Bayes classifier of the restricted classification problem where the learner has only access to the projected input ${\mathbf{X}}_d \in {\mathbb{R}}^d$, rather than the whole trajectory $X$.\
We are now ready to reinterpret ${\widehat}\Phi_d$ as a plug-in classifier. Note that $$\begin{aligned}
\| \mathbf{X}_d - {\widehat}\theta \|_d {\leqslant}\| \mathbf{X}_d- {\widehat}\mu \|_d, 
& \Longleftrightarrow & \frac{\exp\left( \frac{1}{2} \lbrace\| \mathbf{X}_d - {\widehat}\mu \|_d^2 - \| \mathbf{X}_d- {\widehat}\theta \|_d^2 \rbrace \right)}{1+\exp\left( \frac{1}{2} \lbrace\| \mathbf{X}_d - {\widehat}\mu \|_d^2 - \| \mathbf{X}_d- {\widehat}\theta \|_d^2 \rbrace\right)} {\geqslant}\frac{1}{2} \;, \\
& \Longleftrightarrow & {\widehat}\eta_d(X)  {\geqslant}\frac{1}{2} \;,\end{aligned}$$ where the estimated regression function ${\widehat}{\eta}_d$ is defined by $$\label{eq:etadestim}
{\widehat}\eta_d(X):= \frac{\exp\left( \langle {\widehat}\theta - {\widehat}\mu , {\mathbf{X}}_d \rangle_d -\frac{1}{2}\|{\widehat}\theta\|_d^2 + \frac{1}{2}\|{\widehat}\mu\|_d^2 \right)}{1+\exp\left( \langle {\widehat}\theta - {\widehat}\mu, {\mathbf{X}}_d \rangle_d -\frac{1}{2}\|{\widehat}\theta\|_d^2 + \frac{1}{2}\|{\widehat}\mu\|_d^2 \right)} \;.$$ In other words, our classifier ${\widehat}\Phi_d$ can be rewritten as ${\widehat}\Phi_d(X) = \mathds{1}_{\lbrace {\widehat}\eta_d(X) {\geqslant}1/2  \rbrace}$ where ${\widehat}\eta_d$ is an estimator of the ’truncated’ regression function $\eta_d$ introduced in .

[**Proof strategy.**  ]{} In the next sections we upper bound the excess risk of ${\widehat}\Phi_d$. We use the following classical decomposition (all quantities below are defined in Section \[sec:intro\], , and ): $$\mathcal{R}_{f,g}({\widehat}{\Phi}_{d}) - \mathcal{R}_{f,g}(\Phi^\star) = \underbrace{\mathcal{R}_{f,g}({\widehat}{\Phi}_{d}) - \mathcal{R}_{f,g}(\Phi_d^{\star})}_{\textrm{estimation error}} +
\underbrace{\mathcal{R}_{f,g}(\Phi_d^{\star}) - \mathcal{R}_{f,g}(\Phi^\star)}_{\textrm{approximation error}} \,.$$ The first term 