© Institute of Mathematical Statistics, 2014 OPTIMAL LEARNING WITH Q-AGGREGATION

We consider a general supervised learning problem with strongly convex and Lipschitz loss and study the problem of model selection aggregation. In particular, given a finite dictionary functions (learners) together with the prior, we generalize the results obtained by Dai, Rigollet and Zhang [Ann. Statist. 40 (2012) 1878–1905] for Gaussian regression with squared loss and fixed design to this learning setup. Specifically, we prove that the Q-aggregation procedure outputs an estimator that satisfies optimal oracle inequalities both in expectation and with high probability. Our proof techniques somewhat de-part from traditional proofs by making most of the standard arguments on the Laplace transform of the empirical process to be controlled. 1. Introduction and main results. Let X be a probability space and let (X,Y) ∈ X × R be a random couple. Broadly speaking, the goal of statistical learning is to predict Y given X. To achieve this goal, we observe a dataset D = {(X1, Y1),..., (Xn,Yn)} that consists of n independent copies of (X,Y) and use these observations to construct a function (learner) f:X →R such that f (X