(I will write the prediction $\hat{y}$ as $y$.)
$argmin_k \lvert\lvert t_k - y \lvert\lvert = argmin_k (t_k - y)^T (t_k - y) = argmin_k [{t_k}^T t_k - 2 y^T t_k + y^Ty]$.
${t_k}^T{t_k}$ is 1 for any $k$ and $y^Ty$ is not affected by $k$. So, the problem is equivalent to $argmax_k [y^T t_k]$, which picks the position in $y$ with the largest element.