Lebesgue-Contrastive Prediction for Multivariate Conformal Regression

The minimum-volume prediction region with conditional coverage $1-\alpha$ is a superlevel set of the conditional density, known as the conditional highest-density region (HDR), so a scalar score reproducing this level-set geometry suffices for optimal multivariate regression. We propose Lebesgue-contrastive prediction (LECP), which learns such a score by logistic discrimination of observed responses against a Lebesgue reference measure, requiring neither a normalized density model nor an invertible parameterization. An alternating procedure fits the contrastive score and a conditional-quantile level that tracks the target density contour, followed by split conformal calibration on an independent hold-out set. The resulting regions enjoy finite-sample marginal coverage by construction and, under estimation consistency, converge to the conditional HDR. On synthetic and real multivariate regression benchmarks, LECP yields substantial volume reductions over existing methods while maintaining competitive conditional reliability.