Online Conformal Prediction via Universal Portfolio Algorithms

TL;DR

This work develops a regret-to-coverage theory for online conformal prediction by introducing linearized regret and a Fenchel-duality-based bound that implies $1-\alpha$ coverage for online procedures. It then constructs UP-OCP, a parameter-free method that reduces OCP to a two-asset portfolio problem and leverages universal portfolio algorithms to achieve strong finite-time miscoverage guarantees, even as nonconformity scores grow polynomially. The approach is instantiated with a closed-form UP update and demonstrated to yield favorable size/coverage trade-offs across real and synthetic nonstationary data, outperforming several baselines without hyperparameter tuning. Overall, UP-OCP offers robust, adaptive coverage with practical efficiency, making online conformal prediction more reliable in adversarial and nonstationary settings.

Abstract

Online conformal prediction (OCP) seeks prediction intervals that achieve long-run $1-α$ coverage for arbitrary (possibly adversarial) data streams, while remaining as informative as possible. Existing OCP methods often require manual learning-rate tuning to work well, and may also require algorithm-specific analyses. Here, we develop a general regret-to-coverage theory for interval-valued OCP based on the $(1-α)$-pinball loss. Our first contribution is to identify \emph{linearized regret} as a key notion, showing that controlling it implies coverage bounds for any online algorithm. This relies on a black-box reduction that depends only on the Fenchel conjugate of an upper bound on the linearized regret. Building on this theory, we propose UP-OCP, a parameter-free method for OCP, via a reduction to a two-asset portfolio selection problem, leveraging universal portfolio algorithms. We show strong finite-time bounds on the miscoverage of UP-OCP, even for polynomially growing predictions. Extensive experiments support that UP-OCP delivers consistently better size/coverage trade-offs than prior online conformal baselines.

Figures (80)

• Figure 1: Pareto frontiers for average prediction set size on the AXP Dataset, for 50 target miscoverage rates $\alpha$ uniformly from 0.05 to 0.25. Better performance is closer to the bottom-left corner.
• Figure 2: Realized vs. target coverage.
• Figure 3: Pareto frontiers on the synthetic sinusoid. Optimal performance is the bottom-left corner (tightest sets for highest coverage).
• Figure 4: Plot of the realized marginal coverage against the average prediction set size, averaged over 10 independent random seeds. Error bars indicate the standard error of the mean along both axes.
• Figure 5: Observed nonconformity scores for AAPL stock returns.

Theorems & Definitions (27)

• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Definition 4.1: The Conformal Market
• Definition 4.2: Wealth Process
• Theorem 4.3: Regret of UP-OCP
• Theorem 4.4: Coverage bound for UP-OCP
• Lemma A.1
• proof
• proof : Proof of Theorem \ref{['thm:finite-time-bound']}