all $n {\geqslant}N_{s,R}$, the classifier ${\widehat}\Phi_{d_n}$ defined by  with $d_n = \lfloor (R^2 n)^{\frac{1}{2s+1}} \rfloor $ satisfies $$\sup_{\substack{f,g \in {\mathcal{H}_s}(R) \\ \|f-g\| {\geqslant}\Delta}} \left\{ \mathcal{R}_{f,g}({\widehat}\Phi_{d_n}) - \inf_{\Phi} \mathcal{R}_{f,g}(\Phi)  \right\}  {\leqslant}\left\{\begin{array}{ll}
    c \, R^{\frac{1}{2s+1}} n^{-\frac{s}{2s+1}} \log(n) & \textrm{if } \Delta < R^{\frac{1}{2s+1}} n^{-\frac{s}{2s+1}} \log(n) \\[0.2em]
    \displaystyle \frac{c}{\Delta} R^{\frac{2}{2s+1}} n^{-\frac{2s}{2s+1}} \log^2(n) & \textrm{if } \Delta {\geqslant}R^{\frac{1}{2s+1}} n^{-\frac{s}{2s+1}} \log(n)
\end{array}\right.$$

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The proof is postponed to Appendix \[s:pminmax\] and combines Lemmas \[prop:CO\] and \[lem:proj\] from the previous sections. A proof sketch is also provided below.

Note that the two bounds of the right-hand side coincide when $\Delta = R^{1/(2s+1)} \, n^{-s/(2s+1)} \log(n)$. Therefore, there is a continuous transition from a slow rate (when $\Delta$ is small) to a fast rate (when $\Delta$ is large). This leads to the following remark.

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-   Taking $\Delta=0$, we recover the worst-case bound of [@Cadre Corollary 4.4] (where $u=1/s$) up to logarithmic factors. As shown by Theorem \[thm:minimaxlowerbound\] below, this slow rate is unimprovable for a small distance $\| f- g \|$.

-   However, we obtain a significantly faster rate when $\| f- g \|$ is bounded from below. This improved rate is a consequence of the margin behavior (see, [e.g.]{}, [@AT07; @GKM16]), but not of the 