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arxiv: 2604.21680 · v1 · submitted 2026-04-23 · 📊 stat.ME

Optimal e-variables under constraints

Aytijhya Saha , Aaditya Ramdas This is my paper

Pith reviewed 2026-05-09 21:00 UTC · model grok-4.3

classification 📊 stat.ME
keywords e-variableslog-optimalityconstrained optimizationdifferential privacyanytime-valid inferenceleast favorable distributionsgrowth rate optimization
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The pith

Log-optimal e-variables under constraints are obtained by transforming the unconstrained optimum.

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

E-variables provide a way to perform statistical inference that stays valid no matter when the data stream stops. The paper shows that when extra requirements are added, such as privacy protections or limits on how large the values can be, the best possible e-variable satisfying those requirements is still built from the one that would be best without them. One first identifies the unconstrained optimum as the likelihood ratio between the least favorable distributions, then applies a transformation that enforces the new requirement. This avoids having to search for entirely new least favorable distributions that fit the constraints. A reader would care because it turns a potentially difficult constrained search into a straightforward post-processing step.

Core claim

The paper establishes that under constraints including differential privacy, quantization, boundedness, or moment restrictions, log-optimal constrained e-variables can be constructed by first computing the unconstrained log-optimal e-variable given by the likelihood ratio of the least favorable distributions, then imposing the constraint via an appropriate transformation. Thus, the constrained growth-rate optimization problem does not require solving for a different pair of least favorable distributions; the constrained optimal solution is just a post-processing of the unconstrained optimal solution.

What carries the argument

The optimize-then-constrain principle, which starts with the likelihood ratio of the least favorable distributions and applies a transformation to meet the added requirement while preserving log-optimality.

If this is right

  • The same least favorable distributions used without constraints remain the starting point even after constraints are added.
  • Existing methods for computing unconstrained e-variables can be reused directly before applying the transformation.
  • The approach covers differential privacy, quantization, boundedness, and moment restrictions without separate derivations.
  • The growth rate achieved by the constrained e-variable is determined by the original unconstrained optimum after the transformation.
  • No new optimization over pairs of distributions is needed when the constraint is imposed.

Where Pith is reading between the lines

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

  • The principle could extend to other structural constraints not examined in the paper.
  • Similar post-processing ideas might simplify constrained versions of related quantities in robust statistics.
  • Numerical checks on concrete examples could identify the explicit transformations for each constraint type.
  • The result suggests that anytime-valid procedures can incorporate privacy or boundedness requirements more readily.

Load-bearing premise

That an appropriate transformation exists for each listed constraint which preserves log-optimality without requiring a different pair of least favorable distributions.

What would settle it

Finding one of the listed constraints, such as differential privacy, for which the transformed unconstrained optimum is not the true log-optimal constrained e-variable would disprove the central claim.

read the original abstract

E-variables enable safe and anytime-valid inference, with log-optimal e-variables given by the likelihood ratio of the least favorable distributions (LFDs) when they exist in composite settings. While this unconstrained theory is well understood, one may need/wish to impose additional structural constraints, including differential privacy, quantization, boundedness, or moment restrictions. We show that under these constraints, log-optimal constrained e-variables can often be constructed by a simple \emph{optimize-then-constrain} principle: first compute the unconstrained log-optimal e-variable, then impose the constraint via an appropriate transformation. Thus, the constrained growth-rate optimization problem does not require solving for a different LFD pair; the constrained optimal solution is just a post-processing of the unconstrained optimal solution.

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

2 major / 1 minor

Summary. The paper claims that log-optimal constrained e-variables under structural constraints (differential privacy, quantization, boundedness, moment restrictions) can be obtained via an 'optimize-then-constrain' principle: compute the unconstrained log-optimal e-variable as the likelihood ratio of the least favorable distributions (LFDs), then apply a suitable post-processing transformation T to enforce the constraint while preserving the e-variable property and the maximal growth rate E_Q[log T(E)]. The constrained optimization thus does not require solving for a new LFD pair.

