Matching Lower Bounds Beyond Polynomial Weak-Signal Windows

Establish whether the matching minimax lower bounds for semantic-entropy estimation under a known weak pairwise-equivalence channel hold throughout the whole window n\Delta^2\le c_1, including regimes in which n\Delta^2 decays more slowly than every power of n or remains bounded away from zero.

Background

The paper proves matching upper and lower minimax rates for estimating the semantic entropy H(\theta) when the pairwise judge has known channel parameters and the weak-signal quantity n\Delta^2 lies in any fixed polynomial window n\Delta^2\le n^{-\kappa}. Its rates are governed by N=n^K\Delta^{2K-2} and by the number of meaning probabilities that may be rare.

The authors do not establish whether these lower bounds remain valid over the larger window n\Delta^2\le c_1. This unresolved extension would cover signals that approach zero subpolynomially slowly, as well as signals for which n\Delta^2 stays bounded away from zero, and would clarify the full range in which the rarity-dependent minimax rates are sharp.