Average Stability is Invariant to Data Preconditioning. Implications to Exp-concave Empirical Risk Minimization

TL;DR

This paper proves that average stability in generalized linear models remains unchanged under data preconditioning, suggesting regularization may be unnecessary for handling ill-conditioned data in ERM and SGD.

Contribution

It establishes the invariance of average stability to data preconditioning for a broad class of models, impacting analysis of ERM and SGD.

Findings

Stability rate is invariant to data preconditioning.

ERM achieves fast rates controlled by preconditioned stability.

Improved bounds on SGD stability rate.

Abstract

We show that the average stability notion introduced by \cite{kearns1999algorithmic, bousquet2002stability} is invariant to data preconditioning, for a wide class of generalized linear models that includes most of the known exp-concave losses. In other words, when analyzing the stability rate of a given algorithm, we may assume the optimal preconditioning of the data. This implies that, at least from a statistical perspective, explicit regularization is not required in order to compensate for ill-conditioned data, which stands in contrast to a widely common approach that includes a regularization for analyzing the sample complexity of generalized linear models. Several important implications of our findings include: a) We demonstrate that the excess risk of empirical risk minimization (ERM) is controlled by the preconditioned stability rate. This immediately yields a relatively short and elegant proof for the fast rates attained by ERM in our context. b) We strengthen the recent bounds of \cite{hardt2015train} on the stability rate of the Stochastic Gradient Descent algorithm.