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On the Stability of Empirical Risk Minimization in the Presence of Multiple Risk Minimizers

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2 Author(s)
Benjamin I. P. Rubinstein ; Microsoft Research, Silicon Valley, CA, USA ; Aleksandr Simma

Recently, Kutin and Niyogi investigated several notions of algorithmic stability-a property of a learning map conceptually similar to continuity-showing that training stability is sufficient for consistency of empirical risk minimization (ERM) while distribution-free CV-stability is necessary and sufficient for having finite VC-dimension. This paper concerns a phase transition in the training stability of ERM, conjectured by the same authors. Kutin and Niyogi proved that ERM on finite hypothesis spaces containing a unique risk minimizer has training stability that scales exponentially with sample size, and conjectured that the existence of multiple risk minimizers prevents even super-quadratic convergence. We prove this result for the strictly weaker notion of CV-stability, positively resolving the conjecture.

Published in:

IEEE Transactions on Information Theory  (Volume:58 ,  Issue: 7 )