{title2=$\widehat R(h)$}

On labelled examples $Z_1,\ldots,Z_n$, the empirical misclassification risk is
$$
\widehat R(h)=\frac1n\sum_{i=1}^n\ell_{0-1}(h,Z_i).
$$
An <empirical risk minimization>[empirical risk minimizer] chooses a <classifier> attaining the smallest value over its <hypothesis class>.


Empirical misclassification risk

{title2=$\ell_{0-1}$}

= Zero-one loss
{synonym}

For a <classifier> $h$ and a labelled example $z=(x,y)$, the misclassification loss is
$$
\ell_{0-1}(h,z)=\mathbf1_{\{h(x)\ne y\}}.
$$
It is also called zero-one loss because it is zero for a correct prediction and one for an incorrect prediction.


Misclassification loss

{title2=$R(h)$}

The misclassification risk of a <classifier> is its <expected value>[expected] <misclassification loss>,
$$
R(h)=\mathbb P(h(X)\ne Y)=\mathbb E\ell_{0-1}(h,(X,Y)).
$$


Misclassification risk

= Binary classifier
{synonym}

A classifier is a <function> that assigns an input to one of a discrete set of labels. A <binary classifier> has two possible labels.


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 Incoming links: Classifier