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

Zhenisbek Assylbekov; Rustem Takhanov

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arXiv · 2610.20402

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

Zhenisbek Assylbekov·Rustem Takhanov

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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.

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BibTeX RIS

Zhenisbek Assylbekov, Rustem Takhanov. 2026-10-14. Posterior Frequencies and the Cost of Not Fitting: A Correction to a Memorization Lower Bound. https://arxiv.org/abs/2610.20402

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