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arxiv: 2606.19859 · v1 · pith:R4XYRVULnew · submitted 2026-06-18 · 💻 cs.IT · cs.LG· math.IT· math.PR· math.ST· stat.TH

Doeblin Curves

Dongmin Lee , William Lu , Anuran Makur , Japneet Singh This is my paper

Pith reviewed 2026-06-26 15:46 UTC · model grok-4.3

classification 💻 cs.IT cs.LGmath.ITmath.PRmath.STstat.TH
keywords Doeblin curvesinformation contractionMarkov kernelsgeneralization boundsnoisy computationdifferential privacymulti-way contraction
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The pith

Doeblin curves quantify contraction for Markov kernels even when the standard Doeblin coefficient is zero.

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

The paper introduces Doeblin curves as nonlinear functions that measure how a Markov kernel contracts collections of input distributions at given levels of divergence and power. This construction produces explicit multi-way contraction bounds without requiring the usual Doeblin coefficient to stay bounded away from zero. The resulting bounds are then applied to obtain generalization guarantees in noisy optimization, error bounds for noisy circuit computation, and differential privacy statements for iterative algorithms, all in broader group or domain settings than previously available.

Core claim

A Doeblin curve is a nonlinear function that quantifies the contraction behavior of a Markov kernel on collections of input distributions at specific levels of divergence and power. Through a new variational characterization of Doeblin coefficients, the paper derives properties, upper and lower bounds, and power-constrained versions of these curves, showing that they deliver non-vacuous contraction guarantees for channels whose ordinary Doeblin coefficient equals zero.

What carries the argument

Doeblin curve: a nonlinear function that quantifies contraction of a Markov kernel on collections of input distributions at fixed divergence and power levels.

If this is right

  • Generalization bounds for noisy iterative optimization extend to wider domains and group settings.
  • Error bounds for reliable computation with noisy circuits apply beyond single-instance cases.
  • Differential privacy guarantees for online iterative algorithms capture finer contraction at specific operating points.
  • Power-constrained versions of the curves allow contraction statements under input restrictions.

Where Pith is reading between the lines

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

  • The variational characterization may permit numerical approximation of contraction rates for channels too large for direct Doeblin-coefficient computation.
  • Doeblin curves could be composed across product channels to obtain joint contraction statements without separate coefficient calculations.
  • The same nonlinear viewpoint might adapt to other distance measures such as Renyi divergence or Wasserstein distance where linear coefficients also vanish.

Load-bearing premise

Concepts of nonlinear information contraction are enough to define Doeblin curves that still produce usable multi-way contraction bounds when the ordinary Doeblin coefficient is zero.

What would settle it

An explicit channel whose Doeblin coefficient is zero and whose Doeblin curve remains flat (no contraction) at the divergence and power values relevant to a target application would show the claimed non-vacuous bounds do not hold.

Figures

Figures reproduced from arXiv: 2606.19859 by Anuran Makur, Dongmin Lee, Japneet Singh, William Lu.

Figure 1
Figure 1. Figure 1: Randomly sampled joint range F(K; R 5×5 sto ) for a 5 × 5 Markov matrix K, shaded in blue. By submultiplicativity of complementary Doeblin coeffi￾cients [22, Section 4], [19, Theorem 1], we have ρ(WK) ≤ ρ(W)ρ(K). Hence, the Doeblin curve and joint range satisfy the outer bounds FK(t; G) ≤ ρ(K)t, F(K; G) ⊆ {(t, y) ∈ [0, 1]2 : y ≤ ρ(K)t}, (10) and since ρ(K) ≤ 1, we have FK(t; G) ≤ t, F(K; G) ⊆ {(t, y) ∈ [0,… view at source ↗
Figure 2
Figure 2. Figure 2: Average extremal power-constrained Doeblin curves for the Gaussian, [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
read the original abstract

Recent research on Doeblin coefficients has shed light on their usefulness as a multi-way generalization of the Dobrushin contraction coefficient for TV distance, in a separate vein from their classic role in the theory of Markov chain ergodicity. However, strong conditions, such as being bounded away from 0, are typically necessary for Doeblin coefficients to establish the existence of information contraction. Building on recently formulated concepts of nonlinear information contraction, we aim to propose a finer-grained Doeblin-based characterization of multi-way contraction behavior which yields non-vacuous contraction guarantees even for channels whose Doeblin coefficient is 0. To this end, we introduce the notion of a Doeblin curve -- a nonlinear function which quantifies the contraction behavior of a Markov kernel on collections of input distributions at specific levels of divergence and power. Through the course of our analysis, we develop a new variational characterization of Doeblin coefficients, present several properties of Doeblin curves, define several versions of power-constrained Doeblin curves, and derive upper and lower bounds using our aforementioned variational characterization. We then utilize these results in diverse areas, including generalization bounds for noisy iterative optimization, error bounds for reliable computation with noisy circuits, and differential privacy guarantees for online iterative algorithms. In particular, we extend results in these areas to broader domains or group settings, leveraging Doeblin curves to reveal finer-grained contraction phenomena than Doeblin coefficients.

