# ε-Softmax: Approximating One-Hot Vectors for Mitigating Label Noise

arXiv: https://arxiv.org/abs/2508.02387 · HTML: https://mlsanity.com/paper/2508.02387/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 49 references that are in the map.
- First posted on arXiv 4 Aug 2025.

## Builds on (most cited first, 30 of 36)

- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [RandAugment: Practical automated data augmentation with a reduced search space](https://mlsanity.com/paper/1909.13719/) (2019)
- [FixMatch: Simplifying Semi-Supervised Learning with Consistency and Confidence](https://mlsanity.com/paper/2001.07685/) (2020)
- [Co-teaching: Robust Training of Deep Neural Networks with Extremely Noisy Labels](https://mlsanity.com/paper/1804.06872/) (2018)
- [Generalized Cross Entropy Loss for Training Deep Neural Networks with Noisy Labels](https://mlsanity.com/paper/1805.07836/) (2018)
- [A Closer Look at Memorization in Deep Networks](https://mlsanity.com/paper/1706.05394/) (2017)
- [DivideMix: Learning with Noisy Labels as Semi-supervised Learning](https://mlsanity.com/paper/2002.07394/) (2020)
- [Making Deep Neural Networks Robust to Label Noise: a Loss Correction Approach](https://mlsanity.com/paper/1609.03683/) (2016)
- [Symmetric Cross Entropy for Robust Learning with Noisy Labels](https://mlsanity.com/paper/1908.06112/) (2019)
- [Weak-to-Strong Generalization: Eliciting Strong Capabilities With Weak Supervision](https://mlsanity.com/paper/2312.09390/) (2023)
- [Robust Loss Functions under Label Noise for Deep Neural Networks](https://mlsanity.com/paper/1712.09482/) (2017)
- [Early-Learning Regularization Prevents Memorization of Noisy Labels](https://mlsanity.com/paper/2007.00151/) (2020)
- [How does Disagreement Help Generalization against Label Corruption?](https://mlsanity.com/paper/1901.04215/) (2019)
- [Combating noisy labels by agreement: A joint training method with co-regularization](https://mlsanity.com/paper/2003.02752/) (2020)
- [WebVision Database: Visual Learning and Understanding from Web Data](https://mlsanity.com/paper/1708.02862/) (2017)
- [Normalized Loss Functions for Deep Learning with Noisy Labels](https://mlsanity.com/paper/2006.13554/) (2020)
- [Learning with Noisy Labels Revisited: A Study Using Real-World Human Annotations](https://mlsanity.com/paper/2110.12088/) (2021)
- [Are Anchor Points Really Indispensable in Label-Noise Learning?](https://mlsanity.com/paper/1906.00189/) (2019)
- [Learning with Instance-Dependent Label Noise: A Sample Sieve Approach](https://mlsanity.com/paper/2010.02347/) (2020)
- [NLNL: Negative Learning for Noisy Labels](https://mlsanity.com/paper/1908.07387/) (2019)
- [Peer Loss Functions: Learning from Noisy Labels without Knowing Noise Rates](https://mlsanity.com/paper/1910.03231/) (2019)
- [A Survey of Label-noise Representation Learning: Past, Present and Future](https://mlsanity.com/paper/2011.04406/) (2020)
- [Learning with Symmetric Label Noise: The Importance of Being Unhinged](https://mlsanity.com/paper/1505.07634/) (2015)
- [Understanding and Improving Early Stopping for Learning with Noisy Labels](https://mlsanity.com/paper/2106.15853/) (2021)
- [Provably End-to-end Label-Noise Learning without Anchor Points](https://mlsanity.com/paper/2102.02400/) (2021)
- [Robust Training under Label Noise by Over-parameterization](https://mlsanity.com/paper/2202.14026/) (2022)
- [A Second-Order Approach to Learning with Instance-Dependent Label Noise](https://mlsanity.com/paper/2012.11854/) (2020)
- [Generalized Jensen-Shannon Divergence Loss for Learning with Noisy Labels](https://mlsanity.com/paper/2105.04522/) (2021)
- [To Smooth or Not? When Label Smoothing Meets Noisy Labels](https://mlsanity.com/paper/2106.04149/) (2021)
- [Asymmetric Loss Functions for Learning with Noisy Labels](https://mlsanity.com/paper/2106.03110/) (2021)

## Built on by (most cited first, 6 of 6)

- [Optimal Transport for LLM Reward Modeling from Noisy Preference](https://mlsanity.com/paper/2605.06036/) (2026)
- [Foundation Model Priors Enhance Object Focus in Feature Space for Source-Free Object Detection](https://mlsanity.com/paper/2512.17514/) (2025)
- [What's on My Network? Using Large Language Models to Identify Real-World IoT Devices at Scale](https://mlsanity.com/paper/2510.13817/) (2025)
- [Learning to Forget with Information Divergence Reweighted Objectives for Noisy Labels](https://mlsanity.com/paper/2508.06622/) (2025)
- [Evolution-Inspired Sample Competition for Deep Neural Network Optimization](https://mlsanity.com/paper/2604.12568/) (2026)
- [Robust Losses from Univariate Base Functions for Noisy-Label Learning](https://mlsanity.com/paper/2607.16768/) (2026)

## Abstract (arXiv)

Noisy labels pose a common challenge for training accurate deep neural networks. To mitigate label noise, prior studies have proposed various robust loss functions to achieve noise tolerance in the presence of label noise, particularly symmetric losses. However, they usually suffer from the underfitting issue due to the overly strict symmetric condition. In this work, we propose a simple yet effective approach for relaxing the symmetric condition, namely ε-softmax, which simply modifies the outputs of the softmax layer to approximate one-hot vectors with a controllable error ε. Essentially, ε-softmax not only acts as an alternative for the softmax layer, but also implicitly plays the crucial role in modifying the loss function. We prove theoretically that ε-softmax can achieve noise-tolerant learning with controllable excess risk bound for almost any loss function. Recognizing that ε-softmax-enhanced losses may slightly reduce fitting ability on clean datasets, we further incorporate them with one symmetric loss, thereby achieving a better trade-off between robustness and effective learning. Extensive experiments demonstrate the superiority of our method in mitigating synthetic and real-world label noise. The code is available at https://github.com/cswjl/eps-softmax.