$Y=0$, this entails that $\eta_d(X) = {\mathbb{P}}(Y=1|{\mathbf{X}}_d)$ almost surely.

In other words, $\eta_d$ is the regression function of the restricted classification problem where the learner has only access to the projected trajectory ${\mathbf{X}}_d \in {\mathbb{R}}^d$, instead of the whole trajectory $X$. The function $\Phi^*_d = \mathds{1}_{\{\eta_d {\geqslant}1/2\}}$ is the associated Bayes classifier.

 \
We are now ready to compare the risk of our classifier ${\widehat}\Phi_d$ to that of the $d$-dimensional oracle $\Phi_d^\star$. The proof of the next lemma is postponed to Appendix \[s:pproj\]. (The value of $4608$ could most probably be improved.) We recall that $f_d=\Pi_d(f)$ and $g_d=\Pi_d(g)$.

\[lem:proj\] We consider Model (\[eq:model\]). Let $d\in \mathbb{N}^*$ and set $\Delta_d := \| f_d - g_d \|$. Let $0 < {\varepsilon}{\leqslant}1/8$ and $n {\geqslant}27$ such that $$\left(\Delta_d + 2 \sqrt{\frac{d\log(n)}{n}} \, \right) \sqrt{\frac{d\log(n)}{n}} {\leqslant}\frac{{\varepsilon}}{48} \;.
\label{eq:lemma2conditioneps}$$ Then, the classifiers ${\widehat}\Phi_d$ and $\Phi_d^\star$ defined by  and  satisfy $$\mathcal{R}_{f,g}({\widehat}\Phi_d) - \mathcal{R}_{f,g}(\Phi_d^\star) {\leqslant}2 {\varepsilon}\left( 1\wedge \frac{10 {\varepsilon}}{\Delta_d}  \right) + 6 \exp \left(  - \frac{n {\varepsilon}^2}{4608 \, d \log n } \right) + \frac{13}{n} \;.$$

In the same vein as for the approximation error, the estimation error bound above strongly depends on the distance between the two functions $f_d$ and $g_d$ of interest. This is again 