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 / 2 minor

Summary. The paper introduces Doeblin curves, defined as nonlinear functions that quantify multi-way contraction of a Markov kernel on collections of input distributions at fixed divergence and power levels. Building on nonlinear information contraction ideas, it provides a new variational characterization of Doeblin coefficients, derives properties and bounds for (power-constrained) Doeblin curves, and applies the framework to obtain generalization bounds for noisy optimization, error bounds for noisy circuits, and differential privacy guarantees for iterative algorithms, claiming non-vacuous contraction even when the ordinary Doeblin coefficient is zero.

Significance. If the central claims hold, the work would meaningfully extend contraction-coefficient techniques to kernels with vanishing global Doeblin coefficient by replacing a single scalar with a curve indexed by divergence level; the listed applications would then cover broader regimes than standard Doeblin-coefficient arguments permit.

major comments (2)
  1. [§3] §3 (variational characterization): the manuscript must exhibit an explicit channel example (or general construction) in which the Doeblin curve evaluated at a positive divergence level yields a contraction factor strictly less than 1 while the ordinary Doeblin coefficient is exactly zero; without this, the non-vacuous guarantee asserted in the abstract remains unverified.
  2. [§4–5] §4–5 (properties and bounds): the upper and lower bounds derived from the variational form must be shown to be strictly tighter than the trivial bound of 1 at the operating points used in the applications; if the bounds collapse to the classical Doeblin coefficient when the latter is zero, the claimed extension fails.
minor comments (2)
  1. Notation for the power-constrained variants should be introduced once and used consistently; several symbols appear to be redefined across sections.
  2. [§6–8] The applications in §6–8 would benefit from a short table comparing the new bounds against the classical Doeblin-coefficient bounds on the same examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address the two major comments point by point below and will incorporate the requested clarifications and examples into a revised manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (variational characterization): the manuscript must exhibit an explicit channel example (or general construction) in which the Doeblin curve evaluated at a positive divergence level yields a contraction factor strictly less than 1 while the ordinary Doeblin coefficient is exactly zero; without this, the non-vacuous guarantee asserted in the abstract remains unverified.

    Authors: We agree that an explicit example is required to verify the non-vacuous claim. The variational characterization in §3 already implies the existence of such kernels (by taking collections of distributions at positive divergence where the multi-way contraction is strictly less than the global Doeblin coefficient of zero), but we did not include a concrete construction. In the revision we will add a specific example, e.g., a carefully parameterized binary erasure channel or a constructed finite-state kernel whose Doeblin coefficient vanishes while the curve at a chosen positive divergence level yields a factor <1, computed directly from the variational form. revision: yes

  2. Referee: [§4–5] §4–5 (properties and bounds): the upper and lower bounds derived from the variational form must be shown to be strictly tighter than the trivial bound of 1 at the operating points used in the applications; if the bounds collapse to the classical Doeblin coefficient when the latter is zero, the claimed extension fails.

    Authors: We accept that the applications in §6–8 must be accompanied by explicit verification that the derived bounds are strictly less than 1 at the relevant operating points. The power-constrained Doeblin curves are designed precisely so that they remain <1 at positive divergence levels even when the ordinary coefficient is zero; however, the current text does not numerically or analytically confirm this at the exact parameter values used in the generalization, circuit, and DP bounds. In the revision we will insert short calculations or plots in §4–5 (and cross-references in the applications) demonstrating that the variational upper and lower bounds are strictly tighter than 1 at those points. revision: yes

Circularity Check

0 steps flagged

No circularity: Doeblin curves defined from independent nonlinear contraction concepts

full rationale

The paper builds explicitly on 'recently formulated concepts of nonlinear information contraction' to introduce Doeblin curves as a nonlinear function quantifying contraction on collections of input distributions. It develops a new variational characterization, properties, power-constrained versions, and bounds, then applies them to generalization, computation, and privacy. No quoted equations or steps reduce the target contraction guarantee to a fitted parameter, self-definition, or self-citation chain by construction. The variational form is presented as a development enabling finer-grained analysis when the standard Doeblin coefficient is zero, without evidence that it collapses to the ordinary infimum. The derivation chain remains self-contained with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the definition of Doeblin curves and the assumption that nonlinear information contraction concepts extend to level-specific multi-way bounds; no free parameters or external data are mentioned.

axioms (1)
  • domain assumption Nonlinear information contraction concepts recently formulated in the literature
    Abstract states the work builds directly on these concepts to define the curves.
invented entities (1)
  • Doeblin curve no independent evidence
    purpose: Nonlinear function quantifying contraction behavior of a Markov kernel on collections of distributions at specific divergence and power levels
    Newly introduced object whose properties and bounds are developed in the paper.

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