Hinge loss

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 :\ell(y) = \max(0, 1 + \max_{y \ne t} \mathbf{w}_y \mathbf{x} - \mathbf{w}_t \mathbf{x}), where t is the target label, \mathbf{w}_t and \mathbf{w}_y are the model parameters. Weston and Watkins provided a similar definition, but with a sum rather than a max: :\ell(y) = \sum_{y \ne t} \max(0, 1 + \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 with margin rescaling use the following variant, where denotes the SVM's parameters, the SVM's predictions, the joint feature function, and the Hamming loss: :\begin{align} \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{align}. ==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 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 However, since the derivative of the hinge loss at ty = 1 is undefined, smoothed versions may be preferred for optimization, such as Rennie and Srebro's :\ell(y) = \begin{cases} \frac{1}{2} - ty & \text{if} ~~ ty \le 0, \\ \frac{1}{2} (1 - ty)^2 & \text{if} ~~ 0 or the quadratically smoothed :\ell_\gamma(y) = \begin{cases} \frac{1}{2\gamma} \max(0, 1 - ty)^2 & \text{if} ~~ ty \ge 1 - \gamma, \\ 1 - \frac{\gamma}{2} - ty & \text{otherwise} \end{cases} suggested by Zhang. The modified Huber loss L is a special case of this loss function with \gamma = 2, specifically L(t,y) = 4 \ell_2(y). == See also ==