Co-optimization for Adaptive Conformal Prediction

In regression, conformal prediction often suffers from inefficiency under heteroscedasticity and skewness due to fixed, non-adaptive interval centering. We propose CoCP, a new method that parameterizes prediction intervals by a center $m(x)$ and a radius $h(x)$, and alternates between learning $h(x)$ via smooth quantile regression on folded residuals and refining $m(x)$ using a smooth interval loss. This corrects mis-centering and drives the interval toward high-density regions. Beyond finite-sample marginal validity via split-conformal calibration, we prove that CoCP asymptotically achieves conditional coverage and optimal interval length if the base estimators are consistent. Experiments demonstrate that CoCP yields tighter intervals and achieves state-of-the-art conditional coverage reliability.