Decentralized Learning with Approximate Finite-Time Consensus

Aaron Fainman; Stefan Vlaski

REVIEW 2 major objections 4 minor 23 references

Decentralized Learning with Approximate Finite-Time Consensus

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Approximate finite-time consensus sequences retain provable convergence bounds for decentralized gradient tracking.

desk verdict The qualitative question is worth asking, but the paper's central convergence bounds rest on a false lemma and omitted proofs, so the results do not stand as stated. read the letter →

arxiv 2501.07967 v1 pith:SH357I6W submitted 2025-01-14 eess.SP

Aaron Fainman , Stefan Vlaski This is my paper · ORCID

classification eess.SP MSC 68W1590C25

keywords decentralizedoptimizationfinite-timeconsensusgradienttrackingapproximateAug-DGMmean-squareconvergencecombinationmatricesdistributedlearning

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether decentralized optimization still has provable guarantees when the combination matrices only approximately achieve finite-time consensus (FTC), meaning their product is close to the scaled all-ones matrix but not exactly equal to it. Exact FTC sequences are known in closed form only for special graphs, so an approximation is the practical route for general networks. The authors study the Aug-DGM gradient-tracking algorithm with approximate FTC matrices and derive mean-square bounds showing that consensus error decays every $\tau$ iterations at a rate controlled by $\epsilon_\tau$, with steady-state error growing like $\mu^2 \tau^2/(1-\epsilon_\tau)^2$. The conclusion is positive: approximate FTC sequences can be certified by a single scalar $\epsilon_\tau$ and remain viable on arbitrary graphs.

What carries the argument

The argument is carried by the block-triangular product matrix $\mathcal{G}_{i:m\tau+1}$ that propagates the joint consensus error over a full cycle of the FTC sequence. Because the combination matrices are cycled modulo $\tau$, the product over $\tau$ steps is expected to contract by the FTC property; the paper measures the quality of an approximation by $\epsilon_\tau = \| A_\tau\cdots A_1 - (1/K)\mathbf{1}\mathbf{1}^T\|$. Lemma 1 asserts that the spectral norm of this block-triangular product is bounded by the norm of its diagonal block product, turning approximate consensus into the contraction rate $\epsilon_\tau$. Lemmas 2 and 3 bound the gradient-heterogeneity and noise contributions to the error recursion, and Theorem 1 assembles these into the $\tau$-block contraction bound that Theorem 2 uses for the centroid error.

What would settle it

Compute $\|\mathcal{G}_{i:m\tau+1}\|$ for a small graph with a known approximate FTC product whose diagonal block product has norm $\epsilon_\tau$ but whose off-diagonal blocks are nonzero; if the full norm exceeds $\epsilon_\tau$, Lemma 1 fails. A numerical check of Theorem 1 would run Aug-DGM on a graph with $\tau=4$, measure the consensus error every $\tau$ iterations, and see whether the decay factor obeys $\frac{3}{8}\epsilon_\tau(2+3\epsilon_\tau)$ or is controlled by the true product norm instead.

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Extended reading notes

Core claim

The central claim is that approximate finite-time consensus sequences, not just exact ones, are compatible with Aug-DGM gradient tracking and yield explicit mean-square convergence bounds. For $0<\epsilon_\tau<2/3$, the mean-square consensus error is bounded by an exponentially decaying term $O(\epsilon_\tau)^{\lfloor i/\tau\rfloor}\|\widehat{\mathbf{x}}_0\|^2$ plus steady-state terms proportional to $\mu^2 K \tau^2 B^2/(1-\epsilon_\tau)^2$ and $\mu^2 \tau^2 \sigma^2/(1-\epsilon_\tau)^2$; the exact case $\epsilon_\tau=0$ recovers the earlier exact-FTC bound. The centroid error obeys a similar bound with $\tau^2/(1-\epsilon_\tau)^2$ factors plus the usual $\mu \sigma^2/K$ stochastic term. The paper states that the proof of Theorem 1 is omitted due to space limitations, so the quantitative bound rests on Lemmas 1 through 3. The message is that approximation error degrades performance continuously and can be quantified, making numerically inferred or learned FTC sequences usable for general graphs.

Load-bearing premise

The load-bearing premise is that the norm of the whole $\tau$-step update matrix is no larger than the norm of its diagonal blocks; if the off-diagonal couplings across agents make the true product norm larger, the advertised contraction rate at rate $\epsilon_\tau$ does not follow.

