Improved Multi-Dimensional Forecasting for Swap Regret

2026-06-28

AI Summary

This work addresses the problem of providing predictions for an arbitrary number of downstream agents with unknown objectives in a multidimensional outcome space, with the goal of achieving sublinear growth of swap regret for all agents. By integrating techniques from online learning and regret minimization in game theory, the authors design the first polynomial-time algorithm that attains an $\tilde{O}(\sqrt{kT})$ bound on swap regret in the two-dimensional setting, significantly improving upon the previous $\tilde{O}(kT^{5/8})$ result. The approach generalizes to any dimension $d$, yielding a swap regret bound of $\tilde{O}(d\sqrt{kT})$, which also surpasses the best-known $\tilde{O}(T^{2/3})$ bounds for higher dimensions.

Abstract

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 $\tilde{O}(\sqrt{kT})$ swap regret for any downstream agent with $k$ actions. This improves over the previously known bound of $\tilde{O}(kT^{5/8})$ and avoids the exponential in $T$ runtime of prior algorithms in this setting. Our algorithm extends nicely to other low dimensional environments, retaining $\tilde{O}(\sqrt{T})$ 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 $\tilde{O}(d\sqrt{kT})$ 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 $\tilde{O}(T^{2/3})$ dependence and required additional behavioral assumptions.

Problem

• swap regret
• multi-dimensional forecasting
• downstream agents
• unknown objectives
• best response

Innovation

• swap regret
• multi-dimensional forecasting
• sublinear regret
• polynomial-time algorithm
• online learning