In the standard nonparametric regression setting, it is assumed that the model is well-specied, i.e., the unknown regression function belongs to a given functional class. The value of reference characterizing the best estimation is the minimax risk over this class. On the other hand, in Machine Learning no assumption is placed on the regression function; it does not necessarily belong to the specied class and the corresponding value of reference can be written as the minimax regret (mini-max excess risk). The study of the minimax risk and of the minimax regret represents two parallel developments: the former has been analyzed mostly within Nonparametric Statistics, while the second { within Statistical Learning Theory. This talk aims to bring out a connection between these two objects. We introduce a more general risk measure that realizes a smooth transition between the minimax risk and the minimax regret depending on the magnitude of the approximation error. The minimax risk and the minimax regret appear as the two extremities of this scale. The main result shows that, unless the functional class is extremely "massive", the minimax risk and minimax regret have the same asymptotic behavior. Furthermore, the optimal rates for the minimax regret and minimax risk are attained by one and the same procedure called the aggregation-of-leaders estimator while they are not attained by classical procedures such as empirical risk minimization and skeleton aggregation. For very "massive" classes the minimax risk turns out to be of smaller order than the minimax regret. As a by-product of general results, we obtain oracle inequalities for the excess risk
involving Vapnik-Chervonenkis type classes and classes with polynomial growth of empirical entropy without the usual convexity assumption on the class. Finally, for a slightly modied method, we derive a bound on the excess risk of s-sparse convex aggregation providing the optimal rate. This is a joint work with Alexander Rakhlin and Karthik Sridharan.