Optimal learning with Bernstein online aggregation

We introduce a new recursive aggregation procedure called Bernstein Online Aggregation (BOA). Its exponential weights include a second order refinement. The procedure is optimal for the model selection aggregation problem in the bounded iid setting for the square loss: the excess of risk of its batch version achieves the fast rate of convergence log (M)/n in deviation. The BOA procedure is the first online algorithm that satisfies this optimal fast rate. The second order refinement is required to achieve the optimality in deviation as the classical exponential weights cannot be optimal, see Audibert (Advances in neural information processing systems. MIT Press, Cambridge, MA, 2007 ). This refinement is settled thanks to a new stochastic conversion that estimates the cumulative predictive risk in any stochastic environment with observable second order terms. The observable second order term is shown to be sufficiently small to assert the fast rate in the iid setting when the loss is Lipschitz and strongly convex. We also introduce a multiple learning rates version of BOA. This fully adaptive BOA procedure is also optimal, up to a loglog (n) factor.