## Section: New Results

Keywords : Statistical learning theory, support vector machine, model selection.

### Structural risk minimization inductive principle for multi-class discriminant analysis

Participants : Yannick Darcy, Yann Guermeur, Frédéric Sur.

We have continued our study of the generalization error of large margin multi-class discriminant models, laying emphasis on the use of bounds for model selection. A first algorithm of model selection, dedicated to M-SVMs, was based on a bound on the entropy numbers of the evaluation operator [27] . The computation of tighter bounds on those entropy numbers is still a work in progress. It takes the form of the derivation of a generalized formulation of the Maurey-Carl theorem [33] . Those bounds will then be compared with those involving extended Sauer's lemmas and generalized VC dimensions [18] . In parallel, the work on the computation of estimates of the risk based on the leave-one-out procedure has given birth to a first theorem [31] , extending Chapelle's "radius-margin bound". All the aforementioned bounds are progressively incorporated in our M-SVM software, where they can be used to select the soft margin parameter C.