arxiv: 2604.00449 · v2 · submitted 2026-04-01 · 💻 cs.LG · cs.MA· cs.SY· eess.SY

Recognition: 2 theorem links

· Lean Theorem

Convergence of Byzantine-Resilient Gradient Tracking via Probabilistic Edge Dropout

Amirhossein Dezhboro, Arman Adibi, Erfan Amini, Fateme Maleki, Jose E. Ramirez-Marquez

Authors on Pith no claims yet

Pith reviewed 2026-05-13 23:30 UTC · model grok-4.3

classification 💻 cs.LG cs.MAcs.SYeess.SY

keywords Byzantine-resilient optimizationgradient trackingprobabilistic edge dropoutdoubly stochastic mixinglinear convergenceleaky integrationdistributed learningadversarial networks

0 comments

The pith

Probabilistic edge dropout in gradient tracking preserves doubly stochastic mixing and achieves linear convergence to a variance-bounded neighborhood despite Byzantine agents.

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

The paper shows that clipping incoming messages to a fixed radius around the receiver and then dropping edges according to a dual-metric trust score lets gradient tracking retain its linear convergence rate even when some agents send arbitrary messages. When adversaries are fully isolated the neighborhood size depends only on the variance of honest stochastic gradients; when partial connections remain, a leaky integrator keeps the accumulated tracking error bounded by the ratio of the clip radius to the leak factor. The design deliberately avoids robust aggregation rules that destroy the doubly stochastic property of the mixing matrix, so standard convergence arguments continue to apply with only minor modifications. Experiments on MNIST under sign-flip, ALIE, and inner-product attacks confirm that the resulting algorithm outperforms coordinate-wise trimmed mean under stealthy attacks.

Core claim

GT-PD clips each received vector to a ball of radius tau centered at the local state and then applies a probabilistic dropout rule driven by a dual-metric trust score computed separately on the decision and tracking channels; the resulting random mixing matrix remains doubly stochastic in expectation. Under complete isolation (p_b = 0) the iterates converge linearly to a neighborhood whose radius is determined solely by the variance of honest stochastic gradients. For partial isolation the variant GT-PD-L augments the tracker with a leaky integrator whose leak factor controls the steady-state bias, yielding linear convergence to a neighborhood whose size is set by the stochastic variance and

What carries the argument

Probabilistic edge dropout driven by a dual-metric trust score, combined with self-centered projection clipping of radius tau and, for partial isolation, a leaky integrator on the tracking variable.

If this is right

Under complete Byzantine isolation the convergence neighborhood shrinks to the usual stochastic-gradient variance term.
With partial isolation the leaky-integrator variant bounds the neighborhood radius by the product of gradient variance and the clipping-to-leak ratio.
Two-tier dropout with honest probability one introduces no extra variance into the honest consensus dynamics.
The same clipping-plus-dropout layer can be added to other gradient-tracking variants without altering their linear rate.

Where Pith is reading between the lines

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

The same trust-driven dropout could be inserted into other consensus-based methods such as decentralized momentum or second-order trackers.
If the trust score is computed from recent history rather than a single step, the method might tolerate slowly adapting adversaries.
The clip radius tau and leak factor could be adapted locally from observed message norms, potentially tightening the neighborhood without global tuning.

Load-bearing premise

The trust score must separate honest and adversarial messages reliably enough that the expected mixing matrix stays doubly stochastic and unbiased.

What would settle it

Run the algorithm on a small network where adversaries are constructed to produce the same trust scores as honest agents; if the iterates diverge or the neighborhood size grows with attack strength, the separation assumption has failed.

Figures

Figures reproduced from arXiv: 2604.00449 by Amirhossein Dezhboro, Arman Adibi, Erfan Amini, Fateme Maleki, Jose E. Ramirez-Marquez.

