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arxiv: 2606.12720 · v1 · pith:TIP5323Inew · submitted 2026-06-10 · 🧮 math.PR · math.ST· stat.ML· stat.TH

On McDiarmid's Inequality under Dependence via Approximate Tensorization of Entropy

Valentin Roth This is my paper

Pith reviewed 2026-06-27 08:03 UTC · model grok-4.3

classification 🧮 math.PR math.STstat.MLstat.TH
keywords McDiarmid inequalityapproximate tensorization of entropyentropy methodconcentration inequalitiesdependent random variablesGaussian measuresDvoretzky-Kiefer-Wolfowitz inequalitystochastic localization
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The pith

Approximate tensorization of entropy implies McDiarmid's inequality for dependent variables via the entropy method, with the constant scaling as the condition number of the covariance for non-isotropic Gaussians.

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

The paper establishes that the approximate tensorization of entropy property suffices to derive McDiarmid-type concentration bounds even when the underlying variables are dependent. The derivation proceeds through the entropy method and produces explicit constants controlled by the tensorization factor. For non-isotropic Gaussian vectors the bound scales with the condition number of the covariance matrix. The same framework is applied to obtain concentration results for the sign of a Gaussian vector, for dependent Erdős-Rényi graphs, and for a Dvoretzky-Kiefer-Wolfowitz inequality on the empirical distribution function that achieves the 1/sqrt(n) rate under weak dependence.

Core claim

Approximate tensorization of entropy implies McDiarmid's inequality via the Entropy Method. For X ~ N(μ, Σ) this yields a McDiarmid constant of order the condition number of Σ. The ATE property is obtained independently via stochastic localization and also follows from a more general result on the Gibbs sampler for strongly log-concave and log-smooth measures, which extends the concentration statement to that broader class.

What carries the argument

Approximate tensorization of entropy (ATE), the multiplicative control of joint entropy by a sum of conditional entropies that lets the entropy method produce concentration inequalities under dependence.

If this is right

  • McDiarmid-type bounds hold for every measure obeying approximate tensorization of entropy, with the multiplicative constant fixed by the tensorization factor.
  • For non-isotropic Gaussians the McDiarmid constant is of order the condition number of Σ.
  • Concentration inequalities for sign(X) follow directly for Gaussian vectors X.
  • A Dvoretzky-Kiefer-Wolfowitz inequality holds at the expected 1/sqrt(n) rate for observations drawn from any measure with ATE and continuous marginal CDFs.
  • The same concentration applies to Erdős-Rényi graphs whose edges satisfy the ATE property.

Where Pith is reading between the lines

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

  • The stochastic-localization derivation of ATE may extend to other families of log-concave measures beyond Gaussians.
  • The resulting bounds could be used to analyze concentration in additional dependent graph models or in statistical procedures with weakly dependent samples.
  • Connections between ATE and other functional inequalities such as Poincaré or log-Sobolev may yield further concentration statements.
  • Numerical verification on high-condition-number covariance matrices would test whether the predicted scaling is sharp.

Load-bearing premise

The probability measures under study satisfy the approximate tensorization of entropy property.

What would settle it

Exhibit a measure satisfying approximate tensorization of entropy yet violating the corresponding McDiarmid bound, or compute the exact tail constant for a Gaussian vector whose covariance has large condition number and check whether the observed constant exceeds the predicted order.

read the original abstract

We argue that dependent versions of McDiarmid's inequality are a useful but underutilized tool in mathematical statistics, learning theory and theoretical computer science. To make this point, we first highlight that approximate tensorization of entropy (ATE) implies McDiarmid's via the Entropy Method. Second, we derive McDiarmid's inequality for non-isotropic Gaussian random vectors $X \sim \mathcal N(\mu, \Sigma)$ through ATE with a constant of the order of the condition number of $\Sigma$. We both independently obtain this ATE through a simple application of stochastic localization and also discuss how a more general ATE for the Gibbs sampler due to Ascolani et al., 2026 generalizes McDiarmid's-like concentration to strongly log-concave and log-smooth probability measures. We then apply the resulting concentration inequalities to resolve a question on the concentration of $\operatorname{sign}(X)$ posed by Simone Bombari, investigate Erd\H{o}s-R\'enyi graphs under dependence and prove a Dvoretzky-Kiefer-Wolfowitz-type inequality for observations from a joint measure fulfilling ATE and continuous marginal CDFs. For the class of strongly log-concave and log-smooth measures, this result improves upon a prior Dvoretzky-Kiefer-Wolfowitz-type inequality for non-i.i.d. observations due to Bobkov and G\"otze, 2010, by establishing the expected $1/\sqrt{n}$-rate of convergence under weak dependence instead of $n^{-1/3}$.

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

0 major / 3 minor

Summary. The paper claims that approximate tensorization of entropy (ATE) implies McDiarmid-type concentration via the entropy method. It derives an ATE-based McDiarmid inequality for X ~ N(μ, Σ) with constant of order cond(Σ), obtained independently via stochastic localization (and notes a generalization from Ascolani et al. 2026 for strongly log-concave measures). Applications include concentration of sign(X), dependent Erdős-Rényi graphs, and a DKW-type inequality for ATE-satisfying measures with continuous marginals that achieves the 1/√n rate under weak dependence (improving Bobkov-Götze 2010).

Significance. If the derivations hold, the work usefully connects ATE to McDiarmid inequalities under dependence, with the Gaussian result and the improved DKW rate providing concrete tools for statistics and TCS. The independent stochastic-localization derivation of the ATE factor and the explicit applications (sign(X), ER graphs, DKW) are strengths that make the contribution self-contained and falsifiable.

minor comments (3)
  1. [§3] §3 (Gaussian case): the statement that the ATE constant is 'of order the condition number' should include an explicit upper bound (e.g., in terms of λ_max/λ_min) rather than asymptotic order, to make the comparison with isotropic McDiarmid immediate.
  2. [Abstract, §1] Abstract and §1: the citation 'Ascolani et al., 2026' appears to be a forward reference; confirm the year and add a note on whether the present derivation is independent or recovers the same constant.
  3. [§5] §5 (DKW application): the proof sketch that ATE + continuous marginal CDFs yields the 1/√n rate should explicitly cite the entropy-method step that replaces the n^{-1/3} barrier of Bobkov-Götze.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. The report correctly identifies the core contributions: the link from approximate tensorization of entropy (ATE) to McDiarmid-type bounds via the entropy method, the Gaussian result with condition-number dependence obtained independently via stochastic localization, the generalization via Ascolani et al. (2026), and the concrete applications yielding improved rates.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds from the standard implication that approximate tensorization of entropy yields McDiarmid-type bounds via the entropy method, followed by an independent derivation of the ATE factor for non-isotropic Gaussians obtained directly via stochastic localization; this step is self-contained and does not reduce to any fitted input, self-definition, or load-bearing self-citation. The additional reference to Ascolani et al. 2026 supplies a generalization but is not required for the core Gaussian or application results, which rest on the paper's own localization argument and the entropy-method implication. All subsequent applications (sign(X), ER graphs, DKW) follow directly once ATE is granted, with no equations or claims that collapse by construction to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the ATE property holding for the target measures; this is treated as a domain assumption derived via stochastic localization or prior results.

axioms (1)
  • domain assumption Approximate tensorization of entropy holds for the probability measures considered (non-isotropic Gaussians and strongly log-concave log-smooth measures).
    This property is invoked to obtain McDiarmid via the entropy method and is the load-bearing premise for all subsequent inequalities.

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discussion (0)

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