Determine the compound minimax rate for discrete output laws

Determine the compound minimax recalibration risk when the common output law \(\mu\) has \(L\) atoms, as a function of \(m\), \(n\), \(L\), the sup-distance heterogeneity parameter \(\tau\), and the variation parameter \(V\), including the interpolation between a parametric regime for small \(L\) and the nonparametric rate \(\Phi(m,n,\tau,V)\).

Background

The main compound minimax theorem assumes that the common output law is non-atomic. With an atomic law, repeated output values permit direct per-atom averaging and can yield parametric rates; for example, when μ\mu is a point mass and τ=0\tau=0, the pooled label mean has risk at most $1/(4mn)$.

The lower-bound construction in the paper relies on many distinct output locations and therefore does not apply directly to finite grids. The desired result should describe how the effective number of atoms changes the balance between parametric estimation and isotonic nonparametric estimation.

References

Determine the compound minimax rate when $\mu$ has $L$ atoms, as a function of $(m,n,L,\tau,V)$. Per-atom averaging suggests a parametric regime when $L$ is small compared with $(mn){1/3}$, and the answer should interpolate between it and $\Phi$.

— Recalibrating Many Questions with Few Labels: Minimax Rates and Identification Limits  (2610.18488 - Li et al., 13 Oct 2026) in Section 9, Section 9.3; also Appendix “Scope and limitations”

Per-question tuning around a shape estimated from the pooled data is open problem~7. For that class there is numerical evidence (E1) that it also loses at moderate sizes (\cref{fig:e01-adaptivity}(a),(b)).

— Recalibrating Many Questions with Few Labels: Minimax Rates and Identification Limits  (2610.18488 - Li et al., 13 Oct 2026) in Section 9, Section 9.7; also Remark “Scope of Proposition 4.4” and Appendix “Scope and limitations”

A worst-question guarantee, on $\max_jE#1{g_j-g_j}2$, for the adaptive procedure is open problem~8.

— Recalibrating Many Questions with Few Labels: Minimax Rates and Identification Limits  (2610.18488 - Li et al., 13 Oct 2026) in Section 9, Section 9.8; also Appendix “Scope and limitations”