# Online Conformal Prediction: Enforcing monotonicity via Online Optimization

arXiv: https://arxiv.org/abs/2605.12668 · HTML: https://mlsanity.com/paper/2605.12668/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 26 references that are in the map.
- First posted on arXiv 12 May 2026.

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

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Adaptive Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2106.00170/) (2021)
- [Conformal prediction beyond exchangeability](https://mlsanity.com/paper/2202.13415/) (2022)
- [Adaptive Conformal Predictions for Time Series](https://mlsanity.com/paper/2202.07282/) (2022)
- [Conformal Inference for Online Prediction with Arbitrary Distribution Shifts](https://mlsanity.com/paper/2208.08401/) (2022)
- [Safe Planning in Dynamic Environments using Conformal Prediction](https://mlsanity.com/paper/2210.10254/) (2022)
- [Conformal PID Control for Time Series Prediction](https://mlsanity.com/paper/2307.16895/) (2023)
- [Improved Online Conformal Prediction via Strongly Adaptive Online Learning](https://mlsanity.com/paper/2302.07869/) (2023)
- [Fair Conformal Predictors for Applications in Medical Imaging](https://mlsanity.com/paper/2109.04392/) (2021)
- [Exact and Robust Conformal Inference Methods for Predictive Machine Learning With Dependent Data](https://mlsanity.com/paper/1802.06300/) (2018)
- [Practical Adversarial Multivalid Conformal Prediction](https://mlsanity.com/paper/2206.01067/) (2022)
- [Online conformal prediction with decaying step sizes](https://mlsanity.com/paper/2402.01139/) (2024)
- [Federated Conformal Predictors for Distributed Uncertainty Quantification](https://mlsanity.com/paper/2305.17564/) (2023)
- [Nested Conformal Prediction Sets for Classification with Applications to Probation Data](https://mlsanity.com/paper/2104.09358/) (2021)
- [Conformal Prediction for Time-series Forecasting with Change Points](https://mlsanity.com/paper/2509.02844/) (2025)

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

- [Risk-Averse Decision Making with Multi-Level Reliability Guarantees](https://mlsanity.com/paper/2609.11524/) (2026)

## Abstract (arXiv)

Conformal prediction provides a principled framework for uncertainty quantification with finite-sample coverage guarantees. While recent work has extended conformal prediction to online and sequential settings, existing methods typically focus on a single coverage level and do not ensure consistency across multiple confidence levels. In many real-world applications, such as weather forecasting, macroeconomic prediction, and risk management, different users operate under heterogeneous risk tolerances and require calibrated uncertainty estimates across a range of coverage levels. In such settings, it is desirable to produce prediction sets corresponding to different coverage levels that are nested and valid simultaneously. In this paper, we propose two novel online conformal prediction methods that output nested prediction sets across a range of coverage levels, enabling simultaneous uncertainty quantification across the entire risk spectrum. Beyond interpretability, jointly estimating multiple coverage levels is known to improve statistical efficiency in classical quantile regression by enforcing non-crossing constraints and sharing information across quantiles. Our approaches leverage an online optimization perspective with small regret that translates to quantile estimation error control while enforcing nestedness of prediction sets. Empirical results on synthetic and real-world datasets, including applications in forecasting tasks with heterogeneous risk requirements, demonstrate that our method achieves stable coverage across all levels, strictly nested prediction sets, and improved efficiency compared to existing online conformal baselines.