**Figure 2.** Figure 2: Test accuracy of GT-PD, GT-PD-L, CWTM, and unprotected gradient tracking under three [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗

**Figure 3.** Figure 3: Extended diagnostics across three attacks (rows). Column (a): test accuracy. Column (b): consensus [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗

read the original abstract

We study distributed optimization over networks with Byzantine agents that may send arbitrary adversarial messages. We propose \emph{Gradient Tracking with Probabilistic Edge Dropout} (GT-PD), a stochastic gradient tracking method that preserves the convergence properties of gradient tracking under adversarial communication. GT-PD combines two complementary defense layers: a universal self-centered projection that clips each incoming message to a ball of radius $\tau$ around the receiving agent, and a fully decentralized probabilistic dropout rule driven by a dual-metric trust score in the decision and tracking channels. This design bounds adversarial perturbations while preserving the doubly stochastic mixing structure, a property often lost under robust aggregation in decentralized settings. Under complete Byzantine isolation ($p_b=0$), GT-PD converges linearly to a neighborhood determined solely by stochastic gradient variance. For partial isolation ($p_b>0$), we introduce \emph{Gradient Tracking with Probabilistic Edge Dropout and Leaky Integration} (GT-PD-L), which uses a leaky integrator to control the accumulation of tracking errors caused by persistent perturbations and achieves linear convergence to a bounded neighborhood determined by the stochastic variance and the clipping-to-leak ratio. We further show that under two-tier dropout with $p_h=1$, isolating Byzantine agents introduces no additional variance into the honest consensus dynamics. Experiments on MNIST under Sign Flip, ALIE, and Inner Product Manipulation attacks show that GT-PD-L outperforms coordinate-wise trimmed mean by up to 4.3 percentage points under stealth attacks.

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.

Desk Editor's Note private letter to a colleague

GT-PD pairs self-centered clipping with trust-score probabilistic dropout to keep gradient tracking convergent under Byzantines while trying to hold onto doubly stochastic mixing, but that invariance is the part that needs the closest look.

read the letter

The paper introduces GT-PD, which clips incoming messages to a ball around the receiver and then applies a dual-metric trust score to drop edges probabilistically in both the decision and tracking steps. The goal is to bound adversarial influence without breaking the mixing matrix properties that gradient tracking relies on for linear rates. They add a leaky integrator version (GT-PD-L) for cases where Byzantine agents stay partially connected. On MNIST with sign-flip, ALIE, and inner-product attacks, the method beats coordinate-wise trimmed mean by up to 4.3 points under stealth conditions. That combination of clipping plus channel-specific dropout is not a standard extension of prior robust aggregation work, and the claim that two-tier dropout with full honest retention adds no extra variance is a clean observation if it holds. The leaky fix for partial isolation is a practical touch that addresses a real gap in earlier isolation-only schemes. The main soft spot is the mixing guarantee. Gradient tracking convergence proofs need the expected weight matrix to stay doubly stochastic so the consensus error contracts at rate 1 minus the second eigenvalue. Because dropout decisions come from per-node trust scores on incoming messages, it is not immediate that the retention probabilities remain symmetric across each edge or that both channels preserve column sums. The abstract states the structure is maintained by design, but without the explicit expectation calculation or symmetry argument visible, it is hard to judge how much extra work the proof requires. Experiments report accuracy gains yet omit error bars and detailed ablations on the clipping radius and leak factor, which makes it difficult to separate the contribution of the dropout rule from the clipping alone. This work is aimed at people building decentralized optimizers that must tolerate faulty or malicious nodes. It is concrete enough and the claims are specific enough that a serious referee should see it, even if the mixing step and experimental controls will likely need tightening.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes Gradient Tracking with Probabilistic Edge Dropout (GT-PD) for distributed optimization over networks containing Byzantine agents. It combines a self-centered clipping projection of radius τ with a fully decentralized probabilistic edge dropout rule driven by a dual-metric trust score in the decision and tracking channels. The design is claimed to bound adversarial perturbations while preserving the doubly stochastic property of the mixing matrix. Under complete isolation (p_b=0) GT-PD is asserted to converge linearly to a neighborhood determined only by stochastic gradient variance; for partial isolation (p_b>0) the variant GT-PD-L with leaky integration is introduced and claimed to converge linearly to a neighborhood whose size depends on stochastic variance and the clipping-to-leak ratio. Experiments on MNIST under Sign-Flip, ALIE and Inner-Product-Manipulation attacks report accuracy gains of up to 4.3 percentage points over coordinate-wise trimmed mean.

