Optimal learning with Bernstein Online Aggregation

We introduce a new recursive aggregation procedure called Bernstein Online Aggregation (BOA). The exponential weights include an accuracy term and a second order term that is a proxy of the quadratic variation as in Hazan and Kale (2010). This second term stabilizes the procedure that is optimal in different senses. We first obtain optimal regret bounds in the deterministic context. Then, an adaptive version is proved to solve the so-called impossible tuning problem already solved in Gaillard et al. (2014). The second order bounds in the deterministic context are extended to a general stochastic context using the cumulative predictive risk. Such conversion provides the main result of the paper, an inequality of a novel type comparing the procedure with any deterministic aggregation procedure for an integrated criteria. It provides an observable confident interval on the excess of risk of the BOA procedure. To assert the optimality, we consider finally the iid case for strongly convex and Lipschitz continuous losses and we prove that the rate of convergence is of the optimal order given in Tsybakov (2003). The batch version of the BOA procedure is then the first adaptive solution satisfying an optimal oracle inequality with high probability.