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arxiv: 2606.24768 · v1 · pith:F662UVOAnew · submitted 2026-06-23 · 🧮 math.ST · cs.GT· cs.IT· math.IT· math.PR· stat.ME· stat.TH

Strong duality for the GROW criterion

Ashwin Ram , Martin Larsson , Johannes Ruf , Aaditya Ramdas This is my paper

Pith reviewed 2026-06-25 21:45 UTC · model grok-4.3

classification 🧮 math.ST cs.GTcs.ITmath.ITmath.PRstat.MEstat.TH
keywords e-variablesGROW criterionstrong dualityrelative entropycomposite hypothesesbettinghypothesis testingminimax log-optimality
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The pith

The GROW value for bounded e-variables always equals the relative entropy of a weak-* joint information projection pair that exists between any composite null and alternative.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes strong duality results for the GROW criterion used in betting-based hypothesis testing. It shows that for any composite null hypothesis set and any composite alternative, the optimal value of the minimax log criterion equals the relative entropy between a certain pair of distributions. The pair is a weak-* joint information projection that is proven to always exist. This provides a general characterization without restrictions on the sets of distributions, generalizing earlier results limited to simple alternatives. A reader would care because it gives a precise way to find the best betting strategy against uncertain hypotheses.

Core claim

We identify a weak-* joint information projection pair between arbitrary Pcal and Qcal that always exists and show that the GROW value for 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.

What carries the argument

The weak-* joint information projection pair between the composite null Pcal and alternative Qcal, whose relative entropy equals the GROW value for bounded e-variables.

If this is right

  • The GROW criterion for bounded e-variables admits strong duality via relative entropy for arbitrary composite hypotheses.
  • A similar strong duality holds for the REGROW criterion with bounded e-variables and arbitrary bounded offsets.
  • Extensions of the duality to unbounded e-variables hold only under additional assumptions, with counterexamples otherwise.
  • The results are analogous to strong duality statements that swap tests for bounded e-variables, minimax risk for the GROW criterion, and total variation for relative entropy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Optimal bounded e-variables might be located by solving for the joint projection pair instead of directly optimizing the original minimax expression.
  • The same projection technique could apply to other log-optimal betting problems in sequential hypothesis testing.
  • Numerical checks on simple cases such as bounded support distributions could confirm whether the equality holds exactly as predicted.

Load-bearing premise

A weak-* joint information projection pair always exists for any arbitrary composite null and alternative sets of distributions.

What would settle it

A specific pair of composite sets Pcal and Qcal for which either no such weak-* projection pair exists or the GROW value for bounded e-variables does not equal the relative entropy of the pair.

read the original 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 $\P \in \Pcal$. Following Kelly, Breiman, Cover, Shafer, Gr\"unwald 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper presents general strong duality results for the GROW criterion when testing composite hypotheses via e-variables. It generalizes prior work on simple alternatives to arbitrary composite Qcal, claiming that a weak-* joint information projection pair (P*,Q*) between arbitrary Pcal and Qcal always exists, and that the GROW value for bounded e-variables equals the relative entropy D(Q*||P*) with no restrictions on the sets. Analogous strong duality is shown for the REGROW criterion with bounded e-variables and arbitrary bounded offsets. Extensions to unbounded e-variables hold under various assumptions, with counterexamples showing failures without them. The results are positioned as analogous to complete-class theorems, replacing tests with bounded e-variables, minimax risk with GROW, and total variation with relative entropy.

Significance. If the claimed existence of the weak-* joint projection pair and the resulting duality hold in full generality, this provides a parameter-free characterization of the GROW value via relative entropy, extending the theoretical toolkit for log-optimal betting and e-value based inference to fully composite settings. The generality for bounded e-variables without restrictions on Pcal or Qcal would be a clear advance over the cited prior results, with the counterexamples for unbounded cases adding useful boundary clarification.

major comments (1)
  1. [Abstract] Abstract (and main duality theorem): The assertion that a weak-* joint information projection pair always exists for arbitrary Pcal and Qcal is load-bearing for the strong duality claim that GROW equals D(Q*||P*) with no restrictions. The manuscript must supply the explicit argument establishing attainment in the weak* topology, including verification that the relative entropy functional is lower semi-continuous and that the relevant sets admit a minimizer (e.g., via compactness in the dual ball); without this, the equality cannot be guaranteed in the stated generality.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comment on the existence argument. We address the point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and main duality theorem): The assertion that a weak-* joint information projection pair always exists for arbitrary Pcal and Qcal is load-bearing for the strong duality claim that GROW equals D(Q*||P*) with no restrictions. The manuscript must supply the explicit argument establishing attainment in the weak* topology, including verification that the relative entropy functional is lower semi-continuous and that the relevant sets admit a minimizer (e.g., via compactness in the dual ball); without this, the equality cannot be guaranteed in the stated generality.

    Authors: We agree that the existence claim is central and that the current write-up would benefit from a more self-contained verification. In the revision we will add an explicit subsection (or appendix) proving attainment of the weak-* joint information projection pair. The argument will (i) recall that relative entropy is lower semi-continuous with respect to the weak-* topology on probability measures, (ii) establish that the relevant constraint sets are weak-* compact (via the Banach-Alaoglu theorem applied to the dual ball), and (iii) conclude existence of a minimizer by the standard compactness-plus-lsc argument. This will be inserted into the proof of the main duality theorem so that the generality statement is fully justified. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained mathematical proof

full rationale

The paper claims to prove existence of a weak-* joint information projection pair for arbitrary composite sets and to establish equality between the GROW value (for bounded e-variables) and relative entropy of that pair. This is presented as a direct generalization of prior results to the composite case, with the proofs supplied in the manuscript itself rather than reduced to inputs by definition, fitted parameters renamed as predictions, or load-bearing self-citations. Analogous results in cited works are referenced for context but do not substitute for the claimed derivations here. No self-definitional, ansatz-smuggling, or renaming patterns appear in the provided abstract or structure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard background from probability (e-variables, relative entropy) plus the domain assumption that a weak-* joint information projection pair always exists for arbitrary P and Q. No free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Existence of a weak-* joint information projection pair between arbitrary composite null Pcal and alternative Qcal
    Invoked to establish that the GROW value equals the relative entropy of this pair for any Pcal and Qcal.

pith-pipeline@v0.9.1-grok · 5821 in / 1258 out tokens · 26641 ms · 2026-06-25T21:45:13.071831+00:00 · methodology

discussion (0)

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