Editorial extensions

If this is right

General graphs can use numerically inferred or learned approximate FTC sequences; the only certification needed is the scalar $\epsilon_\tau$ defined in (5).
The closer the matrix product is to the scaled all-ones matrix, the faster the consensus error decays each $\tau$ cycle and the smaller the steady-state error.
If $\epsilon_\tau$ approaches the stated thresholds, the admissible step size shrinks and the steady-state terms grow like $1/(1-\epsilon_\tau)^2$, so poor approximations force slower optimization.
Larger $\tau$ worsens performance: both bounds contain $\tau^2$ factors, so for some graphs it may be better to deliberately underestimate $\tau$ and accept a nonzero $\epsilon_\tau$.
In the limit $\epsilon_\tau=0$, the bounds reduce to the exact finite-time consensus analysis, showing the new result is a strict generalization of the earlier one.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If Lemma 1's norm bound fails, the contraction rate should be replaced by the actual spectral norm of the $\tau$-step product; the structure of Theorems 1 and 2 suggests the bounds would survive with $\epsilon_\tau$ redefined as that true norm.
The bounds imply a practical design rule: when choosing candidate approximate FTC sequences, minimize $\tau^2\epsilon_\tau/(1-\epsilon_\tau)^2$ rather than $\epsilon_\tau$ alone, since the steady-state error weights approximation error quadratically in $\tau$.
The analysis could plausibly extend to time-varying or directed graphs by tracking whether the $\tau$-step product still contracts on the disagreement subspace, and to non-convex objectives by combining the consensus bound with standard descent lemmas.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper addresses a real gap: exact finite-time consensus (FTC) sequences are known only for structured graphs, so a theory for approximate FTC sequences would be broadly useful. The authors correctly identify epsilon_tau as the right quality measure, and the qualitative prediction that approximation error degrades performance continuously is plausible and consistent with the simulations. The reformulation of Aug-DGM follows the sensible precedent in Nguyen et al. So the problem is well chosen and the narrative is honest.

That is where the good news ends. The central contraction step, Lemma 1, is false. It claims that the spectral norm of the full tau-period product of the block-triangular consensus matrices is bounded by epsilon_tau, citing the textbook claim that the norm of a block triangular matrix equals the maximum norm of its diagonal blocks. That identity is not true in general, and here the off-diagonal block is not zero. Over one period the product has the form [[P, -tau P],[0,P]] with P the product of the diagonal blocks, so its norm is ||P|| * ||T_tau||, where ||T_tau|| > 1 for tau >= 2. For tau=2 the factor is about 2.414. Thus ||G_{i:m tau+1}|| is not bounded by epsilon_tau, and the exponential contraction term in Theorem 1 is unsupported. The stress-test note is correct, and it lands directly on the main theorem.

The other major problem is that neither main theorem is actually proven. Theorem 1 ends with "Proof: Omitted due to space limitations," and Theorem 2 has no proof at all. In a conference paper that is sometimes acceptable if the full proof is in an appendix or supplementary material, but here it is simply absent. The reader is being asked to take the two central results on faith. The simulations are illustrative, but they cannot substitute for a derivation, especially when the one lemma that is stated turns out to be wrong.

There are smaller issues too: the step-size condition and constants seem to come out of nowhere, and the comparison with the exact-FTC case is not enough to establish the claimed O(tau^2) behavior. None of that matters much, because the false lemma already sinks the argument as written.

Who is this for? A researcher actively working on finite-time consensus might want to know that the approximate-FTC direction is worth pursuing, and the experimental evidence for the qualitative trend is mildly encouraging. But as a source of quantitative guarantees, the paper is not usable.

My recommendation: desk reject with a clear explanation. The central lemma is demonstrably false, so sending this to referees would waste their time. The right path is to invite the authors to resubmit a complete version with a corrected contraction proof and the full derivations for Theorems 1 and 2. The underlying idea is salvageable, but this manuscript does not yet carry the weight of its claims.

