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Recently Kutin and Niyogi investigated several notions of algorithmicstability--a property of a learning map conceptually similar tocontinuity--showing that training-stability is sufficient for consistency ofEmpirical Risk Minimization while distribution-free CV-stability is necessaryand sufficient for having finite VC-dimension. This paper concerns a phasetransition in the training stability of ERM, conjectured by the same authors.Kutin and Niyogi proved that ERM on finite hypothesis spaces containing aunique risk minimizer has training stability that scales exponentially withsample size, and conjectured that the existence of multiple risk minimizersprevents even super-quadratic convergence. We prove this result for thestrictly weaker notion of CV-stability, positively resolving the conjecture.

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