Statistical Learning: Stability is Sufficient for Generalization and Necessary and Sufficient for Consistency of Empirical Risk Minimization

Abstract : 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 leave-one-out stability, called CVEEE(loo) stability. Our main new results are two. We prove that for bounded loss classes CVEEE(loo) stability is (a) sufficient for generalization, that is convergence in probability of the empirical error to the expected error, for any algorithm satisfying it and, (b) necessary and sufficient for generalization and consistency of ERM. Thus CVEEE(loo) stability is a weak form of stability that represents a sufficient condition for generalization for general learning algorithms while subsuming the classical conditions for consistency of ERM. We discuss alternative forms of stability. In particular, we conclude that for ERM a certain form of well-posedness is equivalent to consistency.

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