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On labelled examples $Z_1,\dots ,Z_n$, the empirical misclassification risk isAn empirical risk minimizer chooses a classifier attaining the smallest value over its hypothesis class.

If $h^{\ast }$ minimizes the population risk over a hypothesis class, the excess risk of $h$ is the nonnegative difference $R(h)-R(h^{\ast })$.

Given a hypothesis class $\mathcal{H}$ and a loss function $\ell$, its loss class is the function classIt lets properties of a learning problem be studied through the random fluctuations of the losses themselves.

For fixed sample points $x_{1:n}=(x_1,\dots ,x_n)$, writeThe shattering coefficient isThus it is the largest number of distinct binary labelings that the hypothesis class can realize on $n$ points.

A set of $n$ points is shattered when all $2^n$ labelings are realized. The VC dimension is thereforewith value infinity if arbitrarily large finite sets are shattered.