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.

翻译：共形预测（CP）提供有限样本、分布无关的边际覆盖，但标准共形回归区间在异方差性和偏态性下可能效率低下。特别是常用构造方法（如共形化分位数回归（CQR））往往继承固定的中心概念并强制对称误差，这可能导致区间偏离高密度区域并产生不必要的宽泛集合。我们提出自适应共形预测协同优化（CoCP）框架，通过联合优化中心$m(x)$与半径$h(x)$来学习预测区间。CoCP交替执行以下步骤：（i）通过当前中心处折叠绝对残差的分位数回归学习$h(x)$；（ii）利用可微软覆盖目标精炼$m(x)$，其梯度在当前边界附近集中，无需估计完整条件密度即可有效校正中心偏移。通过采用标准化非共形评分的分割共形校准，保证了有限样本边际有效性。理论分析了软目标的总体不动点，并证明在标准正则性条件下，当估计误差与平滑项趋零时，CoCP渐近逼近目标覆盖水平下的长度最小化条件区间。在合成与真实基准测试中的实验表明，CoCP能持续生成更短的区间，并达到最先进的条件覆盖诊断指标。