# High-dimensional online calibration from harmonic weights

arXiv: https://arxiv.org/abs/2610.07740 · HTML: https://mlsanity.com/paper/2610.07740/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 16 references that are in the map.
- First posted on arXiv 6 Oct 2026.

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

- [Fast swap regret minimization and applications to approximate correlated equilibria](https://mlsanity.com/paper/2310.19647/) (2023)
- [Forecasting for Swap Regret for All Downstream Agents](https://mlsanity.com/paper/2402.08753/) (2024)
- [Breaking the T2/3 Barrier for Sequential Calibration](https://mlsanity.com/paper/2406.13668/) (2024)
- [Calibration Error for Decision Making](https://mlsanity.com/paper/2404.13503/) (2024)
- [High dimensional online calibration in polynomial time](https://mlsanity.com/paper/2504.09096/) (2025)
- [High-Dimensional Calibration from Swap Regret](https://mlsanity.com/paper/2505.21460/) (2025)
- [From External to Swap Regret 2.0: An Efficient Reduction and Oblivious Adversary for Large Action Spaces](https://mlsanity.com/paper/2310.19786/) (2023)
- [Testable and Actionable Calibration for Full Swap Regret](https://mlsanity.com/paper/2605.17749/) (2026)

## Abstract (arXiv)

We study the online calibration of multidimensional forecasts over an arbitrary convex set Ysubseteqℝd relative to an arbitrary error norm |·|L. For forecasting d binary outcomes simultaneously (Y=[0,1]d), we give the first algorithm that achieves varepsilon-calibration in a number of rounds that is polynomial in d for every fixed accuracy. It requires dO(1/varepsilon) rounds, exponentially improving the dimension dependence of previous bounds. For multi-class forecasting (Y=Deltad), we obtain the same dO(1/varepsilon) rate, improving the d^widetildeO(1/varepsilon2) bounds of Peng and Fishelson et al. Our algorithm is simple: on each round, it outputs a harmonically weighted distribution over harmonically smoothed past outcomes. The same algorithm works for every forecast set and norm. More generally, it achieves varepsilon-calibration after exp(O(γ(Y,L)/varepsilon)) rounds, where γ(Y,L) is a geometric parameter defined by a matrix discrepancy problem. The harmonic weights are motivated by the fact that the discrete Hilbert transform matrix achieves the optimal discrepancy up to a universal constant, simultaneously for every L. This optimality result may be of independent interest.