choice of $d_n$ that is oblivious to $\| f- g \|$.

-   Continuous transitions from slow rates to faster rates were already derived in the past. For instance, for any supervised classification problem where the margin $\big|2\eta(X)-1\big| {\geqslant}h$ is almost surely bounded from below, [@MaNe-06-RiskBounds Corollary 3] showed that the excess risk w.r.t. a class of VC-dimension $V$ varies continuously from $\sqrt{V/n}$ to $V/(nh)$ (omitting log factors) as a function of the margin parameter $h$. In a completely different setting, [@RaSrTs-13-MinimaxRegretRisk Theorem 5] analyzed the minimax excess risk for nonparametric regression with well-specified and misspecified models. They showed a continuous transition from slow to faster rates when the distance of the regression function to the statistical model decreases to zero.

We finally note that, though the choice of the parameter $d_n$ does not depend on $\Delta$, it still depends on the (possibly unknown) smoothness parameter $s$. Though designing an adaptive classifier is beyond the scope of this paper, it might be addressed via the Lepski method (see, [e.g.]{}, [@L90]) after adapting it to the classification setting.

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*Sketch of the proof.* We outline the main ingredients. For all $d \in {\mathbb{N}}^*$ and $0 < {\varepsilon}_1,{\varepsilon}_2 {\leqslant}1/8$ satisfying the assumptions of 