Online conformal inference with retrospective adjustment for faster adaptation to distribution shift
Authors: Jungbin Jun, Ilsang Ohn
Organizations: Department of Statistics, Inha University
Abstract
Conformal prediction has emerged as a powerful framework for constructing distribution-free prediction sets with guaranteed coverage assuming only the exchangeability assumption. However, this assumption is often violated in online environments where data distributions evolve over time. Several recent approaches have been proposed to address this limitation, but, typically, they slowly adapt to distribution shifts because they update predictions only in a forward manner, that is, they generate a prediction for a newly observed data point while previously computed predictions are not updated. In this paper, we propose a novel online conformal inference method with retrospective adjustment, which is designed to achieve faster adaptation to distributional shifts. Our method leverages regression approaches with efficient leave-one-out update formulas to retroactively adjust past predictions when new data arrive, thereby aligning the entire set of predictions with the most recent data distribution. Through extensive numerical studies performed on both synthetic and real-world data sets, we show that the proposed approach achieves coverage close to the nominal level while reducing predictive interval width by up to approximately 30% compared to existing online conformal prediction methods, demonstrating improved statistical efficiency alongside faster adaptation.
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.
This article considers an online version of conformal inference, called adaptive conformal inference [ACI] and introduced by Gibbs and Candès (2021): prediction sets are issued sequentially, after observing features and before the outcomes are revealed. These sets are evaluated both in terms of validity (the fraction of rounds where the outcome was lying in the prediction set) and efficiency (the average lengths of the prediction sets). The two criteria point to different directions (validity favors larger sets). We also target a wide range of scenarios, with exchangeable data and arbitrary data (lack of any stochastic guarantees) as two extremes. A series of existing strategies for ACI typically guarantee that empirical coverage converges to the desired level for arbitrary sequences, but they generally lack simultaneous efficiency guarantees. To provide a unified study, we first formulate ACI as a repeated two-player game with finite action sets and vector-valued payoffs encoding validity and efficiency. Building on this reformulation, we introduce a strategy based on Blackwell approachability and on its opportunistic extension by Bernstein et al. (2014) that ensures validity while adapting the efficiency of the prediction intervals to the underlying degree of stochasticity of the opponent player. The resulting guarantee is "best of many worlds": it recovers the relevant efficiency guarantees in exchangeable and adversarial settings, and provides guarantees in intermediate settings that arise in typical applications such as the forecasting of time series.
Sequential conformal prediction (CP) provides valid uncertainty quantification under the assumption of residual exchangeability. However, this assumption is often violated in real-world time series due to temporal dependencies and distributional shifts. While recent methods attempt to approximate exchangeability through reweighting, identifying optimal weights remains an open challenge. To address this limitation, we propose DistMatch, a binning-based method that recursively partitions residuals within a binary tree using the Kolmogorov-Smirnov (KS) statistic. We theoretically show that this partitioning induces approximately exchangeable leaves, thereby avoiding the need for reweighting. By applying quantile regression with online updates within each leaf, DistMatch enables locally adaptive inference and improves robustness to distributional shifts. Extensive experiments demonstrate that DistMatch outperforms existing sequential CP methods.