Optimization-based Online Conformal Prediction for Multi-step Forecasting

Conformal prediction (CP) provides distribution-free coverage guarantees, making it well suited for uncertainty quantification in time series forecasting. However, existing methods often struggle with multi-step settings: they either calibrate horizons independently---ignoring temporal correlations---or enforce strict simultaneous coverage, resulting in overly conservative intervals. In this work, we propose O2CP: Optimization-Based Online Conformal Prediction, a framework that augments a broad family of online CP methods with cross-horizon optimization while preserving their long-term coverage guarantees. We first characterize this family of methods, showing that long-term coverage is preserved as long as, at each forecast horizon, the selected control variable remains within an admissible set around the method's nominal output. Building on this result, O2CP uses a two-layer design: the first layer constructs these admissible sets from the underlying online CP updates, and the second performs constrained optimization across horizons within them, jointly modeling the cross-horizon distributions to minimize a user-specified objective. Extensive experiments on real-world datasets---including autonomous driving, climate forecasting, and public health---demonstrate that O2CP consistently outperforms state-of-the-art baselines, achieving target coverage with significantly sharper prediction intervals and reduced regret over long horizons.