Training-Conditional Coverage Bounds for Uniformly Stable Learning Algorithms
Mehrdad Pournaderi, Yu Xiang
TL;DR
The paper addresses training-conditional coverage guarantees for conformal prediction under uniform stability in regression models realized as convex-regularized empirical risk minimization over reproducing kernel Hilbert spaces. It develops concentration-based bounds for the estimated predictor to derive finite-sample guarantees for full-conformal, jackknife+, and CV+ intervals, with explicit stability- and dimension-dependent terms. In the ridge-regression setting, the authors obtain $n^{-1/2}$-rate bounds that depend on dimension, and contrast these with the slower $n^{-1/5}$ rates from prior $(m,n)$-stability analyses, highlighting practical improvements in coverage accuracy. The results clarify how uniform stability can yield sharper training-conditional guarantees and provide guidance on when conformal prediction intervals are reliable in finite samples, especially for convex RKHS-based learners.
Abstract
The training-conditional coverage performance of the conformal prediction is known to be empirically sound. Recently, there have been efforts to support this observation with theoretical guarantees. The training-conditional coverage bounds for jackknife+ and full-conformal prediction regions have been established via the notion of $(m,n)$-stability by Liang and Barber~[2023]. Although this notion is weaker than uniform stability, it is not clear how to evaluate it for practical models. In this paper, we study the training-conditional coverage bounds of full-conformal, jackknife+, and CV+ prediction regions from a uniform stability perspective which is known to hold for empirical risk minimization over reproducing kernel Hilbert spaces with convex regularization. We derive coverage bounds for finite-dimensional models by a concentration argument for the (estimated) predictor function, and compare the bounds with existing ones under ridge regression.