Significance. If the central claim holds with explicit constructions and proofs for the listed constraints, the result would simplify construction of constrained e-variables by reusing unconstrained LFD theory and transformations, enabling practical anytime-valid inference in privacy-preserving, quantized, or bounded settings without re-deriving saddle points. It builds directly on established LFD concepts from prior literature rather than introducing new free parameters or self-referential definitions.

major comments (2)
  1. [Abstract] Abstract and introduction: The 'optimize-then-constrain' principle is asserted to work for differential privacy, quantization, boundedness, and moment restrictions, but the manuscript must explicitly verify (via derivation or counterexample check) that for each constraint the transformation T simultaneously satisfies E_P[T(E)] ≤ 1, meets the constraint, and achieves the maximal growth rate without the supporting LFD pair changing. The skeptic concern is that some constraints may alter the feasible set enough to shift the saddle point of the growth-rate game, requiring a different LFD pair; this needs to be addressed with a concrete check for at least the DP and moment-restriction cases.
  2. [Introduction / main claim] The central derivation relies on the axiom that 'the constrained optimization problem admits a solution that is a transformation of the unconstrained solution.' This needs to be proven or shown to hold under the paper's assumptions rather than taken as given, especially since the full details of the transformations and their preservation of log-optimality are not verifiable from the abstract alone.
minor comments (1)
  1. Clarify notation for the transformation T and the constrained growth-rate objective in the main text to avoid ambiguity with the unconstrained case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the referee's constructive feedback. We respond to the major comments point by point below. We agree that strengthening the explicit verifications will benefit the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: The 'optimize-then-constrain' principle is asserted to work for differential privacy, quantization, boundedness, and moment restrictions, but the manuscript must explicitly verify (via derivation or counterexample check) that for each constraint the transformation T simultaneously satisfies E_P[T(E)] ≤ 1, meets the constraint, and achieves the maximal growth rate without the supporting LFD pair changing. The skeptic concern is that some constraints may alter the feasible set enough to shift the saddle point of the growth-rate game, requiring a different LFD pair; this needs to be addressed with a concrete check for at least the DP and moment-restriction cases.

    Authors: We appreciate the referee pointing out the need for more explicit verification to address potential concerns about shifts in the LFDs. The manuscript derives the post-processing transformations for each listed constraint and shows that they preserve both the e-variable property and the maximal growth rate using the original LFD pair. This is because the transformations are designed to be applied after computing the likelihood ratio, and the expectation under the null remains bounded by 1 while the growth rate is not reduced. To directly address the skeptic concern, we will add a new subsection with concrete derivations and checks for the differential privacy and moment restriction cases, demonstrating that the saddle point does not change. revision: yes

  2. Referee: [Introduction / main claim] The central derivation relies on the axiom that 'the constrained optimization problem admits a solution that is a transformation of the unconstrained solution.' This needs to be proven or shown to hold under the paper's assumptions rather than taken as given, especially since the full details of the transformations and their preservation of log-optimality are not verifiable from the abstract alone.

    Authors: We clarify that the manuscript does not treat this as an unproven axiom but establishes it through a general argument applicable to the considered constraint classes. The key is that for these structural constraints, there exists a transformation T that maps the unconstrained optimal e-variable to a constrained one without loss of optimality. The full proof and details of T are provided in the body of the paper. Since the abstract is necessarily concise, we will revise the introduction to include a brief proof outline and better signpost the relevant sections where the preservation of log-optimality is shown. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on independent LFD theory

full rationale

The paper's central result is a construction principle showing that constrained log-optimal e-variables are obtained via post-processing transformations applied to the unconstrained log-optimal e-variable (likelihood ratio of LFD pair). This is framed as a theorem to be proven for listed constraints, not a definitional equivalence or fitted parameter renamed as prediction. The unconstrained LFD theory is described as 'well understood' from prior literature without load-bearing self-citations that reduce the claim to unverified author work. No self-definitional steps, ansatz smuggling, or renaming of known results appear in the provided derivation outline. The argument remains externally falsifiable via the growth-rate optimization and e-variable properties.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the prior existence of log-optimal e-variables via LFDs and on the mathematical existence of suitable post-processing transformations for the listed constraint classes.

axioms (2)
  • domain assumption Existence of least favorable distributions (LFDs) that achieve the unconstrained log-optimal e-variable
    The paper explicitly builds on the known result that unconstrained log-optimal e-variables are given by LFD likelihood ratios.
  • ad hoc to paper The constrained optimization problem admits a solution that is a transformation of the unconstrained solution
    This is the central new assumption required for the optimize-then-constrain principle to hold.

pith-pipeline@v0.9.0 · 5414 in / 1198 out tokens · 31573 ms · 2026-05-09T21:00:57.456842+00:00 · methodology

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Reference graph

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