# Improved Multi-Dimensional Forecasting for Swap Regret

arXiv: https://arxiv.org/abs/2606.29533 · HTML: https://mlsanity.com/paper/2606.29533/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 28 references that are in the map.
- First posted on arXiv 28 Jun 2026.

## Builds on (most cited first, 9 of 9)

- [Stronger Calibration Lower Bounds via Sidestepping](https://mlsanity.com/paper/2012.03454/) (2020)
- [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)
- [Econometrics for Learning Agents](https://mlsanity.com/paper/1505.00720/) (2015)
- [Breaking the T2/3 Barrier for Sequential Calibration](https://mlsanity.com/paper/2406.13668/) (2024)
- [Bid Prediction in Repeated Auctions with Learning](https://mlsanity.com/paper/2007.13193/) (2020)
- [Optimal Multiclass U-Calibration Error and Beyond](https://mlsanity.com/paper/2405.19374/) (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)

## Abstract (arXiv)

We study the problem of forecasting for an arbitrary number of downstream agents with unknown objectives, each of whom best responds to the forecaster's predictions. We seek a single forecaster that guarantees sublinear swap regret for all downstream agents simultaneously. For two-dimensional outcome spaces, we give a polynomial time algorithm that guarantees tildeO(√kT) swap regret for any downstream agent with k actions. This improves over the previously known bound of tildeO(kT5/8) and avoids the exponential in T runtime of prior algorithms in this setting. Our algorithm extends nicely to other low dimensional environments, retaining tildeO(√T) downstream swap regret while the exponent of k in the regret bound and the exponent of T in the running time both grow with dimension. For arbitrary dimension d, we give a forecasting algorithm that guarantees tildeO(d√kT) swap regret, assuming the forecaster knows an upper bound k on the number of actions available to any downstream agent, albeit with a much longer runtime. This improves upon previous high dimensional guarantees that had tildeO(T2/3) dependence and required additional behavioral assumptions.