Remove neighboring-density-ratio dependence from single-score minimax bounds

Determine whether the dependence on the upper bound of the neighboring density ratio can be removed from the minimax lower and upper bounds for single concrete-score estimation under the score-entropy loss.

Background

The paper establishes minimax lower and upper bounds for estimating a single concrete score function under the score-entropy loss over distributions whose neighboring density ratios are bounded. Although the bounds match up to polylogarithmic factors when the ratio bound is treated as fixed, they differ by a factor that depends on this bound.

The authors explicitly identify as unresolved whether this dependence is necessary or merely an artifact of the analysis. Resolving the issue would sharpen the statistical characterization of single-score estimation for distributions with varying neighboring density ratios.