损失函数：Hinge Loss（max margin）

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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

{\displaystyle \ell (y)=\max(0,1-t\cdot y)}

Note that y should be the "raw" output of the classifier's decision function, not the predicted class label. For instance, in linear SVMs, {\displaystyle y=\mathbf {w} \cdot \mathbf {x} +b}, where {\displaystyle (\mathbf {w} ,b)} are the parameters of the hyperplane and {\displaystyle \mathbf {x} } 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 {\displaystyle |y|\geq 1}, the hinge loss {\displaystyle \ell (y)=0}, but when they have opposite sign, {\displaystyle \ell (y)} increases linearly with y (one-sided error).

Extensions[edit]

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]

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

In structured prediction, the hinge loss can be further extended to structured output spaces. Structured SVMs with margin rescaling use the following variant, where ydenotes the SVM's parameters, φ the joint feature function, and Δ the Hamming loss:

{\displaystyle {\begin{aligned}\ell (\mathbf {y} )&=\max(0,\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle -\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\\&=\max(0,\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[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 {\displaystyle y=\mathbf {w} \cdot \mathbf {x} } that is given by

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

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 {\displaystyle ty=1} is non-deterministic, smoothed versions may be preferred for optimization, such as Rennie and Srebro's^[5]

{\displaystyle \ell (y)={\begin{cases}{\frac {1}{2}}-ty&{\text{if}}~~ty\leq 0,\\{\frac {1}{2}}(1-ty)^{2}&{\text{if}}~~0

or the quadratically smoothed

{\displaystyle \ell (y)={\frac {1}{2\gamma }}\max(0,1-ty)^{2}}

suggested by Zhang.^[6] The modified Huber loss is a special case of this loss function with {\displaystyle \gamma =2}.^[6]

References[edit]

1. Jump up^ 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. doi:10.1162/089976604773135104. PMID 15070510.
2. Jump up^ 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.doi:10.1007/11494683_28. ISBN 978-3-540-26306-7.
3. Jump up^ Crammer, Koby; Singer, Yoram (2001). "On the algorithmic implementation of multiclass kernel-based vector machines" (PDF). J. Machine Learning Research. 2: 265–292.
4. Jump up^ 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.
5. Jump up^ 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.
6. ^ Jump up to:^a b Zhang, Tong (2004). Solving large scale linear prediction problems using stochastic gradient descent algorithms. ICML.

Categories: 

• Loss functions
• Support vector machines