of the right-hand side (estimation error) measures how close ${\widehat}{\Phi}_{d}$ is to the oracle $\Phi_d^{\star}$ in the trunctated space; we analyse it in Section \[sec:estimationerror\] below. The second term (approximation error) quantifies the statistical loss induced by the $d$-dimensional projection; we study it in Section \[sec:trunc\].

Approximation error {#sec:trunc}
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We first upper bound the approximation error $\mathcal{R}_{f,g}(\Phi_d^{\star}) - \mathcal{R}_{f,g}(\Phi^\star)$, where the two oracle classifiers $\Phi_d^{\star}$ and $\Phi^\star$ are defined by  and  respectively. Comparing the definitions of $\eta$ and $\eta_d$ in  and , we can expect that, for $d$ large enough, $\Pi_d(f) \approx f$ and $\Pi_d(g) \approx g$, so that $\eta_d(X) \approx \eta(X)$ and therefore $\mathcal{R}_{f,g}(\Phi_d^\star) \approx \mathcal{R}_{f,g}(\Phi^\star)$.

Lemma \[prop:CO\] below quantifies this approximation. The proof is postponed to Appendix \[s:pCO\]. We recall that, for notational convenience, we write $f_d=\Pi_d(f)$ and $g_d=\Pi_d(g)$.

\[prop:CO\] Let $X$ be distributed according to Model (\[eq:model\]), and recall that $\Delta := \|f-g\|$. Let $0<{\varepsilon}{\leqslant}1/8$ and $d\in \mathbb{N}^\star$ such that $$\max \left( \| f-f_d\|^2,\| g-g_d \|^2   \right) {\leqslant}\frac{{\varepsilon}^2}{512 \ln(1/{\varepsilon}^2)}\,.
\label{eq:lemma1conditioneps}$$ Then, the two oracle classifiers $\Phi_d^\star$ and $\Phi^\star$ defined by  and  satisfy $$\mathcal{R}_{f,g}(\Phi_d^\star) - \mathcal{R}_{f,g}(\Phi^\star) {\leqslant}12 {\varepsilon}^2 + 2 {\varepsilon}\left( 1 \wedge \frac{10{\varepsilon}}{\Delta} \right) \,.$$

We stress that the distance $\Delta$ between $f$ and $g$ has 