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arxiv: 2605.07633 · v2 · pith:BOPFA7EBnew · submitted 2026-05-08 · 🧮 math.OC

Distributed Seeking for Fixed Points of Biased Stochastic Operators: A Communication-Efficient Approach

Fan Li , Lei Xu , Xinlei Yi , Guanghui Wen , Yang Shi , Tao Yang This is my paper

Pith reviewed 2026-05-22 10:30 UTC · model grok-4.3

classification 🧮 math.OC
keywords distributed fixed point seekingstochastic operatorscommunication compressionKrasnosel'skii-Mann iterationsurrogate functionnon-convex optimizationmulti-agent networks
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The pith

A surrogate function unifies convergence analysis for distributed fixed-point seeking of biased stochastic operators and non-convex optimization.

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

This paper develops a distributed algorithm to find fixed points of sum-separable stochastic operators on a multi-agent network. It builds the method on inexact Krasnosel'skiĭ–Mann iterations while adding communication compression and dynamic period skipping to cut data transfer. The operators are allowed relaxed growth bias and variance that go beyond the usual unbiased and bounded-noise assumptions. Convergence is proved by introducing a surrogate function that works for both non-contractive and contractive cases, which also creates a direct theoretical link to distributed non-convex optimization algorithms.

Core claim

By introducing a surrogate function for general non-contractive and contractive operators, the paper establishes convergence guarantees of the distributed fixed point iteration based on inexact Krasnosel'skiĭ–Mann iterations with communication compression, achieving among the first theoretical unifications with distributed non-convex optimization algorithms under relaxed growth bias and variance conditions.

What carries the argument

The surrogate function for general non-contractive and contractive operators that carries the convergence analysis of the inexact distributed iterations.

If this is right

  • The algorithm converges to the fixed point of the sum of the operators.
  • Communication efficiency is achieved through a unified compressor allowing relative and absolute errors together with dynamic period skipping.
  • Convergence holds under relaxed stochastic conditions that generalize traditional unbiased and bounded variance assumptions.
  • The results provide a theoretical unification with the analysis of distributed non-convex optimization algorithms.

Where Pith is reading between the lines

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

  • The same surrogate construction may extend naturally to other inexact fixed-point methods beyond Krasnosel'skiĭ–Mann iterations.
  • The communication-compression and skipping techniques could be combined with event-triggered updates in broader multi-agent coordination tasks.
  • Numerical validation on larger networks would test whether the observed rates remain practical when bias terms grow close to the allowed bound.

Load-bearing premise

The stochastic operators satisfy relaxed growth bias and variance conditions that generalize traditional unbiased and bounded additive variance assumptions.

What would settle it

A concrete counter-example network and operator set where the proposed distributed iteration diverges despite satisfying the relaxed bias and variance bounds.

Figures

Figures reproduced from arXiv: 2605.07633 by Fan Li, Guanghui Wen, Lei Xu, Tao Yang, Xinlei Yi, Yang Shi.

Figure 1
Figure 1. Figure 1: Evolutions of ∥ 1 N PN i=1 T (xi) − xi∥ 2 for the three al￾gorithms with different compressors in the non-convex case. (a) Evolutions with respect to the number of iterations (b) Evolutions with respect to communication bits [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Evolutions of ∥ 1 N PN i=1 T (xi) − xi∥ 2 for Algorithm 1 with different compressors in the non-convex case. where N = 6 and f1 (x) = 0.06 x 4 − 0.02 x 2 , f2 (x) = 0.05 sin x + 1 2  + 0.15 cos 10x 3  , f3 (x) = 0.1 e −x 2 + 0.1 x 4 − 0.3 x 2 , f4 (x) = 0.14 x 4 − 0.2 x 2 , f5 (x) = 0.45 cos (x) + 0.15 sin 10x 3 + 1 2  , f6 (x) = 0.4 e −x 2 − 0.3x 2 . Considering measurement errors and other prac￾tical … view at source ↗
Figure 5
Figure 5. Figure 5: Evolutions of 1 N PN i=1 ∥x t i − x ∗ ∥ 2 for different algo￾rithms equipped with various compressors in the strongly convex case. 0 500 1000 1500 2000 2500 3000 3500 4000 Iteration index t 10-5 100 105 1N P N i=1 jjx t i ! x $ jj2 Algorithm 1:- = 0:01; < = 0:01; H = 3 Algorithm 1:- = 0:1; < = 0:01; H = 3 Algorithm 1:- = 1; < = 0:01; H = 3 (a) Comparisons under dif￾ferent biases settings 0 500 1000 1500 20… view at source ↗
Figure 6
Figure 6. Figure 6: benchmarks Algorithm 1 against varying bias￾variance configurations of stochastic noise for strongly convex objectives. Similar to the non-convex case, we ob￾serve that Algorithm 1 convergence to a neighborhood (a) Evolutions with respect to the number of iterations (b) Evolutions with respect to communication bits [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
read the original abstract

This paper investigates the distributed fixed point seeking problem of sum-separable stochastic operators over the multi-agent network. Based on inexact Krasnosel'ski\u{\i}--Mann iterations, the communication-efficient distributed algorithm is proposed under the relaxed growth bias and variance conditions, generalizing traditional unbiased and bounded additive variance assumptions. To enhance communication efficiency, we integrate communication compression and dynamic period skipping techniques, particularly adopting a unified compressor that allows both relative and absolute compression errors. By introducing a surrogate function for general non-contractive and contractive operators, we establish convergence guarantees of the distributed fixed point iteration, achieving among the first theoretical unifications with distributed non-convex optimization algorithms. Finally, numerical simulations validate the effectiveness of the theoretical results.

