Establish tightness theory for more general loss functions

Establish the tightness theory of Occupancy-based Quantile Risk Control for more general loss functions that do not necessarily satisfy the ideal continuity and density conditions required by the current convergence analysis.

Background

The paper proves that the OQRC risk-control bound converges to the optimal bound at rate Op(n−1/2)\mathcal{O}_p(n^{-1/2}) under conditions including a density bounded away from zero. The appendix explains that the false-negative proportion used in the experiments is not continuous and does not satisfy this lower-density condition.

A synthetic experiment illustrates convergence under a more regular loss construction, but the authors do not prove the corresponding tightness result for broader classes of loss functions. Extending the tightness theory to such losses remains unresolved.

References

We leave the establishment of the OQRC tightness theory for more general loss functions to future work.

— Occupancy-based Quantile Risk Control  (2609.03104 - Shi et al., 2 Sep 2026) in Appendix, Section "Tightness of OQRC" (Section \ref{sec:tightness_theory})

Adaptive conformal inference gives long-run coverage for a future calibration score $s_{t}$. It does not, without further argument, yield a bound on the distance between two population functionals, $|CVaR{p_\theta}\alpha-CVaR{P}\alpha|\le q_t(\delta')$. Closing that gap requires (i) a precise statement of how the score of Definition~\ref{def:score} identifies the population CVaR difference, in particular a monotone link between score quantiles and that difference, and (ii) an argument that each adaptively proposed refinement attempt is individually valid at level $\delta/B$ before the union bound is applied over attempts. We have neither, and statement~\ref{thm:certified} should be read as conditional on that link, which is why it is not labelled a theorem.

— Verify Claims, Not Scores: Evidence-Based Verification of Modular Agents  (2610.01348 - Alzahrani, 1 Oct 2026) in Appendix, Section 14, 'Conditional validity and unidentified steps', paragraph 'What we have not proved'