Feedback-Enhanced Online Multiple Testing with Applications to Conformal Selection

We study online multiple testing with feedback, where decisions are made sequentially and the true state of the hypothesis is revealed after the decision has been made, either instantly or with a delay. We propose GAIF, a feedback-enhanced generalized alpha-investing framework that dynamically adjusts thresholds using revealed outcomes, ensuring finite-sample false discovery rate (FDR)/marginal FDR control. Extending GAIF to online conformal testing, we construct independent conformal $p$-values and introduce a feedback-driven model selection criterion to identify the best model/score, thereby improving statistical power. We demonstrate the effectiveness of our methods through numerical simulations and real-data applications.

CAS: A General Algorithm for Online Selective Conformal Prediction with FCR Control

We study the problem of post-selection predictive inference in an online fashion. To avoid devoting resources to unimportant units, a preliminary selection of the current individual before reporting its prediction interval is common and meaningful in online predictive tasks. Since the online selection causes a temporal multiplicity in the selected prediction intervals, it is important to control the real-time false coverage-statement rate (FCR) to measure the averaged miscoverage error. We develop a general framework named CAS (Calibration after Adaptive Selection) that can wrap around any prediction model and online selection rule to output post-selection prediction intervals. If the current individual is selected, we first perform an adaptive selection on historical data to construct a calibration set, then output a conformal prediction interval for the unobserved label. We provide tractable constructions for the calibration set for popular online selection rules. We proved that CAS can achieve an exact selection-conditional coverage guarantee in the finite-sample and distribution-free regimes. For the decision-driven selection rule, including most online multiple-testing procedures, CAS can exactly control the real-time FCR below the target level without any distributional assumptions. For the online selection with symmetric thresholds, we establish the error bound for the control gap of FCR under mild distributional assumptions. To account for the distribution shift in online data, we also embed CAS into some recent dynamic conformal prediction methods and examine the long-run FCR control. Numerical results on both synthetic and real data corroborate that CAS can effectively control FCR around the target level and yield more narrowed prediction intervals over existing baselines across various settings.