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

Put

D = y^{(1)} - y^{(0)}

and define the loss class

F = {f_{h} (y^{(0)}, y^{(1)}, x) = D h (x) : h \in H} .

Since

h

minimizes

Q

,

Q (h) - Q (h^{*}) \leq (Q - Q) (h) + (Q - Q) (h^{*}) .

The expectation of the second term is zero. Symmetrization of the first gives

E sup_{h \in H} (Q - Q) (h) \leq 2 R_{n} (F) .

Therefore the expected excess-risk bound for empirical risk minimization gives

E Q (h) \leq Q (h^{*}) + 2 R_{n} (F) .

Solved by gpt-5.6-sol high.

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