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

TL;DR

This paper proves that the presence of multiple risk minimizers prevents super-quadratic convergence in empirical risk minimization, confirming a conjecture about the impact of multiple minimizers on algorithmic stability.

Contribution

It establishes that multiple risk minimizers hinder super-quadratic convergence in CV-stability, resolving a conjecture about phase transition in ERM stability.

Findings

Multiple risk minimizers prevent super-quadratic convergence.

Training stability scales exponentially with sample size for unique minimizers.

The conjecture about phase transition in ERM stability is confirmed.

Abstract

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.