Although conformal prediction provides robust marginal coverage guarantees, achieving reliable conditional coverage for specific inputs remains challenging. While 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 target 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 nonconformity 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，但针对特定输入实现可靠的保形条件覆盖仍具有挑战性。虽然有限样本下无法实现精确的无分布条件覆盖，但近期研究聚焦于改进标准保形程序的条件覆盖能力。与针对条件覆盖松弛概念的方法不同，我们通过优化许多保形方法所依赖的分位数回归组件，直接针对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锐化替代目标：一种密度加权弹球损失函数，其中权重由非一致性分数在真实分位数处的条件密度给出。我们提出三头分位数网络，通过使用辅助分位数水平 $1-α\pm δ$ 的有限差分估计这些权重，随后通过优化加权损失微调中心分位数。我们提供了理论分析，给出了刻画由此产生的超额风险的确切非渐近保证。在多样化高维真实世界数据集上的广泛实验表明，条件覆盖性能显著提升。