# PolyLoss: A Polynomial Expansion Perspective of Classification Loss Functions

arXiv: https://arxiv.org/abs/2204.12511 · HTML: https://mlsanity.com/paper/2204.12511/

## Key facts

- Cited by 63 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 42 references that are in the map.
- First posted on arXiv 26 Apr 2022.

## Builds on (most cited first, 30 of 34)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Mask R-CNN](https://mlsanity.com/paper/1703.06870/) (2017)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [SSD: Single Shot MultiBox Detector](https://mlsanity.com/paper/1512.02325/) (2015)
- [Scalability in Perception for Autonomous Driving: Waymo Open Dataset](https://mlsanity.com/paper/1912.04838/) (2019)
- [PointPillars: Fast Encoders for Object Detection from Point Clouds](https://mlsanity.com/paper/1812.05784/) (2018)
- [Class-Balanced Loss Based on Effective Number of Samples](https://mlsanity.com/paper/1901.05555/) (2019)
- [EfficientNetV2: Smaller Models and Faster Training](https://mlsanity.com/paper/2104.00298/) (2021)
- [EfficientDet: Scalable and Efficient Object Detection](https://mlsanity.com/paper/1911.09070/) (2019)
- [Generalized Cross Entropy Loss for Training Deep Neural Networks with Noisy Labels](https://mlsanity.com/paper/1805.07836/) (2018)
- [PV-RCNN: Point-Voxel Feature Set Abstraction for 3D Object Detection](https://mlsanity.com/paper/1912.13192/) (2019)
- [CornerNet: Detecting Objects as Paired Keypoints](https://mlsanity.com/paper/1808.01244/) (2018)
- [Focal Loss for Dense Object Detection](https://mlsanity.com/paper/1708.02002/) (2017)
- [Training Region-based Object Detectors with Online Hard Example Mining](https://mlsanity.com/paper/1604.03540/) (2016)
- [Symmetric Cross Entropy for Robust Learning with Noisy Labels](https://mlsanity.com/paper/1908.06112/) (2019)
- [Regularizing Neural Networks by Penalizing Confident Output Distributions](https://mlsanity.com/paper/1701.06548/) (2017)
- [Robust Loss Functions under Label Noise for Deep Neural Networks](https://mlsanity.com/paper/1712.09482/) (2017)
- [Bag of Tricks for Image Classification with Convolutional Neural Networks](https://mlsanity.com/paper/1812.01187/) (2018)
- [ResNet strikes back: An improved training procedure in timm](https://mlsanity.com/paper/2110.00476/) (2021)
- [Rethinking Pre-training and Self-training](https://mlsanity.com/paper/2006.06882/) (2020)
- [Hierarchical Multi-Scale Attention for Semantic Segmentation](https://mlsanity.com/paper/2005.10821/) (2020)
- [Revisiting ResNets: Improved Training and Scaling Strategies](https://mlsanity.com/paper/2103.07579/) (2021)
- [Imbalance Problems in Object Detection: A Review](https://mlsanity.com/paper/1909.00169/) (2019)
- [RSN: Range Sparse Net for Efficient, Accurate LiDAR 3D Object Detection](https://mlsanity.com/paper/2106.13365/) (2021)
- [Making Risk Minimization Tolerant to Label Noise](https://mlsanity.com/paper/1403.3610/) (2014)
- [SpineNet: Learning Scale-Permuted Backbone for Recognition and Localization](https://mlsanity.com/paper/1912.05027/) (2019)
- [AM-LFS: AutoML for Loss Function Search](https://mlsanity.com/paper/1905.07375/) (2019)
- [Improved Training Speed, Accuracy, and Data Utilization Through Loss Function Optimization](https://mlsanity.com/paper/1905.11528/) (2019)
- [Loss Max-Pooling for Semantic Image Segmentation](https://mlsanity.com/paper/1704.02966/) (2017)

