Algorithmic stability and sanity-check bounds for leave-one-out cross-validation

Area of Science:

• Machine Learning
• Statistical Learning Theory
• Algorithmic Stability

Background:

• Leave-one-out cross-validation (LOOCV) is a common method for estimating generalization error.
• Prior bounds on LOOCV error relied on strong hypothesis stability, limiting their applicability.
• Assurance that LOOCV error is not substantially worse than training error has been limited.

Purpose of the Study:

• To derive sanity-check bounds for the error of the LOOCV estimate of generalization error.
• To provide assurance that LOOCV performance is not considerably worse than training error estimates.
• To extend LOOCV error bounds to a wider range of learning algorithms.

Main Methods:

• Introduction of a new, weaker notion of "error stability".
• Application of error stability to derive sanity-check bounds for LOOCV.
• Analysis of bounds for training error minimization and Bayesian algorithms.
• Derivation of lower bounds to demonstrate the necessity of error stability.

Main Results:

• Sanity-check bounds for LOOCV error are established.
• Error stability is shown to be applicable to broader classes of algorithms beyond local methods.
• Lower bounds confirm the necessity of error stability and highlight dependence on Vapnik-Chervonenkis dimension for certain algorithms.

Conclusions:

• The new error stability notion enables broader theoretical guarantees for LOOCV.
• The findings provide crucial assurance for the reliability of LOOCV estimates.
• Theoretical limitations exist for training error minimization algorithms, necessitating consideration of hypothesis class complexity.