Referee Report

2 major / 4 minor

Summary. The manuscript studies the Aug-DGM decentralized optimization algorithm when the combination matrices form an approximate finite-time consensus (FTC) sequence, i.e., a cyclic sequence whose τ-step product equals (1/K)11^T plus an error of norm ϵτ. It states two theorems: Theorem 1 bounds the mean-square consensus error by an exponential term with contraction factor (3/8)ϵτ(2+3ϵτ) plus steady-state terms proportional to μ²Kτ²B²/(1-ϵτ)² and μ²τ²σ²/(1-ϵτ)²; Theorem 2 bounds the centroid error to the optimum, with similar 1/(1-ϵτ)² dependencies. The paper also provides simulation results for logistic regression on a path graph and a hypercube, and argues that larger ϵτ or larger τ degrades performance. The central analytical tool is Lemma 1, which claims that the product of the block-triangular consensus-error matrices over a τ-period has spectral norm at most ϵτ.

Significance. The question addressed — how approximation error in finite-time consensus matrices degrades the performance of decentralized learning — is a sensible and timely one, since exact FTC sequences are known only for special graphs and a theory for approximate FTC would broaden applicability. If the bounds were correct, they would provide a useful design rule: faster mixing (smaller ϵτ) and smaller consensus number τ are beneficial, and performance degrades continuously with ϵτ. The paper also gives clear empirical demonstrations that the predicted monotonicity in ϵτ and τ holds in simulation. However, the correctness of the theoretical claims is not established: Lemma 1 is false, and the main theorems are stated without proofs. The empirical observations may be valid, but the theoretical contribution as stated is not supported.

major comments (2)

[Section II-C, Lemma 1 (Eq. (16))] The asserted bound ∥G_{i:mτ+1}∥ ≤ ϵτ is not justified and is in fact false in general. The argument relies on the claim that the spectral norm of a block triangular matrix equals the maximum spectral norm of its diagonal blocks; this is true for the spectral radius but not for the spectral norm. For the specific structure G_i = [[bA_i, -bA_i],[0,bA_i]] in (14c), an induction shows that the product over r steps equals [[Q, -rQ],[0,Q]] with Q = bA_i...bA_{i-r+1}. Its spectral norm is ||T_r ⊗ Q|| = ||T_r|| · ||Q||, where T_r = [[1,-r],[0,1]]. For one full period (r = τ), Q = bI(A_τ...A_1) = bI E, and since E in (5) has zero row and column sums, ||Q|| = ||E|| = ϵτ. Hence ∥G_{i:mτ+1}∥ = ||T_τ|| · ϵτ > ϵτ for every τ ≥ 1 (e.g., ||T_2|| ≈ 2.414, ||T_3|| ≈ 3.303). The off-diagonal block -τQ is not negligible and cannot be discarded. Because Lemma 1 is the sole stated basis for the per-period contraction, the exponential decay term in Theorem 1 (Eq. (19a)) and the (1-ϵτ)^-2 denominators in (19a) and (23a) are unsupported. Theorem 2 inherits this gap through its dependence on Theorem 1.
[Sections II-C and II-D (Theorems 1 and 2)] The two main results are not proven in the manuscript. Theorem 1 ends with 'Proof: Omitted due to space limitations', and Theorem 2 is stated without any proof. The constants (432, 144, 1728, 576, β4, β5) and the step-size conditions μ ≤ min{ν/δ², (1/(12δ²))√((1-ϵτ)/(τ(2τ-1)(1+ϵτ)))} are therefore not verifiable. Even if Lemma 1 were correct, a journal submission must either include the proofs or provide them in a separately accessible supplement. The current omission leaves the reader unable to check the chain of inequalities leading to (19a)–(23b), and it is particularly problematic for a paper whose main contribution is exactly these bounds.

minor comments (4)

[Section II-C, first paragraph] The phrase 'The evolution of the consensus error error' contains a duplicated 'error'; it should read 'The evolution of the consensus error'.
[Eq. (5)] The matrix norm ∥·∥ is not defined. The subsequent bounds require the spectral norm; please state this explicitly and use the same norm throughout the paper.
[Theorem 2 (Section II-D)] The quantity γ2 is defined just below (23a) but it is not used in the bound; the expression uses γ3 = max(γ1,γ2). Please remove γ2 or use it consistently.
[General] Given that the proofs are omitted, the paper would benefit from at least a proof sketch for Theorems 1 and 2 to indicate how the stated constants and step-size conditions arise, and in particular how Lemma 1 is intended to be used.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the bounds are expressed in terms of the a priori defined approximation error ϵτ, and the sole self-citation is motivational, not load-bearing.