## Built on by (most cited first, 30 of 63)

- [PointNeXt: Revisiting PointNet++ with Improved Training and Scaling Strategies](https://mlsanity.com/paper/2206.04670/) (2022)
- [YOLOv6: A Single-Stage Object Detection Framework for Industrial Applications](https://mlsanity.com/paper/2209.02976/) (2022)
- [RankT5: Fine-Tuning T5 for Text Ranking with Ranking Losses](https://mlsanity.com/paper/2210.10634/) (2022)
- [Test-Time Adaptation via Conjugate Pseudo-labels](https://mlsanity.com/paper/2207.09640/) (2022)
- [In Search of Lost Online Test-time Adaptation: A Survey](https://mlsanity.com/paper/2310.20199/) (2023)
- [Do Not Blindly Imitate the Teacher: Using Perturbed Loss for Knowledge Distillation](https://mlsanity.com/paper/2305.05010/) (2023)
- [A Theoretical Analysis of the Learning Dynamics under Class Imbalance](https://mlsanity.com/paper/2207.00391/) (2022)
- [Multi-Label Retinal Disease Classification using Transformers](https://mlsanity.com/paper/2207.02335/) (2022)
- [MobileVOS: Real-Time Video Object Segmentation Contrastive Learning meets Knowledge Distillation](https://mlsanity.com/paper/2303.07815/) (2023)
- [Learning Sample Difficulty from Pre-trained Models for Reliable Prediction](https://mlsanity.com/paper/2304.10127/) (2023)
- [Learning Dense Hand Contact Estimation from Imbalanced Data](https://mlsanity.com/paper/2505.11152/) (2025)
- [Temporal Label Smoothing for Early Event Prediction](https://mlsanity.com/paper/2208.13764/) (2022)
- [GradTree: Learning Axis-Aligned Decision Trees with Gradient Descent](https://mlsanity.com/paper/2305.03515/) (2023)
- [AdaptiveClick: Clicks-aware Transformer with Adaptive Focal Loss for Interactive Image Segmentation](https://mlsanity.com/paper/2305.04276/) (2023)
- [Test-Time Adaptation Induces Stronger Accuracy and Agreement-on-the-Line](https://mlsanity.com/paper/2310.04941/) (2023)
- [SLaM: Student-Label Mixing for Distillation with Unlabeled Examples](https://mlsanity.com/paper/2302.03806/) (2023)
- [Adapting Decoder-Based Language Models for Diverse Encoder Downstream Tasks](https://mlsanity.com/paper/2503.02656/) (2025)
- [Learning Symbolic Model-Agnostic Loss Functions via Meta-Learning](https://mlsanity.com/paper/2209.08907/) (2022)
- [Dynamics-Aware Loss for Learning with Label Noise](https://mlsanity.com/paper/2303.11562/) (2023)
- [Enhancing Boundary Segmentation for Topological Accuracy with Skeleton-based Methods](https://mlsanity.com/paper/2404.18539/) (2024)
- [Stochastic Re-weighted Gradient Descent via Distributionally Robust Optimization](https://mlsanity.com/paper/2306.09222/) (2023)
- [MaxEnt Loss: Constrained Maximum Entropy for Calibration under Out-of-Distribution Shift](https://mlsanity.com/paper/2310.17159/) (2023)
- [Robust Lane Detection through Self Pre-training with Masked Sequential Autoencoders and Fine-tuning with Customized PolyLoss](https://mlsanity.com/paper/2305.17271/) (2023)
- [Transformer-Based Speech Synthesizer Attribution in an Open Set Scenario](https://mlsanity.com/paper/2210.07546/) (2022)
- [Meta-Learning Adaptive Loss Functions](https://mlsanity.com/paper/2301.13247/) (2023)
- [Asymmetric Polynomial Loss For Multi-Label Classification](https://mlsanity.com/paper/2304.05361/) (2023)
- [Mask Focal Loss: A unifying framework for dense crowd counting with canonical object detection networks](https://mlsanity.com/paper/2212.11542/) (2022)
- [Learning under Label Noise through Few-Shot Human-in-the-Loop Refinement](https://mlsanity.com/paper/2401.14107/) (2024)
- [Fuzzy-aware Loss for Source-free Domain Adaptation in Visual Emotion Recognition](https://mlsanity.com/paper/2501.15519/) (2025)
- [Neural Loss Function Evolution for Large-Scale Image Classifier Convolutional Neural Networks](https://mlsanity.com/paper/2403.08793/) (2024)

## Abstract (arXiv)

Cross-entropy loss and focal loss are the most common choices when training deep neural networks for classification problems. Generally speaking, however, a good loss function can take on much more flexible forms, and should be tailored for different tasks and datasets. Motivated by how functions can be approximated via Taylor expansion, we propose a simple framework, named PolyLoss, to view and design loss functions as a linear combination of polynomial functions. Our PolyLoss allows the importance of different polynomial bases to be easily adjusted depending on the targeting tasks and datasets, while naturally subsuming the aforementioned cross-entropy loss and focal loss as special cases. Extensive experimental results show that the optimal choice within the PolyLoss is indeed dependent on the task and dataset. Simply by introducing one extra hyperparameter and adding one line of code, our Poly-1 formulation outperforms the cross-entropy loss and focal loss on 2D image classification, instance segmentation, object detection, and 3D object detection tasks, sometimes by a large margin.