Efficient Online Conformal Selection with Limited Feedback

We address the problem of conformal selection, where an agent must select a low-cost subset of options to ensure that at least one "success" is identified at a pre-specified target rate φ. While traditional online conformal prediction focuses on maintaining validity for the observed sequence, minimizing the resource cost (efficiency) of such selections, especially under limited feedback, remains a significant challenge. In this work, we consider highly restricted "bandit" feedback, where the agent only observes feedback about the subset it selected, and not the true label, point, or outcomes of unchosen options. We demonstrate that the simple Adaptive Conformal Inference (ACI) update rule, when applied to the appropriate control parameter or dual variable and paired with explicit boundary actions, is both adversarially valid, ensuring the success target is met on average for any input sequence (and hence under distribution shifts), and stochastically efficient, achieving sublinear efficiency regret for i.i.d. inputs against an optimal stochastic benchmark. The key algorithmic idea is to avoid the projected updates standard in constrained bandits: projections break the exact telescoping identity behind ACI validity, whereas boundary actions stabilize the unprojected update through actual decisions. We show these guarantees under canonical models capturing bandit feedback via a unified algorithmic technique and Lyapunov-based analysis. Our approach handles more general settings than prior work, while requiring significantly less feedback, and provides a new theoretical bridge between efficient online learning with limited feedback and distribution-free uncertainty quantification.