Past exam of the advanced topics in statistical machine learning course of the University of Oxford / 2025 / 2 / c / iii / Solution

For $h$ in the Reproducing kernel Hilbert space ${\mathcal{H}}_k$, take the directional derivative:The reproducing property says $h(x_i)=⟨k(x_i,\cdot ),h{⟩}_{{\mathcal{H}}_k}$. HenceEach kernel section belongs to the Reproducing kernel Hilbert space, so this finite sum is an admissible element. Uniqueness of a vector with these inner products gives the functional gradient