Significance. If the convergence statements and the preservation of doubly-stochastic mixing are rigorously established, the work would constitute a meaningful advance in Byzantine-resilient decentralized optimization. Maintaining linear rates without resorting to aggregation operators that destroy the mixing matrix is a desirable property; the two-tier dropout and leaky-integrator mechanisms address persistent perturbations in a decentralized manner. The experimental results, if supported by proper statistical controls, would indicate practical utility.

major comments (2)

[Method description of the probabilistic dropout rule] The central claim that the trust-score-driven dropout preserves the doubly stochastic structure of the mixing matrix (required for the consensus-error contraction at rate 1-λ₂(W)) is asserted but not verified. No explicit argument or lemma shows that the per-edge retention probabilities remain symmetric across (i,j) pairs and that both rows and columns of E[W] continue to sum to one after asymmetric edge removal induced by the dual-metric trust score. This property is load-bearing for all linear-convergence statements.
[Convergence analysis for GT-PD-L] §4 (convergence analysis): the linear convergence of GT-PD-L to a neighborhood whose radius depends on the clipping-to-leak ratio is stated without an explicit error-bound derivation. The abstract supplies no steps showing how the leaky integrator controls accumulation of tracking errors under persistent perturbations, nor the precise dependence of the neighborhood size on τ and the leak factor.

minor comments (3)

[Abstract] The abstract introduces free parameters τ and the leak factor without stating selection guidelines or sensitivity results; a short discussion or table of default values would improve reproducibility.
[Abstract] The phrase 'two-tier dropout with p_h=1' is used without defining p_h or explaining why it introduces no additional variance into the honest consensus dynamics; a brief clarifying sentence would help.
[Experiments] Experimental results report accuracy improvements but omit error bars, ablation studies on trust-score hyperparameters, and details of the underlying network topology, limiting assessment of statistical significance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and the recommendation for major revision. The comments highlight important points where the manuscript's claims require stronger formal support. We address each major comment below and will revise the manuscript to incorporate explicit verifications and derivations as outlined.

read point-by-point responses

Referee: [Method description of the probabilistic dropout rule] The central claim that the trust-score-driven dropout preserves the doubly stochastic structure of the mixing matrix (required for the consensus-error contraction at rate 1-λ₂(W)) is asserted but not verified. No explicit argument or lemma shows that the per-edge retention probabilities remain symmetric across (i,j) pairs and that both rows and columns of E[W] continue to sum to one after asymmetric edge removal induced by the dual-metric trust score. This property is load-bearing for all linear-convergence statements.

Authors: We agree that the preservation of the doubly stochastic property under the dual-metric trust score requires an explicit lemma rather than an assertion. In the revised manuscript we will add a dedicated lemma (placed in Section 3) proving that (i) the per-edge retention probability computed from the dual-metric trust score is symmetric for every pair (i,j), and (ii) the resulting expected mixing matrix E[W] remains row- and column-stochastic. The proof will rely on the local but reciprocal nature of the trust-score computation and will directly support the contraction rate 1-λ₂(W) used in the subsequent convergence arguments. revision: yes
Referee: [Convergence analysis for GT-PD-L] §4 (convergence analysis): the linear convergence of GT-PD-L to a neighborhood whose radius depends on the clipping-to-leak ratio is stated without an explicit error-bound derivation. The abstract supplies no steps showing how the leaky integrator controls accumulation of tracking errors under persistent perturbations, nor the precise dependence of the neighborhood size on τ and the leak factor.

Authors: We acknowledge that the current Section 4 states the linear convergence result for GT-PD-L without a complete error-bound derivation. In the revision we will expand the analysis to include a step-by-step derivation that (i) bounds the accumulation of tracking errors via the contraction property of the leaky integrator, (ii) shows how persistent perturbations are controlled by the clipping radius τ, and (iii) explicitly expresses the radius of the convergence neighborhood in terms of the stochastic gradient variance, τ, and the leak factor. The updated proof will be self-contained and will also be reflected in a revised abstract statement. revision: yes

Circularity Check

0 steps flagged

No significant circularity; convergence claims rest on independent mixing preservation and standard GT analysis

full rationale

The paper asserts that its probabilistic dropout rule, driven by the dual-metric trust score, preserves the doubly stochastic property of the mixing matrix by design, enabling the standard linear convergence arguments for gradient tracking to apply with neighborhoods determined by stochastic variance (and clipping-to-leak ratio for the leaky variant). No equation reduces a claimed rate or neighborhood size to a fitted constant or self-referential quantity, and the central claims do not rely on load-bearing self-citations or imported uniqueness theorems. The derivation chain remains self-contained against external benchmarks for mixing-matrix contraction and SGD neighborhoods.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the preservation of doubly stochastic mixing under stochastic dropout and on the existence of a trust metric that separates Byzantine from honest messages; clipping radius tau and leak factor appear as design parameters whose values affect the final neighborhood size.

