Feedback-Enhanced Online Multiple Testing with Applications to Conformal Selection

This work studies online multiple testing with feedback, where decisions are made sequentially, and the true state of the hypothesis is revealed after decisions are made, either instantly or with a delay, and under either full or bandit feedback. We propose Generalized alpha-investing with feedback (GAIF) along with its adaptive variants, a feedback-enhanced framework that dynamically adjusts thresholds using revealed outcomes, ensuring finite-sample false discovery rate (FDR)/marginal FDR (mFDR) control. We further extend GAIF to online conformal testing by constructing valid conformal $p$-values and developing feedback-enhanced testing rules with finite-sample mFDR control. We also propose a feedback-driven score selection criterion to adaptively choose the candidate score that is most effective for the testing procedure, together with a theoretical analysis of its optimality. Numerical simulations and real-data applications demonstrate the effectiveness of our methods.