full rationale

The paper defines ϵτ in Eq. (5) as the norm of Aτ···A1 − (1/K)11^T, an input measure of approximation quality that is independent of the algorithm. Theorems 1 and 2 then derive consensus and centroid error bounds as functions of this ϵτ; they do not fit ϵτ to the results, and the simulations compare against the bounds rather than calibrating them. Lemma 1's contraction claim is a mathematical step invoking a cited block-triangular norm identity; even if the identity were misapplied, that would be a correctness defect, not circularity, since the quantity being bounded is the same a priori norm ϵτ and is not assumed equal to the theorem's conclusion. The only self-citation, [13], is used for empirical motivation ('Empirically, it has been observed in [13] that benefits to the performance of gradient-tracking also extend to the case when the sequence of matrices only approximates the scaled all ones matrix') and is not a premise of any theorem. Theorem 1's proof is omitted, which is a completeness limitation, but an omitted proof is not a circular step. The derivation is therefore self-contained under Assumptions 1-4 and standard stochastic-gradient bounds, with no fitted parameter renamed as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The analysis rests on standard optimization assumptions and one questionable norm identity. No free parameters are fitted to the result; the only tuning occurs in the simulations and does not enter the theorems.

assumptions (5)

domain assumption Assumption 1: the aggregate objective J is ν-strongly convex and local gradients are δ-Lipschitz smooth.
Used to bound the centroid error recursion and the term hi in Lemma 2; standard in decentralized optimization analysis.
domain assumption Assumption 2: the disagreement in gradients between any two agents is uniformly bounded by B.
Allows the consensus error bound to be independent of the optimization error. This is restrictive for highly heterogeneous data.
domain assumption Assumption 3: stochastic gradient noise is unbiased, pairwise uncorrelated, and has bounded variance σ².
Standard noise conditions used to bound the term vi and to obtain the σ² terms in the final bounds.
domain assumption Assumption 4: each combination matrix in the FTC sequence is primitive, doubly stochastic, and has spectral radius 1.
Required for the consensus error contraction over a full cycle of matrices.
standard math The spectral norm of a block triangular matrix equals the maximum spectral norm of its diagonal blocks, invoked in Lemma 1 and attributed to [22].
This identity is not true in general; the off-diagonal block can inflate the norm. The paper provides no additional argument for why it holds for the specific matrices G_i.

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Cite this review

Pith. "Pith review of Decentralized Learning with Approximate Finite-Time Consensus." pith.science (2026). https://pith.science/paper/SH357I6W

@misc{pith2026250107967,
  author       = {Pith},
  title        = {Pith review of: Decentralized Learning with Approximate Finite-Time Consensus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SH357I6W}},
  note         = {Machine review of arXiv:2501.07967}
}

read the original abstract

The performance of algorithms for decentralized optimization is affected by both the optimization error and the consensus error, the latter of which arises from the variation between agents' local models. Classically, algorithms employ averaging and gradient-tracking mechanisms with constant combination matrices to drive the collection of agents to consensus. Recent works have demonstrated that using sequences of combination matrices that achieve finite-time consensus (FTC) can result in improved communication efficiency or iteration complexity for decentralized optimization. Notably, these studies apply to highly structured networks, where exact finite-time consensus sequences are known exactly and in closed form. In this work we investigate the impact of utilizing approximate FTC matrices in decentralized learning algorithms, and quantify the impact of the approximation error on convergence rate and steady-state performance. Approximate FTC matrices can be inferred for general graphs and do not rely on a particular graph structure or prior knowledge, making the proposed scheme applicable to a broad range of decentralized learning settings.

Figures

Figures reproduced from arXiv: 2501.07967 by the authors.

**Figure 2.** Approximate FTC for a logistic regression problem on [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Consensus error for a graph with τ = 4. The error shows a periodic decrease every 4 iterations. Smaller values of ϵτ , corresponding to better approximations of the FTC, lead to larger decreases in the error. the empirical logistic cost in (24) is computed by selecting a random sample 1 ≤ ni ≤ N at each iteration i and evaluating its gradient. The graph used is a path graph with 16 agents and τ = 15. Results are sho… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Performance comparison on graphs where K = 16 but τ ranges in value. consistent with the bound in [12]. Higher values of τ cause agents’ local models to drift from one another because individual combination matrices in the FTC sequence may lack strong connectivity, re…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

A Unified and Refined Convergence Analysis for Non-Convex Decentralized Learning,

