarial Training](https://mlsanity.com/paper/2010.00467/) (2020)

## Built on by (most cited first, 30 of 210)

- [RigorLLM: Resilient Guardrails for Large Language Models against Undesired Content](https://mlsanity.com/paper/2403.13031/) (2024)
- [Predictor-Rejector Multi-Class Abstention: Theoretical Analysis and Algorithms](https://mlsanity.com/paper/2310.14772/) (2023)
- [Theoretically Grounded Loss Functions and Algorithms for Score-Based Multi-Class Abstention](https://mlsanity.com/paper/2310.14770/) (2023)
- [Regression with Multi-Expert Deferral](https://mlsanity.com/paper/2403.19494/) (2024)
- [Ranking with Abstention](https://mlsanity.com/paper/2307.02035/) (2023)
- [Realizable H-Consistent and Bayes-Consistent Loss Functions for Learning to Defer](https://mlsanity.com/paper/2407.13732/) (2024)
- [A Universal Growth Rate for Learning with Smooth Surrogate Losses](https://mlsanity.com/paper/2405.05968/) (2024)
- [Cardinality-Aware Set Prediction and Top-k Classification](https://mlsanity.com/paper/2407.07140/) (2024)
- [Multi-Label Learning with Stronger Consistency Guarantees](https://mlsanity.com/paper/2407.13746/) (2024)
- [H-Consistency Guarantees for Regression](https://mlsanity.com/paper/2403.19480/) (2024)
- [Enhanced H-Consistency Bounds](https://mlsanity.com/paper/2407.13722/) (2024)
- [A Two-Stage Learning-to-Defer Approach for Multi-Task Learning](https://mlsanity.com/paper/2410.15729/) (2024)
- [Adversarial Robustness in Two-Stage Learning-to-Defer: Algorithms and Guarantees](https://mlsanity.com/paper/2502.01027/) (2025)
- [Mastering Multiple-Expert Routing: Realizable H-Consistency and Strong Guarantees for Learning to Defer](https://mlsanity.com/paper/2506.20650/) (2025)
- [Optimal Query Allocation in Extractive QA with LLMs: A Learning-to-Defer Framework with Theoretical Guarantees](https://mlsanity.com/paper/2410.15761/) (2024)
- [Balancing the Scales: A Theoretical and Algorithmic Framework for Learning from Imbalanced Data](https://mlsanity.com/paper/2502.10381/) (2025)
- [V2-SAM: Marrying SAM2 with Multi-Prompt Experts for Cross-View Object Correspondence](https://mlsanity.com/paper/2511.20886/) (2025)
- [PrivImage: Differentially Private Synthetic Image Generation using Diffusion Models with Semantic-Aware Pretraining](https://mlsanity.com/paper/2311.12850/) (2023)
- [Budgeted Multiple-Expert Deferral](https://mlsanity.com/paper/2510.26706/) (2025)
- [Understanding Probe Behaviors through Variational Bounds of Mutual Information](https://mlsanity.com/paper/2312.10019/) (2023)
- [K-frames: Scene-Driven Any-k Keyframe Selection for long video understanding](https://mlsanity.com/paper/2510.13891/) (2025)
- [SeqVLA: Sequential Task Execution for Long-Horizon Manipulation with Completion-Aware Vision-Language-Action Model](https://mlsanity.com/paper/2509.14138/) (2025)
- [ReGuLaR: Variational Latent Reasoning Guided by Rendered Chain-of-Thought](https://mlsanity.com/paper/2601.23184/) (2026)
- [SPikE-SSM: A Sparse, Precise, and Efficient Spiking State Space Model for Long Sequences Learning](https://mlsanity.com/paper/2410.17268/) (2024)
- [DART: Disease-aware Image-Text Alignment and Self-correcting Re-alignment for Trustworthy Radiology Report Generation](https://mlsanity.com/paper/2504.11786/) (2025)
- [Improved Balanced Classification with Theoretically Grounded Loss Functions](https://mlsanity.com/paper/2512.23947/) (2025)
- [Universal Visuo-Tactile Video Understanding for Embodied Interaction](https://mlsanity.com/paper/2505.22566/) (2025)
- [Concentrate Attention: Towards Domain-Generalizable Prompt Optimization for Language Models](https://mlsanity.com/paper/2406.10584/) (2024)
- [Why Ask One When You Can Ask k? Learning-to-Defer to the Top-k Experts](https://mlsanity.com/paper/2504.12988/) (2025)
- [Parameter-Selective Continual Test-Time Adaptation](https://mlsanity.com/paper/2407.02253/) (2024)

## Abstract (arXiv)

Cross-entropy is a widely used loss function in applications. It coincides with the logistic loss applied to the outputs of a neural network, when the softmax is used. But, what guarantees can we rely on when using cross-entropy as a surrogate loss? We present a theoretical analysis of a broad family of loss functions, comp-sum losses, that includes cross-entropy (or logistic loss), generalized cross-entropy, the mean absolute error and other cross-entropy-like loss functions. We give the first H-consistency bounds for these loss functions. These are non-asymptotic guarantees that upper bound the zero-one loss estimation error in terms of the estimation error of a surrogate loss, for the specific hypothesis set H used. We further show that our bounds are tight. These bounds depend on quantities called minimizability gaps. To make them more explicit, we give a specific analysis of these gaps for comp-sum losses. We also introduce a new family of loss functions, smooth adversarial comp-sum losses, that are derived from their comp-sum counterparts by adding in a related smooth term. We show that these loss functions are beneficial in the adversarial setting by proving that they admit H-consistency bounds. This leads to new adversarial robustness algorithms that consist of minimizing a regularized smooth adversarial comp-sum loss. While our main purpose is a theoretical analysis, we also present an extensive empirical analysis comparing comp-sum losses. We further report the results of a series of experiments demonstrating that our adversarial robustness algorithms outperform the current state-of-the-art, while also achieving a superior non-adversarial accuracy.