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Lhaut_08351600_2021.pdf
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- Abstract
- Motivated by the classification problem in statistical learning, we propose different bounds on the potential errors committed by an algorithm which minimizes an empirical version of the misclassification risk, inside a given class of functions. It leads us to the study of the maximal deviation between a probability measure and its empirical counterpart inside a class of Borelian sets on a Euclidian space. Even though this class could be very large (typically, she is uncountable), we associate the combinatorial quantities coming from the Vapnik and Chervonenkis theory with inequalities obtained from the Chernoff's method to provide exponential bounds on the maximal deviation, whenever the class has finite VC dimension. In particular, it leads to a generalization of the Glivenko-Cantelli's theorem to VC classes. We then propose some new results concerning the adaptation of those inequalities to the extreme value theory setting, where the probability of encountered events is very small. We illustrate and compare our different bounds in various graphics.