Colorful Pinball: Density-Weighted Quantile Regression for Conditional Guarantee of Conformal Prediction

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.

💡 Research Summary

This paper tackles the long‑standing limitation of split conformal prediction: while it guarantees marginal coverage at a user‑specified level (1-\alpha), it provides no guarantees for the conditional coverage (\pi(x)=\Pr(Y\in C_\alpha(x)\mid X=x)) of individual test points. Exact distribution‑free conditional coverage is impossible with finite data, yet practitioners need reliable per‑instance uncertainty estimates, especially in safety‑critical domains.

The authors adopt the Mean Squared Conditional Error (MSCE) as the performance metric, defined as the expected squared deviation of the conditional coverage from the target level. Prior work (Kiyani et al., 2024) showed that MSCE can be upper‑bounded by the excess risk of standard quantile regression with the pinball loss, but that bound depends on a global Lipschitz constant of the conditional CDF and is therefore overly loose in practice.

To obtain a sharper surrogate, the paper performs a second‑order Taylor expansion of the CDF around the true (\tau)-quantile (q_\tau(x)) (with (\tau=1-\alpha)). The expansion reveals that the leading term of MSCE is proportional to the conditional density of the conformity score evaluated at the true quantile, (f_{S|X}(q_\tau(x))). Concretely,
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