free parameters (2)

clipping radius tau
Bounds adversarial perturbations; value must be chosen relative to honest gradient magnitudes.
leak factor in GT-PD-L
Controls accumulation of tracking errors; appears as a tunable ratio with clipping radius.

axioms (2)

domain assumption Probabilistic edge dropout driven by dual-metric trust score preserves doubly stochastic mixing matrix
Invoked to retain standard linear convergence analysis of gradient tracking.
domain assumption Trust score reliably identifies Byzantine agents without systematic bias on honest channels
Required for the dropout rule to isolate adversaries while keeping honest consensus dynamics unchanged.

pith-pipeline@v0.9.0 · 5581 in / 1381 out tokens · 45575 ms · 2026-05-13T23:30:11.074639+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

This design bounds adversarial perturbations while preserving the doubly stochastic mixing structure, a property often lost under robust aggregation in decentralized settings.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 1 (Doubly stochastic property). If W is symmetric, then W^k defined by (2) is symmetric doubly stochastic almost surely.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[1]

Push-pull gradient methods for distributed optimization in networks,

S. Pu, W. Shi, J. Xu, and A. Nedic, “Push-pull gradient methods for distributed optimization in networks,” IEEE Trans. Autom. Control, vol. 66, no. 1, pp. 1–16, 2021

work page 2021
[2]

A linear algorithm for optimization over directed graphs with geometric convergence,

R. Xin, S. Kar, and U. A. Khan, “A linear algorithm for optimization over directed graphs with geometric convergence,”IEEE Control Syst. Lett., vol. 2, no. 3, pp. 315–320, 2018

work page 2018
[3]

Achieving geometric convergence for distributed optimization over time-varying graphs,

A. Nedic, A. Olshevsky, and W. Shi, “Achieving geometric convergence for distributed optimization over time-varying graphs,”SIAM J. Optim., vol. 27, no. 4, pp. 2597–2633, 2017

work page 2017
[4]

The Byzantine generals problem,

L. Lamport, R. Shostak, and M. Pease, “The Byzantine generals problem,”ACM Trans. Program. Lang. Syst., vol. 4, no. 3, pp. 382–401, 1982

work page 1982
[5]

Distributed function calculation via linear iterative strategies in the presence of malicious agents,

S. Sundaram and C. N. Hadjicostis, “Distributed function calculation via linear iterative strategies in the presence of malicious agents,”IEEE Trans. Autom. Control, vol. 56, no. 7, pp. 1495–1508, 2011

work page 2011
[6]

Resilient asymptotic consensus in robust networks,

H. J. LeBlanc, H. Zhang, X. Koutsoukos, and S. Sundaram, “Resilient asymptotic consensus in robust networks,”IEEE J. Sel. Areas Commun., vol. 31, no. 4, pp. 766–781, 2013

work page 2013
[7]

Byzantine-resilient distributed learning: Towards optimal statistical rates,

D. Yin, Y . Chen, R. Kannan, and P. Bartlett, “Byzantine-resilient distributed learning: Towards optimal statistical rates,” inProc. Int. Conf. Mach. Learn. (ICML), 2018

work page 2018
[8]

Machine learning with adversaries: Byzantine tolerant gradient descent,

P. Blanchard, E. M. El Mhamdi, R. Guerraoui, and J. Stainer, “Machine learning with adversaries: Byzantine tolerant gradient descent,” inProc. Adv. Neural Inf. Process. Syst. (NeurIPS), 2017

work page 2017
[9]

Agreement over random networks,

Y . Hatano and M. Mesbahi, “Agreement over random networks,”IEEE Trans. Autom. Control, vol. 50, no. 11, pp. 1867–1872, 2005

work page 2005
[10]

Consensus over ergodic stationary graph processes,

A. Tahbaz-Salehi and A. Jadbabaie, “Consensus over ergodic stationary graph processes,”IEEE Trans. Autom. Control, vol. 55, no. 1, pp. 225–230, 2010

work page 2010
[11]

Decentralized stochastic optimization and gossip algorithms with compressed communication,

A. Koloskova, S. Stich, and M. Jaggi, “Decentralized stochastic optimization and gossip algorithms with compressed communication,” inProc. Int. Conf. Mach. Learn. (ICML), 2019

work page 2019
[12]

Fault-tolerant multi-agent optimization: Optimal iterative distributed algo- rithms,

