While conformal prediction provides robust marginal coverage guarantees, achieving reliable conditional coverage for specific inputs remains challenging. Although exact distribution-free conditional coverage is impossible with finite samples, recent work has focused on improving the conditional coverage of standard conformal procedures. Distinct from approaches that target relaxed notions of conditional coverage, we directly minimize the mean squared error of conditional coverage by refining the quantile regression components that underpin many conformal methods. Leveraging a Taylor expansion, we derive a sharp surrogate objective for quantile regression: a density-weighted pinball loss, where the weights are given by the conditional density of the conformity score evaluated at the true quantile. We propose a three-headed quantile network that estimates these weights via finite differences using auxiliary quantile levels at \(1-α\pm δ\), subsequently fine-tuning the central quantile by optimizing the weighted loss. We provide a theoretical analysis with exact non-asymptotic guarantees characterizing the resulting excess risk. Extensive experiments on diverse high-dimensional real-world datasets demonstrate remarkable improvements in conditional coverage performance.

翻译：尽管保形预测提供了稳健的边际覆盖保证，但为特定输入实现可靠的条件覆盖仍然具有挑战性。虽然基于有限样本的精确无分布条件覆盖是不可能的，但近期研究聚焦于改进标准保形程序的条件覆盖性能。不同于那些针对松弛条件覆盖概念的方法，我们通过优化支撑许多保形方法的分位数回归组件，直接最小化条件覆盖的均方误差。利用泰勒展开，我们推导出一个精确的分位数回归代理目标：密度加权的弹球损失，其中权重由在真实分位数处评估的符合性得分的条件密度给出。我们提出一个三头分位数网络，通过使用辅助分位数水平 \(1-α\pm δ\) 的有限差分来估计这些权重，随后通过优化加权损失对中心分位数进行微调。我们提供了理论分析，并给出了表征所得超额风险的精确非渐近保证。在多样化高维真实世界数据集上的大量实验表明，该方法在条件覆盖性能方面取得了显著提升。