Hinge loss
![](http://178.128.105.246/content-http-upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Hinge_loss_vs_zero_one_loss.svg/220px-Hinge_loss_vs_zero_one_loss.svg.png)
In machine learning, the hinge loss is a loss function used for training classifiers. The hinge loss is used for "maximum-margin" classification, most notably for support vector machines (SVMs).[1]
For an intended output t = ±1 and a classifier score y, the hinge loss of the prediction y is defined as
Note that should be the "raw" output of the classifier's decision function, not the predicted class label. For instance, in linear SVMs, , where are the parameters of the hyperplane and is the input variable(s).
When t and y have the same sign (meaning y predicts the right class) and , the hinge loss . When they have opposite signs, increases linearly with y, and similarly if , even if it has the same sign (correct prediction, but not by enough margin).
Extensions
[edit]While binary SVMs are commonly extended to multiclass classification in a one-vs.-all or one-vs.-one fashion,[2] it is also possible to extend the hinge loss itself for such an end. Several different variations of multiclass hinge loss have been proposed.[3] For example, Crammer and Singer[4] defined it for a linear classifier as[5]
- ,
where is the target label, and are the model parameters.
Weston and Watkins provided a similar definition, but with a sum rather than a max:[6][3]
- .
In structured prediction, the hinge loss can be further extended to structured output spaces. Structured SVMs with margin rescaling use the following variant, where w denotes the SVM's parameters, y the SVM's predictions, φ the joint feature function, and Δ the Hamming loss:
- .
Optimization
[edit]The hinge loss is a convex function, so many of the usual convex optimizers used in machine learning can work with it. It is not differentiable, but has a subgradient with respect to model parameters w of a linear SVM with score function that is given by
![](http://178.128.105.246/content-http-upload.wikimedia.org/wikipedia/commons/thumb/8/83/Hinge_loss_variants.svg/220px-Hinge_loss_variants.svg.png)
However, since the derivative of the hinge loss at is undefined, smoothed versions may be preferred for optimization, such as Rennie and Srebro's[7]
or the quadratically smoothed
suggested by Zhang.[8] The modified Huber loss is a special case of this loss function with , specifically .
See also
[edit]References
[edit]- ^ Rosasco, L.; De Vito, E. D.; Caponnetto, A.; Piana, M.; Verri, A. (2004). "Are Loss Functions All the Same?" (PDF). Neural Computation. 16 (5): 1063–1076. CiteSeerX 10.1.1.109.6786. doi:10.1162/089976604773135104. PMID 15070510.
- ^ Duan, K. B.; Keerthi, S. S. (2005). "Which Is the Best Multiclass SVM Method? An Empirical Study" (PDF). Multiple Classifier Systems. LNCS. Vol. 3541. pp. 278–285. CiteSeerX 10.1.1.110.6789. doi:10.1007/11494683_28. ISBN 978-3-540-26306-7.
- ^ a b Doğan, Ürün; Glasmachers, Tobias; Igel, Christian (2016). "A Unified View on Multi-class Support Vector Classification" (PDF). Journal of Machine Learning Research. 17: 1–32.
- ^ Crammer, Koby; Singer, Yoram (2001). "On the algorithmic implementation of multiclass kernel-based vector machines" (PDF). Journal of Machine Learning Research. 2: 265–292.
- ^ Moore, Robert C.; DeNero, John (2011). "L1 and L2 regularization for multiclass hinge loss models" (PDF). Proc. Symp. on Machine Learning in Speech and Language Processing. Archived from the original (PDF) on 2017-08-28. Retrieved 2013-10-23.
- ^ Weston, Jason; Watkins, Chris (1999). "Support Vector Machines for Multi-Class Pattern Recognition" (PDF). European Symposium on Artificial Neural Networks. Archived from the original (PDF) on 2018-05-05. Retrieved 2017-03-01.
- ^ Rennie, Jason D. M.; Srebro, Nathan (2005). Loss Functions for Preference Levels: Regression with Discrete Ordered Labels (PDF). Proc. IJCAI Multidisciplinary Workshop on Advances in Preference Handling.
- ^ Zhang, Tong (2004). Solving large scale linear prediction problems using stochastic gradient descent algorithms (PDF). ICML.