model, but derive refined risk bounds (with a different approach) that generalize the worst-case bounds of [@Cadre], as indicated below. [**Some notation.**  ]{} We introduce some notation and definitions in order to present our contributions below. In our setting, a *classifier* $\Phi$ is a measurable function, possibly depending on the sample $\mathcal{S}_n$, that maps each new input $X=(X(t))_{t\in[0,1]}$ to a label in $\{0,1\}$. The *risk* associated with each classifier $\Phi$ depends on $f$ and $g$ and is defined by: $$\mathcal{R}_{f,g}(\Phi) := {\mathbb{P}}(\Phi(X) \not = Y) \,,$$ where the expectation is taken with respect to all sources of randomness ([i.e.]{}, both the sample $\mathcal{S}_n$ and the pair $(X,Y)$). The goal of the learner is to construct a classifier ${\widehat}{\Phi}$ based on the sample $\mathcal{S}_n$ that mimics the *Bayes classifier* $$\Phi^\star = {\operatornamewithlimits{arg\,min}}_\Phi \mathcal{R}_{f,g}(\Phi) \,,
\label{eq:oracle}$$ where the infimum is taken over all possible classifiers (the oracle $\Phi^\star$ is impractical since $f$ and $g$ and thus $\mathbb{P}_{X,Y}$ are unknown). We measure the quality of ${\widehat}{\Phi}$ through its *worst-case excess risk* $$\label{eq:max}
 \sup_{(f,g) \in \mathcal{E}} \left\{\mathcal{R}_{f,g}(\Phi) - \inf_{\Phi} \mathcal{R}_{f,g}(\Phi)\right\}$$ over some set $\mathcal{E}$ of pairs of functions. In the sequel we focus on Sobolev classes ${\mathcal{H}_s}(R)$ (see ) and consider subsets $\mathcal{E} \subseteq 