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arXiv · 2606.24768

Strong duality for the GROW criterion

Ashwin Ram·Martin Larsson·Johannes Ruf·Aaditya Ramdas
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Abstract

This paper presents general strong duality results when testing hypotheses by betting against them. A bet is an e-variable for a composite null hypothesis $\Pcal$: a nonnegative random variable $X$ whose expected value is at most one under every $P \in \mathcal P$. Following Kelly, Breiman, Cover, Shafer, and Grunwald et al. (2024), we study a natural minimax \emph{log-optimality} criterion: given a composite alternative $\Qcal$, we characterize the ``GROW value'' $\sup_{X} \inf_{Q} \E_{Q}[\log X]$. This paper generalizes the results of Larsson et al. (2025) from (arbitrary $\mathcal P$ and) simple $\mathcal Q$ to arbitrary $\mathcal Q$. We prove that there always exists a minimizing information-projection pair between the weak-$*$ closures of the convex hulls of arbitrary $\mathcal P$ and $\mathcal Q$, and show that the GROW value for \emph{bounded} e-variables always equals their relative entropy. We also prove a similarly general strong duality for the REGROW criterion with bounded e-variables and arbitrary bounded offsets. Under various assumptions our results extend to unbounded e-variables, and examples show that without any assumptions such extensions fail. Our results are analogous to those in Larsson et al. (2026), swapping tests for bounded e-variables, minimax risk for the GROW criterion, and total variation for relative entropy.

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Ashwin Ram, Martin Larsson, Johannes Ruf, Aaditya Ramdas. 2026-07-20. Strong duality for the GROW criterion. https://arxiv.org/abs/2606.24768

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