Abstract
We present a new family of model selection algorithms based on the resampling heuristics. It can be used in several frameworks, do not require any knowledge about the unknown law of the data, and may be seen as a generalization of local Rademacher complexities and $V$-fold cross-validation. In the case example of least-square regression on histograms, we prove oracle inequalities, and that these algorithms are naturally adaptive to both the smoothness of the regression function and the variability of the noise level. Then, interpretating $V$-fold cross-validation in terms of penalization, we enlighten the question of choosing $V$. Finally, a simulation study illustrates the strength of resampling penalization algorithms against some classical ones, in particular with heteroscedastic data.
Highlights
IntroductionWhen the final goal is prediction, model selection can be seen more generally as the question of choosing between the outcomes of several prediction algorithms
In the last decades, model selection has received much interest
We show that Resampling Penalization can be used to build an estimator adaptive to the smoothness of s in this framework
Summary
When the final goal is prediction, model selection can be seen more generally as the question of choosing between the outcomes of several prediction algorithms. With such a general formulation, a natural and classical answer is the following. Estimate the prediction error for each model or algorithm; second, select the model minimizing this criterion. Model selection procedures mainly differ on the way of estimating the prediction error. Minimizing the empirical risk can fail dramatically: the empirical risk is strongly biased for models involving a number of parameters growing with the sample size because the same data are used for building predictors and for comparing them
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