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

Let $J(t)$ be the $n\times d$ Jacobian matrix with row $J_i(t)={\nabla }_{\theta }f(\theta (t),x_i{)}^T$. Differentiating the squared-error loss function givesThe parameter gradient flow is consequently $\dot{\theta }=J^T(y-u)$. The chain rule then giveswhere, explicitly,It is a Gram matrix and therefore a positive semidefinite matrix: $v^{T}H(t)v=\Vert J(t{)}^{T}v{\Vert }_2^2\ge 0$. These derivatives are understood wherever the network is differentiable; for piecewise differentiable activations the same calculation applies along differentiable portions of the trajectory.