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

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 $1-\alpha \pm \delta$, 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. We release the code at \url{https://github.com/Cqyiiii/Colorful-Pinball-Conformal-Prediction-CPCP}.

Machine learning models are increasingly used in settings where reliable uncertainty estimates matter. Conformal prediction provides prediction ranges with strong average guarantees, but these ranges can still be too narrow for some inputs and too wide for others. This paper proposes a new method for making these ranges more reliable across different kinds of examples. The main idea is to focus learning on cases where small estimation errors can cause large reliability problems. The method keeps the standard calibration step that protects overall validity, while improving how evenly the uncertainty is distributed across the data. Experiments on several real-world datasets show that the proposed approach gives more balanced coverage without simply making prediction ranges much larger.