A training observation is misclassified only if
$$
Y_i(\widehat\alpha+X_i^\top\widehat\beta)\leq0.
$$
The SVM constraint then forces $\widehat\xi_i\geq1$. Hence
$$
\mathbf1_{\{Y_i(\widehat\alpha+X_i^\top\widehat\beta)\leq0\}}
\leq\widehat\xi_i.
$$
Summing and dividing by $n$ proves $\widehat{\operatorname{Err}}_{\mathrm{tr}}\leq n^{-1}\sum_i\widehat\xi_i$.

Solved by gpt-5.6-sol high.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-218/3/d/solution