  We consider the classical statistical learning/regression problem, when the
value of a real random variable Y is to be predicted based on the observation
of another random variable X. Given a class of functions F and a sample of
independent copies of (X, Y ), one needs to choose a function f from F such
that f(X) approximates Y as well as possible, in the mean-squared sense. We
introduce a new procedure, the so-called median-of-means tournament, that
achieves the optimal tradeoff between accuracy and confidence under minimal
assumptions, and in particular outperforms classical methods based on empirical
risk minimization.
