Posterior Frequencies and the Cost of Not Fitting: A Correction to a Memorization Lower Bound

Abstract:Feldman's memorization lower bound assigns a frequency coefficient based on how often a point appears in the training sample. We give an exact three-point counterexample to its conditional and averaged forms, using independent fair labels and a permutation-equivariant learner. A sequence of sampling priors makes the ratio of the proposed penalty to actual excess risk grow exponentially, ruling out a universal positive constant-factor repair. The priors vary with sample size, the offending profiles become rare, and both quantities vanish. Retaining the original model, we obtain an exact replacement using full-sample posterior probabilities. Under exchangeability, these depend on multiplicity and the complete frequency profile. We characterize the largest multiplicity-only coefficients valid for unrestricted learners. The original coefficient has the posterior-mean form of Good's exact frequency formula and is an occupancy-weighted average of the corrected coefficients.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2610.20402 [stat.ML]

(or arXiv:2610.20402v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2610.20402 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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