Strong duality for the GROW criterion

2026-06-23

AI Summary

This study investigates the construction of e-variables with growth-rate optimality (GROW) in composite hypothesis testing and their associated duality theory. For arbitrary composite null and alternative hypotheses, it establishes strong duality for bounded e-variables under the GROW criterion without imposing any structural restrictions on the hypothesis sets. By leveraging pairs of joint information projections in the weak* topology and employing tools from functional analysis, information projection, and relative entropy, the work proves that the GROW value always equals the relative entropy between the corresponding projection pair, and further extends this result to the REGROW criterion. The analysis also precisely delineates the boundary conditions under which these conclusions hold or fail in the unbounded e-variable setting.

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 $\mathcal{P}$: a nonnegative random variable $X$ whose expected value is at most one under every $¶\in \Pcal$. Following Kelly, Breiman, Cover, Shafer, Grünwald and others, 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 \cite{larsson2025numeraire} from (arbitrary $\Pcal$ and) simple $\Qcal$ to arbitrary $\Qcal$. We identify a weak-$*$ joint information projection pair between arbitrary $\Pcal$ and $\Qcal$ that always exists and show that the GROW value for \emph{bounded} e-variables always equals the relative entropy of this pair, without any restrictions on $\Pcal$ or $\Qcal$. 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~\cite{larsson2026complete}, swapping tests for bounded e-variables, minimax risk for the GROW criterion, and total variation for relative entropy.

Problem

• strong duality
• GROW criterion
• e-variable
• composite hypothesis
• relative entropy

Innovation

• strong duality
• GROW criterion
• e-variable
• relative entropy
• joint information projection