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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 1

Put $D=y^{(1)}-y^{(0)}$ and define the loss class
$$
\mathcal F=\{f_h(y^{(0)},y^{(1)},x)=D\,h(x):h\in H\}.
$$
Since $\widehat h$ minimizes $\widehat Q$,
$$
\begin{aligned}
Q(\widehat h)-Q(h^*)
&\leq (Q-\widehat Q)(\widehat h)
 +(\widehat Q-Q)(h^*).
\end{aligned}
$$
The expectation of the second term is zero. Symmetrization of the first gives
$$
\mathbb E\sup_{h\in H}(Q-\widehat Q)(h)
\leq2R_n(\mathcal F).
$$
Therefore the <expected excess-risk bound for empirical risk minimization> gives
$$
\boxed{\mathbb E Q(\widehat h)\leq Q(h^*)+2R_n(\mathcal F).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 31K / b / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 31K / b / i / Solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 31K / b / i / Solution

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