[hal-00973918] 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 the first exponential weights algorithm that exhibits a second order bound with excess losses that appears first 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 Then we obtain an observable estimate of 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 optimal rate of aggregation of Tsybakov 2003 is achieved The batch version of the BOA procedure is then the first adaptive explicit algorithm that satisfies an optimal oracle inequality with high probability



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