Group-Conditional Conformal Risk Control: Exact Guarantees, Limits, and Overlapping Extensions

We study group-conditional conformal risk control (CRC) for disjoint and overlapping groups under bounded monotone losses. For disjoint groups, we establish exact finite-sample conditional risk control, two-sided bounds on the group-restricted risk, and fundamental limitations arising from local support and calibration-rank granularity. For overlapping groups, we study two exactly valid approaches: group-restricted calibration followed by union deployment, and direct calibration of the deployed prediction-set family. We characterize the union's limiting reconciliation gap, identify conditions for its disappearance, and establish sharp lower bounds for structured losses in specified product-form models. We also demonstrate a strict size advantage for direct calibration in an ordinal model. Experiments show that direct calibration can reduce prediction-set size, but its efficiency depends on group structure, loss type, and shape fitting. Results on historical diabetic-retinopathy scores further illustrate these tradeoffs. Overall, local support, reconciliation slack, and threshold heterogeneity determine efficiency, with neither approach uniformly dominating.