# One vs. Rest¶

The One-versus-Rest (or One-versus-All = OvA) Strategy is a Method to use Binary (Two-Class) Classifiers (such as Support Vector Machines ) for classifying multiple (more than two) Classes.

#### One-vs.-rest¶

##### Inputs¶
• $L \text{ , a learner (training algorithm for binary classifiers) }$
• $\text{ samples } \vec{X}$
• $\text{ labels } \vec{y} \\ \text{ where } y_i \in {1,\ldots,K} \text{ is the label for the sample } X_i$
##### Output¶
• $\text{ a list of classifiers } f_k \text{ for } k \in {1,\ldots,K}$
##### Procedure¶

$\text{ For each } k \text{ in } {1,\ldots,K}$

\begin{align} &\text{ Construct a new label vector } \vec{z} \text{ where } \begin{cases} z_i = 1, & \text{ if } y_i = k \\ z_i = 0, & \text{ otherwise } \end{cases}\\ &\text{ Apply } L \text{ to } \vec{X}, \vec{z} \text{ to obtain } f_k \\ \end{align}\\

Making decisions means applying all classifiers to an unseen sample and predicting the label for which the corresponding classifier reports the highest confidence score:

\hat{y} = \underset{k \in \{1 \ldots K\}}{\text{argmax}}\; f_k(x)