Learning theory: stability is sufficient for generalization and necessary and sufficient for consistency of empirical risk minimization

Sayan Mukherjee, Partha Niyogi, Tomaso Poggio, Ryan Rifkin
2006 Advances in Computational Mathematics  
Solutions of learning problems by Empirical Risk Minimization (ERM) -and almost-ERM when the minimizer does not exist -need to be consistent, so that they may be predictive. They also need to be well-posed in the sense of being stable, so that they might be used robustly. We propose a statistical form of stability, defined as leave-one-out (LOO) stability. We prove that for bounded loss classes LOO stability is (a) sufficient for generalization, that is convergence in probability of the
more » ... l error to the expected error, for any algorithm satisfying it and, (b) necessary and sufficient for consistency of ERM. Thus LOO stability is a weak form of stability that represents a sufficient condition for generalization for symmetric learning algorithms while subsuming the classical conditions for consistency of ERM. In particular, we conclude that a certain form of well-posedness and consistency are equivalent for ERM.
doi:10.1007/s10444-004-7634-z fatcat:64426vw54fd5vn2thn6apemeui