Strong duality for the GROW criterion

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 ∈ mathcal P. Following Kelly, Breiman, Cover, Shafer, and Grunwald et al. (2024), we study a natural minimax log-optimality criterion: given a composite alternative Qcal, we characterize the ``GROW value'' supX infQ EQ[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 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.