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Abstract:
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In recent works with D.M. Mason we have developed strong invariance principles which allow one to take advantage of the nice Gaussian nature of empirical processes. Let consider the usual setting in functional data analysis, non-parametric regression, high dimensional classification, model selection and learning theory : an empirical risk minimizer f_n is selected among a class F of functions, large compared to the size n of the available i.i.d. sample with law P. Thanks to the Gaussian approximation we evaluate the true L_1 or L_2 volume of a r-neighborhood N of f_n in terms of F and the deterministic empirical L_1 radius r. The ERM algorithm is stable if this volume is small with high probability. This is the case when r is small enough compared to a quantity depending on F, P and square root of n. We also study the small probability of catching the true minimizer in N.
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