Hinge loss

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Plot of hinge loss (blue) vs. zero-one loss (misclassification, green: y < 0) for t = 1 and variable y. Note that the hinge loss penalizes predictions y < 1, corresponding to the notion of a margin in a support vector machine.

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 y 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 point to classify.

It can be seen that when t and y have the same sign (meaning y predicts the right class) and , the hinge loss , but when they have opposite sign, increases linearly with y (one-sided error).

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]

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

Plot of three variants of the hinge loss as a function of z = ty: the "ordinary" variant (blue), its square (green), and the piece-wise smooth version by Rennie and Srebro (red).

However, since the derivative of the hinge loss at is non-deterministic, 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 .[8]

References[edit]

  1. ^ 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. PMID 15070510. doi:10.1162/089976604773135104. 
  2. ^ Duan, K. B.; Keerthi, S. S. (2005). "Which Is the Best Multiclass SVM Method? An Empirical Study". Multiple Classifier Systems (PDF). LNCS. 3541. pp. 278–285. ISBN 978-3-540-26306-7. doi:10.1007/11494683_28. 
  3. ^ a b Doğan, Ürün; Glasmachers, Tobias; Igel, Christian (2016). "A Unified View on Multi-class Support Vector Classification" (PDF). J. Machine Learning Research. 17: 1–32. 
  4. ^ Crammer, Koby; Singer, Yoram (2001). "On the algorithmic implementation of multiclass kernel-based vector machines" (PDF). J. Machine Learning Research. 2: 265–292. 
  5. ^ 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. 
  6. ^ Weston, Jason; Watkins, Chris (1999). "Support Vector Machines for Multi-Class Pattern Recognition" (PDF). European Symposium on Artificial Neural Networks. 
  7. ^ 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. 
  8. ^ a b Zhang, Tong (2004). Solving large scale linear prediction problems using stochastic gradient descent algorithms. ICML.