  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 depart from traditional proofs by making most of the
standard arguments on the Laplace transform of the empirical process to be
controlled.
