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：尽管共形预测提供了稳健的边缘覆盖保证，但针对特定输入实现可靠的条件覆盖仍然具有挑战性。虽然基于有限样本实现精确的无分布条件覆盖是不可能的，但近期研究聚焦于改进标准共形程序的条件覆盖性能。与针对松弛条件覆盖概念的方法不同，我们通过优化支撑众多共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，我们推导出分位数回归的精确代理目标函数：一种密度加权的弹球损失函数，其权重由在真实分位数处评估的符合度得分的条件密度给出。我们提出一种三头分位数网络，该网络通过使用辅助分位数水平 \(1-α\pm δ\) 进行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，其中包含刻画所得超额风险的精确非渐近保证。在多样化的高维真实数据集上进行的大量实验表明，该方法在条件覆盖性能方面取得了显著提升。