Consider the problem in Exercise 16a above. Show that [ left|ell_{mathscr{T}}left(g_{mathscr{T}}^{mathscr{G}} ight)-ellleft(g^{mathscr{G}} ight) ight| leqslant 2 sup

Consider the problem in Exercise 16a above. Show that

\[ \left|\ell_{\mathscr{T}}\left(g_{\mathscr{T}}^{\mathscr{G}}\right)-\ell\left(g^{\mathscr{G}}\right)\right| \leqslant 2 \sup _{g \in \mathscr{G}}\left|\ell_{\mathscr{T}}(g)-\ell(g)\right|+\ell_{\mathscr{T}}\left(g^{\mathscr{C}}\right)-\ell\left(g^{\mathscr{G}}\right) . \]

From this, conclude:

\[ \mathbb{E}\left|\ell_{\mathscr{T}}\left(g_{\mathscr{T}}^{\mathscr{G}}\right)-\ell\left(g^{\mathscr{G}}\right)\right| \leqslant 2 \mathbb{E} \sup _{g \in \mathscr{G}}\left|\ell_{\mathscr{T}}(g)-\ell(g)\right| \]

The last bound allows us to assess how close the training \(\ell_{\mathscr{T}}\left(g_{\mathscr{T}}^{\mathscr{G}}\right)\) is to the optimal risk \(\ell\left(g^{\mathscr{G}}\right)\) within class \(\mathscr{G}\).