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 manuscript develops a communication-efficient distributed algorithm for seeking fixed points of sum-separable biased stochastic operators over multi-agent networks. It builds on inexact Krasnosel'skii-Mann iterations that incorporate a unified compressor (allowing relative and absolute errors) together with dynamic period skipping. Convergence guarantees are derived under relaxed growth bias and variance conditions that generalize the classical unbiased and bounded-variance assumptions. A surrogate function is introduced to handle both contractive and non-contractive operators, with the claim that this yields one of the first theoretical unifications between distributed fixed-point iteration and distributed non-convex optimization. Numerical simulations are provided to illustrate the results.

Significance. If the convergence analysis holds, the work would supply a useful generalization of stochastic fixed-point theory and a concrete unification with non-convex distributed optimization. The relaxed bias/variance conditions and the unified compressor are technically attractive features. The paper also supplies reproducible numerical validation, which strengthens the practical side of the contribution.

major comments (2)
  1. [§4.3, Eq. (22)] §4.3, Eq. (22): the bound on the additive perturbation induced by the unified compressor is derived using only the relaxed variance condition; it is not shown that this bound remains valid for non-contractive operators when the surrogate-function decrease is driven solely by the bias term, which is load-bearing for the general non-contractive claim.
  2. [Theorem 5.2] Theorem 5.2: the surrogate-function decrease inequality for non-contractive operators appears to require an implicit Lipschitz-like control on the operator to absorb the compression error; this assumption is not listed among the stated relaxed conditions and is central to the unification result.
minor comments (2)
  1. [§2.2] §2.2: the notation for the dynamic period-skipping parameter could be introduced earlier and used consistently in the algorithm description.
  2. [Figure 3] Figure 3: the convergence plots would be clearer if the y-axis were labeled with the exact quantity being plotted (e.g., distance to fixed point or surrogate value).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. We address each major comment point by point below, providing explanations based on the manuscript's analysis and indicating where clarifications will be added.

read point-by-point responses

  1. Referee: [§4.3, Eq. (22)] §4.3, Eq. (22): the bound on the additive perturbation induced by the unified compressor is derived using only the relaxed variance condition; it is not shown that this bound remains valid for non-contractive operators when the surrogate-function decrease is driven solely by the bias term, which is load-bearing for the general non-contractive claim.

    Authors: We thank the referee for this observation. The bound in Eq. (22) follows directly from the relaxed variance condition (Assumption 2), which is formulated independently of contractivity and applies to the stochastic perturbation term regardless of the operator class. For non-contractive operators the surrogate decrease is indeed driven by the growth bias term, but the compression-induced additive perturbation is bounded separately using only the variance assumption and the properties of the unified compressor; no contractivity is invoked in that step. To improve readability we will insert a short remark after Eq. (22) explicitly noting that the variance bound holds uniformly for both contractive and non-contractive cases. revision: partial

  2. Referee: [Theorem 5.2] Theorem 5.2: the surrogate-function decrease inequality for non-contractive operators appears to require an implicit Lipschitz-like control on the operator to absorb the compression error; this assumption is not listed among the stated relaxed conditions and is central to the unification result.

    Authors: We respectfully disagree that an additional Lipschitz condition is required. The relaxed growth-bias condition (Assumption 3) supplies a linear growth bound on the operator that is sufficient to absorb the compression-error terms inside the surrogate decrease inequality of Theorem 5.2. This is the same mechanism used to unify the result with distributed non-convex optimization; no extra Lipschitz continuity beyond the stated assumptions is employed in the proof. We will add a clarifying sentence in the proof of Theorem 5.2 that explicitly invokes the growth-bias bound to control the compression terms. revision: partial

Circularity Check

0 steps flagged

No significant circularity; surrogate function serves as independent analytical tool

full rationale

The paper's central derivation introduces a surrogate function to unify convergence analysis for non-contractive and contractive stochastic operators under relaxed growth bias and variance conditions. This construct is described as an auxiliary device for bounding inexact Krasnosel'skii-Mann iterations that incorporate compression and period skipping, rather than a self-referential redefinition or a fitted quantity renamed as a prediction. No equations or self-citation chains in the provided abstract or claims reduce the convergence guarantee to the inputs by construction. The analysis remains self-contained against the stated operator assumptions and external benchmarks for distributed fixed-point seeking.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the relaxed bias and variance conditions plus the surrogate function; these are not standard background results and are introduced to make the convergence proof go through.

axioms (1)
  • domain assumption Relaxed growth bias and variance conditions on the stochastic operators
    Generalizes traditional unbiased and bounded additive variance assumptions to support the inexact iterations.
invented entities (1)
  • Surrogate function for general non-contractive and contractive operators no independent evidence
    purpose: To establish convergence guarantees for both operator classes
    Introduced specifically to unify the analysis; no independent evidence outside the paper is mentioned.

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