Hinge loss

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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

$\ell (y)=\max(0,1-t\cdot y)$

Note that y should be the "raw" output of the SVM's decision function, not the predicted class label. E.g., in linear SVMs, $y={\mathbf {w}}\cdot {\mathbf {x}}+b$ .

It can be seen that when $t$ and $y$ have the same sign (meaning $y$ predicts the right class) and $y\geq 1$ , $\ell (y)=0$ , but when they have opposite sign, $\ell (y)$ increases linearly with $y$ (one-sided error).

Extensions

While SVMs are commonly extended to multiclass classification in a one-vs.-all or one-vs.-one fashion,^[2] there exists a "true" multiclass version of the hinge loss due to Crammer and Singer,^[3] defined for a linear classifier as^[4]

$\ell (y)=\max(0,1+\max _{{y\neq t}}{\mathbf {w}}_{y}{\mathbf {x}}-{\mathbf {w}}_{t}{\mathbf {x}})$

In structured prediction, the hinge loss can be further extended to structured output spaces. Structured SVMs use the following variant, where w denotes the SVM's parameters, φ the joint feature function, and Δ the Hamming loss:^[5]

${\begin{aligned}\ell ({\mathbf {y}})&=\Delta ({\mathbf {y}},{\mathbf {t}})+\langle {\mathbf {w}},\phi ({\mathbf {x}},{\mathbf {y}})\rangle -\langle {\mathbf {w}},\phi ({\mathbf {x}},{\mathbf {t}})\rangle \\&=\max _{{y\in {\mathcal {Y}}}}\left(\Delta ({\mathbf {y}},{\mathbf {t}}+\langle {\mathbf {w}},\phi ({\mathbf {x}},{\mathbf {y}})\rangle )\right)-\langle {\mathbf {w}},\phi ({\mathbf {x}},{\mathbf {t}})\rangle \end{aligned}}$

Optimization

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 ${\mathbf {w}}$ of a linear SVM with score function $y={\mathbf {w}}\cdot {\mathbf {x}}$ that is given by

${\frac {\partial \ell }{\partial w_{i}}}={\begin{cases}-t\cdot x_{i}&{\text{if }}t\cdot y<1\\0&{\text{otherwise}}\end{cases}}$

References

↑ Rosasco, L.; De Vito, E. D.; Caponnetto, A.; Piana, M.; Verri, A. (2004). "Are Loss Functions All the Same?". Neural Computation 16 (5): 1063–1076. doi:10.1162/089976604773135104. PMID 15070510.
↑ Duan, K. B.; Keerthi, S. S. (2005). "Which Is the Best Multiclass SVM Method? An Empirical Study". Multiple Classifier Systems. Lecture Notes in Computer Science 3541. p. 278. doi:10.1007/11494683_28. ISBN 978-3-540-26306-7.
↑ Crammer, Koby; and Singer, Yoram (2001). "On the algorithmic implementation of multiclass kernel-based vector machines". J. Machine Learning Research 2: 265–292.
↑ Robert C. Moore and John DeNero (2011). "L₁ and L₂ regularization for multiclass hinge loss models". Proc. Symp. on Machine Learning in Speech and Language Processing.
↑ Alex Spengler, Antoine Bordes and Patrick Gallinari (2010). "A comparison of discriminative classifiers for web news content extraction". RIAO.

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