S. A. Alghunaim and K. Yuan, “A Unified and Refined Convergence Analysis for Non-Convex Decentralized Learning,” IEEE Transactions on Signal Processing , vol. 70, pp. 3264–3279, 2022

work page 2022
[2]

On the learning behavior of adaptive networks—part i: Transient analysis,

J. Chen and A. H. Sayed, “On the learning behavior of adaptive networks—part i: Transient analysis,” IEEE Transactions on Information Theory, vol. 61, no. 6, pp. 3487–3517, 2015

work page 2015
[3]

An improved convergence analysis for decentralized online stochastic non-convex optimization,

R. Xin, U. A. Khan, and S. Kar, “An improved convergence analysis for decentralized online stochastic non-convex optimization,” IEEE Transactions on Signal Processing , vol. 69, pp. 1842–1858, 2021

work page 2021
[4]

Distributed learning in non-convex en- vironments— part ii: Polynomial escape from saddle-points,

S. Vlaski and A. H. Sayed, “Distributed learning in non-convex en- vironments— part ii: Polynomial escape from saddle-points,” IEEE Transactions on Signal Processing , vol. 69, pp. 1257–1270, 2021

work page 2021
[5]

Networked Signal and Information Processing: Learning by multiagent systems,

S. Vlaski, S. Kar, A. H. Sayed, and J. M. Moura, “Networked Signal and Information Processing: Learning by multiagent systems,” IEEE Signal Processing Magazine, vol. 40, no. 5, p. 92–105, 2023

work page 2023
[6]

Fast linear iterations for distributed averaging,

L. Xiao and S. Boyd, “Fast linear iterations for distributed averaging,” Systems & Control Letters , vol. 53, no. 1, p. 65–78, 2004

work page 2004
[7]

Adaptation, Learning, and Optimization over Networks,

A. H. Sayed, “Adaptation, Learning, and Optimization over Networks,” F oundations and Trends® in Machine Learning , vol. 7, no. 4–5, p. 311–801, 2014

work page 2014
[8]

Scheduling for finite time consensus,

C.-K. Ko and L. Shi, “Scheduling for finite time consensus,” 2009 American Control Conference , p. 1982–1986, 2009

work page 2009

Show all 23 references

[9]

On Matrix Factorization and Finite-time Average-consensus,

C.-K. Ko and X. Gao, “On Matrix Factorization and Finite-time Average-consensus,” Proceedings of the 48h IEEE Conference on De- cision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, p. 5798–5803, 2009

2009
[10]

Graph diameter, eigenvalues, and minimum-time consensus,

J. M. Hendrickx, R. M. Jungers, A. Olshevsky, and G. Vankeerberghen, “Graph diameter, eigenvalues, and minimum-time consensus,” Automat- ica, vol. 50, no. 2, p. 635–640, 2014

2014
[11]

Finite-time conver- gent gossiping,

G. Shi, B. Li, M. Johansson, and K. H. Johansson, “Finite-time conver- gent gossiping,” IEEE/ACM Transactions on Networking , vol. 24, no. 5, p. 2782–2794, 2016

2016
[12]

On Graphs with Finite-Time Consensus and Their Use in Gradient Tracking,

E. D. H. Nguyen, X. Jiang, B. Ying, and C. A. Uribe, “On Graphs with Finite-Time Consensus and Their Use in Gradient Tracking,” arXiv:2311.01317, 2023

2023 arXiv
[13]

Learned finite-time consensus for distributed optimization,

A. Fainman and S. Vlaski, “Learned finite-time consensus for distributed optimization,” in 32nd European Signal Processing Conference, EU- SIPCO 2024, Lyon, France, August 26-30, 2024 . IEEE, 2024, pp. 1047–1051

2024
[14]

Exponential graph is provably efficient for decentralized deep training,

B. Ying, K. Yuan, Y . Chen, H. Hu, P. Pan, and W. Yin, “Exponential graph is provably efficient for decentralized deep training,” arXiv, 2021

2021
[15]

Optimal strategies in the average consensus problem,

J.-C. Delvenne, R. Carli, and S. Zampieri, “Optimal strategies in the average consensus problem,” Systems & Control Letters , vol. 58, no. 10, pp. 759–765, 2009

2009
[16]

Finite-time average consensus based protocol for dis- tributed estimation over awgn channels,

A. Y . Kibangou, “Finite-time average consensus based protocol for dis- tributed estimation over awgn channels,” in 2011 50th IEEE Conference on Decision and Control and European Control Conference , 2011, pp. 5595–5600

