Training-Conditional Coverage Bounds for Uniformly Stable Learning Algorithms

TL;DR

This paper investigates training-conditional coverage bounds for conformal prediction methods, establishing new theoretical guarantees based on uniform stability for finite-dimensional models, especially in the context of empirical risk minimization with kernel methods.

Contribution

It introduces uniform stability-based training-conditional coverage bounds for conformal prediction, extending theoretical understanding beyond previous weaker stability notions.

Findings

Derived coverage bounds for finite-dimensional models using concentration arguments.

Compared new bounds with existing ones under ridge regression.

Validated the bounds' applicability to empirical risk minimization in kernel spaces.

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.