Sequential Confidence Sets for Coverage-Constrained Conformal Model Selection
Jing Li, Haibin Zhu
Abstract
Modern conformal forecasting systems often maintain several adaptive pipelines that differ in base forecasters, conformity scores, calibration windows, and update rules. Comparing them is difficult because coverage is a hard constraint, whereas efficiency should be optimized only among feasible pipelines. We formulate this problem as sequential inference for a stochastic constrained argmin. At each time, the target is the set of minimum-cost pipelines satisfying multiple prefix-average conditional miscoverage constraints. We introduce Coverage-Constrained Sequential Model Confidence Sets (CC-SMCS), which separate certifiably feasible, possibly feasible, and possibly constrained-optimal pipelines. Using simultaneous martingale confidence sequences, CC-SMCS projects a rectangular confidence region onto the constrained argmin and admits an exact closed-form rule. With probability at least $1-δ$, it contains every constrained-optimal pipeline simultaneously over all times. This finite-sample guarantee requires no stationarity or mixing assumptions and remains valid under data-dependent stopping. We also establish an impossibility result for safe certification at the coverage boundary and extend the construction to delayed multi-horizon feedback and outcome-dependent efficiency objectives.