Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 218 / 3

The ${\xi }_i$ are slack variables of a support vector machine. The solid line is the support-vector-machine decision boundaryThe dashed lines are the two support-vector-machine margin boundariesEach is at perpendicular distance $1/\Vert \hat{\beta }{\Vert }_2$ from the decision boundary, so the full margin width is $2/\Vert \hat{\beta }{\Vert }_2$.

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For a monarch point, ${\hat{\xi }}_i=\max \{0,1-(\hat{\alpha }+X_i^{\top }\hat{\beta })\}$. Point P2 lies well beyond the monarch-side dashed margin, so a plausible value is $0$. P1 lies between the decision boundary and that margin, so a plausible value is about $0.4$. P3 lies on the wrong side of the decision boundary, so its slack exceeds one; about $1.3$ is plausible. Points P1 and P3 are support vectors, while P2 is not.

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Increasing $C$ raises the cost of slack variables of a support vector machine. The fit therefore generally accepts fewer margin violations and misclassifications, at the price of a larger $\Vert \hat{\beta }{\Vert }_2$ and hence a narrower support-vector-machine margin. Fewer observations will generally lie on or inside the narrower margin, so the number of support vectors tends to decrease. These are qualitative tendencies; individual counts need not vary monotonically for every data set.

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A training observation is misclassified only ifThe SVM constraint then forces ${\hat{\xi }}_i\ge 1$. HenceSumming and dividing by $n$ proves ${\widehat{\mathrm{Err}}}_{\mathrm{tr}}\le n^{-1}{\sum }_i{\hat{\xi }}_i$.

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The code computes Leave-one-out cross-validation. Ifthen observation $i$ is not a support vector. Removing it leaves the optimum unchanged, and the resulting classifier still classifies it correctly. A leave-one-out error can therefore occur only for an observation on or inside the margin, which provesThus the fraction of training observations on or inside the margin is an upper bound on leave-one-out error. One can refit only after deleting support vectors, reusing the full fit for every other observation. Alternatively, K-fold cross-validation needs only $K$ fits and is often preferable for larger data sets.

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