  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 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.
