# Optimization-based Online Conformal Prediction for Multi-step Forecasting

arXiv: https://arxiv.org/abs/2508.13362 · HTML: https://mlsanity.com/paper/2508.13362/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 38 references that are in the map.
- First posted on arXiv 18 Aug 2025.

## Builds on (most cited first, 15 of 15)

- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Modeling Long- and Short-Term Temporal Patterns with Deep Neural Networks](https://mlsanity.com/paper/1703.07015/) (2017)
- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Argoverse: 3D Tracking and Forecasting with Rich Maps](https://mlsanity.com/paper/1911.02620/) (2019)
- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [Adaptive Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2106.00170/) (2021)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [Conformal Inference for Online Prediction with Arbitrary Distribution Shifts](https://mlsanity.com/paper/2208.08401/) (2022)
- [Strongly Adaptive Online Learning](https://mlsanity.com/paper/1502.07073/) (2015)
- [Sequential Predictive Conformal Inference for Time Series](https://mlsanity.com/paper/2212.03463/) (2022)
- [Probabilistic Conformal Prediction Using Conditional Random Samples](https://mlsanity.com/paper/2206.06584/) (2022)
- [Bellman Conformal Inference: Calibrating Prediction Intervals For Time Series](https://mlsanity.com/paper/2402.05203/) (2024)
- [Neural Conformal Control for Time Series Forecasting](https://mlsanity.com/paper/2412.18144/) (2024)
- [Dynamic Regret of Strongly Adaptive Methods](https://mlsanity.com/paper/1701.07570/) (2017)

## Built on by (most cited first, 1 of 1)

- [Adaptive Conformal Inference Under Delayed Feedback: Coverage Guarantees and a Delay-to-Memory Diagnostic](https://mlsanity.com/paper/2609.07251/) (2026)

## Abstract (arXiv)

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.