Determine the monotone-map identification-regret base

Prove that, for priors on nondecreasing maps from [0,1] to [0,1], the identification regret has the same exponential base as for constant-map priors, namely that \(\lim_{n\to\infty}(\sup_G \operatorname{Reg}_n)^{1/n}=1/\beta_*\), and establish the quantitative polynomial comparison \(\sup_G\operatorname{Reg}_n\le \operatorname{poly}(n)\sup_G\operatorname{Reg}_n\) between monotone-map and constant-map classes; determine the unresolved polynomial prefactor in the constant-map bounds.

Background

The paper proves that the identification regret for priors on monotone recalibration maps is bounded above by 2−n2^{-n} and bounded below, through constant-map priors, by approximately β∗−n/n\beta_*^{-n}/\sqrt n, where β∗=2e2Cat/π≈3.583\beta_*=2e^{2\mathrm{Cat}/\pi}\approx3.583. For constant maps, the exponential base is known exactly, but the available upper and lower bounds still differ by a factor of order n\sqrt n.

The authors conjecture that allowing arbitrary monotone maps does not increase the exponential base beyond the constant-map value. Their proof controls the unweighted diameter of the unidentified moment, whereas the conjectured result requires a weighted inequality accounting for rare label patterns.

References

Prove \cref{conj:monotone-base}: for priors on monotone maps the identification regret decays with base $$, as for constant maps, rather than with base $2$. A quantitative form would be $\sup_{G}Reg_n\le\poly(n)\sup_{G}Reg_n$. The upper bound of \cref{thm:monotone-eb} uses only the diameter of the unidentified moment, which \cref{lem:ch1} controls on sorted vectors; the base $$ comes instead from a weighted inequality, the Szegő-type bound of \cref{thm:betastar}, in which the moment range is traded against the probabilities of the label patterns. The natural route is a weighted analogue of \cref{lem:ch1} on the ordered simplex, and we do not know one. Even for constant maps the polynomial prefactor is open: our bounds differ by $72\sqrt n$, and the numerics of \cref{fig:e03-barrier} suggest a prefactor that decays, but slowly.

— Recalibrating Many Questions with Few Labels: Minimax Rates and Identification Limits  (2610.18488 - Li et al., 13 Oct 2026) in Section 9, Section 9.1, and Conjecture 9.1 (Section “The monotone base is β*”)

The matching finite-$m$ upper bound, an empirical-Bayes procedure with excess at most $2{-n}+o(1)$ as $m\to\infty$, is open problem~2 of \cref{sec:open}.

— Recalibrating Many Questions with Few Labels: Minimax Rates and Identification Limits  (2610.18488 - Li et al., 13 Oct 2026) in Section 9, Section 9.2; also Section “Every finite number of questions” and Appendix “Scope and limitations”

Proving $\sup_HW_n(H)\ge2{-n}/\poly(n)$ is open problem~9.

— Recalibrating Many Questions with Few Labels: Minimax Rates and Identification Limits  (2610.18488 - Li et al., 13 Oct 2026) in Section 9, Section 9.9; also Remark “Skibinsky’s gap appears to decay with base 2”