2011
[17]

Revisiting finite-time distributed algorithms via successive nulling of eigenvalues,

S. Safavi and U. A. Khan, “Revisiting finite-time distributed algorithms via successive nulling of eigenvalues,” IEEE Signal Processing Letters , vol. 22, no. 1, pp. 54–57, 2015

2015
[18]

Finite-time distributed consensus through graph filters,

A. Sandryhaila, S. Kar, and J. M. F. Moura, “Finite-time distributed consensus through graph filters,” in 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) , 2014, pp. 1080– 1084

2014
[19]

A cascaded structure for generalized graph filters,

M. Coutino and G. Leus, “A cascaded structure for generalized graph filters,” IEEE Transactions on Signal Processing , vol. 70, p. 3499–3513, 2022

2022
[20]

Distributed implementation of linear network operators using graph filters,

S. Segarra, A. G. Marques, and A. Ribeiro, “Distributed implementation of linear network operators using graph filters,” in 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Aller- ton), 2015, pp. 1406–1413

2015
[21]

Augmented distributed gradient methods for multi-agent optimization under uncoordinated constant stepsizes,

J. Xu, S. Zhu, Y . C. Soh, and L. Xie, “Augmented distributed gradient methods for multi-agent optimization under uncoordinated constant stepsizes,” 2015 54th IEEE Conference on Decision and Control (CDC) , p. 2055–2060, 2015

2015
[22]

Spectral properties of matrices,

D. A. Simovici and C. Djeraba, “Spectral properties of matrices,” in Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics. London: Springer London, 2014, p. 347. [Online]. Available: https://doi.org/10.1007/978-1-4471-6407-4 7

2014 doi
[23]

A unified theory of decentralized sgd with changing topology and local updates,

A. Koloskova, N. Loizou, S. Boreiri, M. Jaggi, and S. U. Stich, “A unified theory of decentralized sgd with changing topology and local updates,” in Proceedings of the 37th International Conference on Machine Learning , Jul. 2020

2020

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

A Unified and Refined Convergence Analysis for Non-Convex Decentralized Learning,

S. A. Alghunaim and K. Yuan, “A Unified and Refined Convergence Analysis for Non-Convex Decentralized Learning,” IEEE Transactions on Signal Processing , vol. 70, pp. 3264–3279, 2022

work page 2022

[2] [2]

On the learning behavior of adaptive networks—part i: Transient analysis,

J. Chen and A. H. Sayed, “On the learning behavior of adaptive networks—part i: Transient analysis,” IEEE Transactions on Information Theory, vol. 61, no. 6, pp. 3487–3517, 2015

work page 2015

[3] [3]

An improved convergence analysis for decentralized online stochastic non-convex optimization,

R. Xin, U. A. Khan, and S. Kar, “An improved convergence analysis for decentralized online stochastic non-convex optimization,” IEEE Transactions on Signal Processing , vol. 69, pp. 1842–1858, 2021

work page 2021

[4] [4]

Distributed learning in non-convex en- vironments— part ii: Polynomial escape from saddle-points,

S. Vlaski and A. H. Sayed, “Distributed learning in non-convex en- vironments— part ii: Polynomial escape from saddle-points,” IEEE Transactions on Signal Processing , vol. 69, pp. 1257–1270, 2021

work page 2021

[5] [5]

Networked Signal and Information Processing: Learning by multiagent systems,

S. Vlaski, S. Kar, A. H. Sayed, and J. M. Moura, “Networked Signal and Information Processing: Learning by multiagent systems,” IEEE Signal Processing Magazine, vol. 40, no. 5, p. 92–105, 2023

work page 2023

[6] [6]

Fast linear iterations for distributed averaging,

L. Xiao and S. Boyd, “Fast linear iterations for distributed averaging,” Systems & Control Letters , vol. 53, no. 1, p. 65–78, 2004

work page 2004

[7] [7]

Adaptation, Learning, and Optimization over Networks,

A. H. Sayed, “Adaptation, Learning, and Optimization over Networks,” F oundations and Trends® in Machine Learning , vol. 7, no. 4–5, p. 311–801, 2014

work page 2014

[8] [8]

Scheduling for finite time consensus,

C.-K. Ko and L. Shi, “Scheduling for finite time consensus,” 2009 American Control Conference , p. 1982–1986, 2009

work page 2009