Simultaneous Coverage and Efficiency Guarantee in Online Conformal Prediction

2026-07-29

AI Summary

This work addresses a critical limitation of existing online conformal prediction methods, which under distribution shift only control signed coverage error and thus risk persistent one-sided miscoverage while lacking effective constraints on prediction set size. The paper proposes a unified online learning framework that simultaneously guarantees absolute coverage validity and prediction efficiency across three settings: adversarial, stochastic, and covariate-dependent shifts. Its core contributions include the first joint control of non-canceling coverage bias and efficiency; the design of a sliding-window quantile tracker combined with a partitioned ACI algorithm; and matching minimax lower bounds establishing rate optimality. The method provides distribution-free, convexity-free theoretical guarantees, achieves optimal convergence rates in the stochastic setting, and ensures simultaneous coverage and efficiency relative to dynamic oracle thresholds under covariate-dependent shifts.

Abstract

Adaptive conformal inference (ACI) of Gibbs and Cand{è}s and its variants are the standard approach to online conformal prediction under distribution shift, but they suffer from three fundamental limitations. First, their guarantees control only the \emph{signed} long-run coverage error: persistent miscoverage in one direction can be masked by compensating errors later, so a method can satisfy the theoretical guarantee while being badly wrong for extended periods. Second, existing guarantees say nothing about prediction-set size, so validity can be achieved trivially at the cost of unduly wide prediction sets. Third, the efficiency guarantees that do exist compare against a \emph{fixed} predictor chosen in hindsight, a benchmark that becomes increasingly less meaningful once the data-generating distribution shifts, since the very notion of an optimal threshold then changes over time. We consider a unified online learning framework that simultaneously controls absolute, non-cancelling coverage violation and prediction-set efficiency against a dynamically evolving benchmark for three important models. In the fully adversarial setting, exploiting the fact that the standard ACI update is exactly projected online gradient descent on the pinball loss, we derive simultaneous coverage and efficiency guarantees for arbitrary monotone Lipschitz efficiency objectives, with no distributional or {\it convexity} assumptions. In the stochastic setting with full-score feedback, we propose a sliding-window quantile tracker and establish a matching minimax lower bound showing our algorithm is rate-optimal. In the covariate-dependent stochastic setting, we develop a partitioned ACI algorithm that tracks a function-valued oracle threshold, and derive simultaneous coverage and efficiency guarantees.

Problem

• online conformal prediction
• coverage guarantee
• prediction efficiency
• distribution shift
• dynamic benchmark

Innovation

• online conformal prediction
• simultaneous coverage and efficiency
• adaptive conformal inference
• distribution shift
• minimax optimality