Kullback-Leibler aggregation and misspecified generalized linear models

The last decade has witnessed a growing interest in the general problem of aggregation, which turned out to be a flexible way to capture many statistical learning setups. Originally introduced in the regression framework by Nemirovski (2000) and Juditsky and Nemirovski (2000) as an extension of the problem of model selection, aggregation became a mature statistical field with the papers of Tsybakov (2003) and Yang (2004) where optimal rates of aggregation were derived. Subsequent applications to density estimation [Rigollet and Tsybakov (2007)] and classification [Belomestny and Spokoiny(2007)] constitute other illustrations of the generality and versatility of aggregation methods. The general problem of aggregation can be described as follows. Consider a finite family H (hereafter called dictionary) of candidates for a certain statistical task.