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). 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), and similarly if , even if it has the same sign (correct prediction, but not by enough margin).
While binary SVMs are commonly extended to multiclass classification in a one-vs.-all or one-vs.-one fashion, it is also possible to extend the hinge loss itself for such an end. Several different variations of multiclass hinge loss have been proposed. For example, Crammer and Singer defined it for a linear classifier as
Where the target label, and the model parameters.
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:
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
or the quadratically smoothed
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