Co-optimization for Adaptive Conformal Prediction

Conformal prediction (CP) provides finite-sample, distribution-free marginal coverage, but standard conformal regression intervals can be inefficient under heteroscedasticity and skewness. In particular, popular constructions such as conformalized quantile regression (CQR) often inherit a fixed notion of center and enforce equal-tailed errors, which can displace the interval away from high-density regions and produce unnecessarily wide sets. We propose Co-optimization for Adaptive Conformal Prediction (CoCP), a framework that learns prediction intervals by jointly optimizing a center $m(x)$ and a radius $h(x)$.CoCP alternates between (i) learning $h(x)$ via quantile regression on the folded absolute residual around the current center, and (ii) refining $m(x)$ with a differentiable soft-coverage objective whose gradients concentrate near the current boundaries, effectively correcting mis-centering without estimating the full conditional density. Finite-sample marginal validity is guaranteed by split-conformal calibration with a normalized nonconformity score. Theory characterizes the population fixed point of the soft objective and shows that, under standard regularity conditions, CoCP asymptotically approaches the length-minimizing conditional interval at the target coverage level as the estimation error and smoothing vanish. Experiments on synthetic and real benchmarks demonstrate that CoCP yields consistently shorter intervals and achieves state-of-the-art conditional-coverage diagnostics.

Adaptive Conformal Inference by Particle Filtering under Hidden Markov Models

Conformal inference is a statistical method used to construct prediction sets for point predictors, providing reliable uncertainty quantification with probability guarantees. This method utilizes historical labeled data to estimate the conformity or nonconformity between predictions and true labels. However, conducting conformal inference for hidden states under hidden Markov models (HMMs) presents a significant challenge, as the hidden state data is unavailable, resulting in the absence of a true label set to serve as a conformal calibration set. This paper proposes an adaptive conformal inference framework that leverages a particle filtering approach to address this issue. Rather than directly focusing on the unobservable hidden state, we innovatively use weighted particles as an approximation of the actual posterior distribution of the hidden state. Our goal is to produce prediction sets that encompass these particles to achieve a specific aggregate weight sum, referred to as the aggregated coverage level. The proposed framework can adapt online to the time-varying distribution of data and achieve the defined marginal aggregated coverage level in both one-step and multi-step inference over the long term. We verify the effectiveness of this approach through a real-time target localization simulation study.