Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 218 / 4 / b / Solution

A separating hyperplane for signed data satisfies $Y_i({\gamma }_0+X_i^T\beta )>0$ for every $i$; its geometric set is $\{x:{\gamma }_0+x^T\beta =0\}$. The plot marks three support vectors. At the shown fit they lie on the two support-vector-machine margin boundaries, so their signed functional margins satisfyThe solid line is the decision hyperplane $X^{\ast T}\hat{\gamma }=0$, while the dashed parallel lines are $X^{\ast T}\hat{\gamma }=1$ and $X^{\ast T}\hat{\gamma }=-1$.

If $\lambda =0$, every parameter vector with all margins at least one has zero hinge loss. Scaling or changing a separating vector can therefore give another minimizer, so the objective need not select the displayed maximum-margin direction or the same three lines. Positive quadratic regularization selects a finite, minimum-norm compromise.