Conformal Candidate Certification for Offline Model-Based Optimization

Uncertainty quantification is essential when deploying machine learning models in safety-critical applications. Online conformal prediction (OCP) provides theoretically principled uncertainty quantification for arbitrary black-box classifiers and non-i.i.d. data streams by constructing prediction sets that are guaranteed to contain the true label at a user-specified frequency. OCP usually updates prediction sets using feedback from previously deployed predictions. We instead study an OCP setting beyond feedback: on each round, the learner can either output a prediction set or query the correct label, but not both. Thus, no deployed prediction is ever evaluated directly. We reduce this problem to a partial monitoring game in which prediction actions return no observation and a separate query action reveals the label. The reward function is constructed in a way that encourages the learner to output small prediction sets while ensuring that the correct label is covered with a sufficiently high probability. To solve this game, we develop OCP with queries (OCPQ) by adapting the label efficient forecaster of Cesa-Bianchi, Lugosi, and Stoltz (2004) to our setting. For any black box classifier and any (non-i.i.d.) oblivious data stream of length $T$, OCPQ has $O(T^{2/3})$ expected regret and expected coverage at least $β-O(T^{-1/3})$ for a user-defined $β$, while querying only an expected $T^{-1/3}$ fraction of rounds. This provides coverage comparable to bandit-based OCP methods while requiring no feedback from deployed prediction sets. Experiments on real-world datasets further demonstrate the effectiveness of our approach.

Bayesian conformal optimisation methods often use the same held-out data both to search for efficient prediction sets and to certify coverage or risk. This coupling is natural for high-probability risk-control guarantees, but it is not necessary when the target is standard finite-sample marginal conformal coverage. We propose Decoupled Conformal Optimisation (DCO), a train-tune-calibrate design principle that uses an independent tuning split for efficiency-oriented structural selection and a fresh calibration split for the final conformal quantile. Conditional on the tuned structure, standard split-conformal exchangeability yields finite-sample marginal coverage for any candidate class, without a confidence parameter or multiple-testing correction. DCO therefore targets a different finite-sample guarantee from PAC-style methods: marginal conformal coverage rather than high-probability risk control. Under consistency assumptions on the coupled risk bound, the two approaches nevertheless converge to the same population threshold. Across classification and regression benchmarks, including ImageNet-A, CIFAR-100, Diabetes, California Housing, and Concrete, DCO tracks the nominal coverage level closely while often reducing average prediction-set size or interval width relative to PAC-style calibration. On ImageNet-A, for example, the average set size decreases from $26.52$ to $25.26$ and the 95th-percentile set size from $58.95$ to $53.73$; on Diabetes, the average interval width decreases from $2.098$ to $1.914$.

Conformal prediction is a framework that provides valid uncertainty quantification for general models with exchangeable data. However, in the online learning and time-series settings, exchangeability is not satisfied. Existing online conformal methods, such as adaptive conformal inference (ACI), can achieve long-run validity, yet they remain inefficient under covariate heterogeneity because they rely on global calibration. We propose \emph{Online Localized Conformal Prediction (OLCP)}, which combines online adaptation with covariate-dependent localization to better reflect heterogeneity. To reduce sensitivity to the localization bandwidth, we further develop \emph{OLCP-Hedge}, which performs bandwidth selection as an online expert aggregation problem using a constrained online convex optimization framework. Importantly, we provide coverage guarantees for both algorithms and demonstrate through simulations and real-data experiments that the proposed methods attain valid long-run coverage with narrower prediction sets than existing baselines.