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**Sample text**

4 The version here includes the regularization term for b suggested in a footnote of [10]. Generalization Bounds for Some Ordinal Regression Algorithms 19 6 Margin Bound for Chu & Keerthi’s Algorithm We consider now a different approach to analyzing ordinal regression algorithms that learn both a real-valued function f : X→R and a set of thresholds b1 ≤ . . ≤ br−1 ≤ br = ∞, and then make label predictions according to the prediction rule gf,b defined in Eq. (13) (recall that b is the threshold vector (b1 , .

The approach relies on linear-by-parts approximations of the ROC curve which correspond to ﬁnitedimensional (piecewise constant) approximations of optimal scoring functions. As the ROC curve provides a performance measure of functional nature, the approximation can be conceived in a variety of ways depending on the topology equipping the space of ROC curves. For instance, the AUC is related to the L1 distance but we will also consider convergence to the optimal ROC curve in a stronger sense described by the L∞ -distance.

There exists a sequence of piecewise constant scoring functions (sN )N ≥1 such that, for any N ≥ 1, sN ∈ SN and: d1 (s∗ , sN ) ≤ C · N −2 and d∞ (s∗ , sN ) ≤ C · N −2 , where the constant C depends only on the distribution. The approximation rate of O(N −2 ) is actually reached by any piecewise linear approximation provided that the mesh length is of order O(N −1 ). This result is well-known folklore in approximation theory, see [DL93]. We underline that the 28 S. Cl´emen¸con and N. Vayatis piecewise linear approximation method we describe next is adaptive in the sense that breakpoints are not ﬁxed in advance and strongly depend on the target curve (which suggests that this scheme possibly yields a sharper constant C).