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Using the harmless normalization $1/(2n)$, the squared-loss empirical risk for linear prediction iswith explicit gradientStarting from any ${\beta }_0\in C$, projected gradient descent with step sizes ${\eta }_k>0$ is

For a fully explicit update, write ${\tilde{\beta }}_{k+1}=(u_k,t_k)$ and ${\rho }_k=\Vert u_k{\Vert }_2$. The projection from part (e) isEvery iterate therefore lies in the prescribed hypothesis class, and any convergent run under the standard convex-optimization step-size conditions targets its empirical-risk minimizer.