# Faster online calibration without randomization: interval forecasts and the power of two choices

arXiv: https://arxiv.org/abs/2204.13087 · HTML: https://mlsanity.com/paper/2204.13087/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 44 references that are in the map.
- First posted on arXiv 27 Apr 2022.

## Builds on (most cited first, 8 of 8)

- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [An Optimal Algorithm for Bandit and Zero-Order Convex Optimization with Two-Point Feedback](https://mlsanity.com/paper/1507.08752/) (2015)
- [Distribution-free binary classification: prediction sets, confidence intervals and calibration](https://mlsanity.com/paper/2006.10564/) (2020)
- [Top-label calibration and multiclass-to-binary reductions](https://mlsanity.com/paper/2107.08353/) (2021)
- [Distribution-free calibration guarantees for histogram binning without sample splitting](https://mlsanity.com/paper/2105.04656/) (2021)
- [Stronger Calibration Lower Bounds via Sidestepping](https://mlsanity.com/paper/2012.03454/) (2020)
- [Fast Rates for Online Prediction with Abstention](https://mlsanity.com/paper/2001.10623/) (2020)
- [The power of online thinning in reducing discrepancy](https://mlsanity.com/paper/1608.02895/) (2016)

## Built on by (most cited first, 4 of 4)

- [On the Distance from Calibration in Sequential Prediction](https://mlsanity.com/paper/2402.07458/) (2024)
- [Online Platt Scaling with Calibeating](https://mlsanity.com/paper/2305.00070/) (2023)
- [Calibrated Probabilistic Forecasts for Arbitrary Sequences](https://mlsanity.com/paper/2409.19157/) (2024)
- [An Elementary Predictor Obtaining 2√T+1 Distance to Calibration](https://mlsanity.com/paper/2402.11410/) (2024)

## Abstract (arXiv)

We study the problem of making calibrated probabilistic forecasts for a binary sequence generated by an adversarial nature. Following the seminal paper of Foster and Vohra (1998), nature is often modeled as an adaptive adversary who sees all activity of the forecaster except the randomization that the forecaster may deploy. A number of papers have proposed randomized forecasting strategies that achieve an ε-calibration error rate of O(1/√T), which we prove is tight in general. On the other hand, it is well known that it is not possible to be calibrated without randomization, or if nature also sees the forecaster's randomization; in both cases the calibration error could be Omega(1). Inspired by the equally seminal works on the "power of two choices" and imprecise probability theory, we study a small variant of the standard online calibration problem. The adversary gives the forecaster the option of making two nearby probabilistic forecasts, or equivalently an interval forecast of small width, and the endpoint closest to the revealed outcome is used to judge calibration. This power of two choices, or imprecise forecast, accords the forecaster with significant power -- we show that a faster ε-calibration rate of O(1/T) can be achieved even without deploying any randomization.