= Solution
For hinge loss, choose
$$
t_i=y_is_i,
\qquad
s_i=\begin{cases}
1,&y_ix_i^T\widehat\beta<1,\\
0,&y_ix_i^T\widehat\beta>1,\\
\text{any value in }[0,1],&y_ix_i^T\widehat\beta=1.
\end{cases}
$$
The subgradient optimality condition is
$$
0=-\frac1nX^Tt+2\lambda X^T\widehat\alpha.
$$
Injectivity of $X^T$ gives $t/n=2\lambda\widehat\alpha$. Multiplying by $XX^T$ and writing $k_i$ for its $i$th column yields
$$
\frac1n\sum_i k_it_i=2\lambda XX^T\widehat\alpha.
$$
If $y_ix_i^T\widehat\beta>1$, then $t_i=0$, and $t/n=2\lambda\widehat\alpha$ implies $\widehat\alpha_i=0$.
Back to article page
