An Efficient Black-Box Reduction from Online Learning to Multicalibration, and a New Route to $\Phi$-Regret Minimization

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arXiv:2604.19592v2 Announce Type: replace Abstract: We give a Gordon-Greenwald-Marks (GGM) style black-box reduction from online learning to online multicalibration. Concretely, we show that to achieve high-dimensional multicalibration with respect to a class of functions $\mathcal{H}$, it suffices to combine any no-regret learner over $\mathcal {H} $ with an expected variational inequality (EVI) solver. We also prove a converse statement showing that efficient multicalibration implies efficient EVI solving, highlighting how EVIs in multicalibration mirror the role of fixed points in the GGM result for $\Phi$-regret. This first set of results addresses the high-dimensional analogue of the open question in Garg, Jung, Reingold, and Roth (SODA '24), showing that oracle-efficient online multicalibration with $\sqrt{T}$-type guarantees is possible in full generality. Furthermore, our GGM-style reduction unifies the analyses of existing algorithms, transfers guarantees from online learning to multicalibration for challenging environments with delayed observations or censored outcomes, and yields the first efficient black-box reduction between online learning and multiclass omniprediction. Our second main result is a fine-grained reduction from high-dimensional online multicalibration to (contextual) $\Phi$-regret minimization. Together with our first result, this establishes a new route from external regret to $\Phi$-regret that bypasses sophisticated fixed-point or semi-separation machinery, dramatically simplifies a result of Daskalakis, Farina, Fishelson, Pipis, and …

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