L. Su and N. H. Vaidya, “Fault-tolerant multi-agent optimization: Optimal iterative distributed algo- rithms,” inProc. ACM Symp. Princ. Distrib. Comput. (PODC), pp. 425–434, 2016

work page 2016
[13]

ByRDiE: Byzantine-resilient distributed coordinate descent for decentralized learning,

Z. Yang and W. U. Bajwa, “ByRDiE: Byzantine-resilient distributed coordinate descent for decentralized learning,”IEEE Trans. Signal Inf. Process. Netw., vol. 5, no. 4, pp. 611–627, 2019

work page 2019
[14]

Nesterov,Introductory Lectures on Convex Optimization: A Basic Course

Y . Nesterov,Introductory Lectures on Convex Optimization: A Basic Course. Boston, MA: Kluwer Academic Publishers, 2004

work page 2004
[15]

A little is enough: Circumventing defenses for distributed learning,

M. Baruch, G. Baruch, and Y . Goldberg, “A little is enough: Circumventing defenses for distributed learning,” inProc. Adv. Neural Inf. Process. Syst. (NeurIPS), 2019

work page 2019
[16]

Analysis and design of optimization algorithms via integral quadratic constraints,

L. Lessard, B. Recht, and A. Packard, “Analysis and design of optimization algorithms via integral quadratic constraints,”SIAM J. Optim., vol. 26, no. 1, pp. 57–95, 2016

work page 2016
[17]

Byzantine-resilient decentralized stochastic optimization with robust aggregation rules,

Z. Wu, T. Chen, and Q. Ling, “Byzantine-resilient decentralized stochastic optimization with robust aggregation rules,”IEEE Trans. Signal Process., 2023. 31

work page 2023
[18]

Efficient Byzantine-resilient decentralized learning: Sparse M-estimation with robust aggrega- tion techniques,

C. J. Li, “Efficient Byzantine-resilient decentralized learning: Sparse M-estimation with robust aggrega- tion techniques,”arXiv preprint arXiv:2410.01736, 2024

work page arXiv 2024
[19]

Byzantine-robust decentralized federated learning,

M. Fang, P. Khanduri, Z. Zhang, Y . Liu, and J. Liu, “Byzantine-robust decentralized federated learning,” arXiv preprint arXiv:2406.10416, 2024

work page arXiv 2024
[20]

On the geometric convergence of Byzantine-resilient dis- tributed optimization algorithms,

K. Kuwaranancharoen and S. Sundaram, “On the geometric convergence of Byzantine-resilient dis- tributed optimization algorithms,”SIAM J. Optim., vol. 35, no. 1, pp. 210–239, 2025

work page 2025
[21]

Byzantine-robust decentralized stochastic optimization over static and time-varying networks,

J. Peng, W. Li, and Q. Ling, “Byzantine-robust decentralized stochastic optimization over static and time-varying networks,”Signal Processing, vol. 183, p. 108020, 2021

work page 2021
[22]

Byzantine-Robust Decentralized Learning via ClippedGossip,

L. He, S. P. Karimireddy, and M. Jaggi, “Byzantine-Robust Decentralized Learning via ClippedGossip,” arXiv preprint arXiv:2202.01545, 2023

work page arXiv 2023
[23]

Credibility Assessment Based Byzantine- Resilient Decentralized Learning,

J. Hou, F. Wang, C. Wei, H. Huang, Y . Hu, and N. Gui, “Credibility Assessment Based Byzantine- Resilient Decentralized Learning,”IEEE Transactions on Dependable and Secure Computing, pp. 1–12, 2022

work page 2022
[24]

Byzantine-robust decentralized federated learning via dual-domain clustering and trust bootstrapping,

P. Sun, X. Liu, Z. Wang, and B. Liu, “Byzantine-robust decentralized federated learning via dual-domain clustering and trust bootstrapping,” inProc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit. (CVPR), pp. 24756–24765, 2024

work page 2024
[25]

Fall of empires: Breaking Byzantine-tolerant SGD by inner product manipulation,

C. Xie, S. Koyejo, and I. Gupta, “Fall of empires: Breaking Byzantine-tolerant SGD by inner product manipulation,” arXiv preprint arXiv:1903.03936, 2019. 32 A Appendix In this appendix, we provide extended diagnostic evaluations of the GT-PD and GT-PD-L algorithms across the considered threat models. As demonstrated in Fig. 3, the dual-layer defense archi...

work page arXiv 1903