Pith. sign in

REVIEW 3 major objections 2 minor 55 references

Towards Serverless Semi-Decentralized Federated Learning with Heterogeneous Optimizers

T0 review · 3 major / 2 minor · reviewed 2026-06-28 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read SSD-FL forms clusters in one lightweight D2D phase then runs fully serverless decentralized training by splitting rounds into intra- and inter-cluster regimes with effective loss functions.

desk verdict SSD-FL adds a serverless semi-decentralized structure and Cheeger-driven clustering with a heterogeneity metric to handle optimizer differences in decentralized FL, but the convergence claims rest on effective loss functions whose coupling to heterogeneous dynamics is not clearly secured by the given description. read the letter →

arxiv 2606.06687 v1 pith:XCSTWLKR submitted 2026-06-04 cs.LG cs.DCcs.NIcs.SYeess.SY

Su Wang , Mung Chiang , H. Vincent Poor This is my paper
classification cs.LGcs.DCcs.NIcs.SYeess.SY
keywords serverlessfederatedlearningsemi-decentralizedFLheterogeneousoptimizersclusterformationCheegerinequalityeffectivelossfunctionsdecentralizedmachineD2Dinitialization
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

The paper introduces SSD-FL to handle decentralized federated learning when devices use different local optimizers and no central server stays available. Cluster formation happens once via device-to-device communication, after which training proceeds without infrastructure by dividing each global round into intra-cluster and inter-cluster stages. Novel effective loss functions combine each device's optimizer with graph-based regularization to drive both local progress and global consensus. An iterative clustering procedure built on the Cheeger inequality selects the number and makeup of clusters using a heterogeneity score. Experiments across graphs, datasets, and optimizer mixes show faster convergence and lower communication cost than prior decentralized methods.

What carries the argument

effective loss functions that integrate device-specific ML optimizers with network graph-based regularization, used to segment training into intra-cluster and inter-cluster regimes

What would settle it

A run on a ring graph with three distinct local optimizers where the devices fail to reach model consensus after the prescribed number of rounds would falsify the convergence claim.

Watch

Extended reading notes

Core claim

SSD-FL segments global rounds into intra-cluster and inter-cluster regimes, ensuring global convergence and consensus through novel effective loss functions that integrate device-specific ML optimizers with network graph-based regularization, and leverages the consensus gap via the Cheeger inequality to develop an iterative clustering algorithm evaluated against derived convergence and consensus bounds that incorporate a scoring metric for data and optimizer heterogeneity.

Load-bearing premise

The novel effective loss functions that integrate device-specific ML optimizers with network graph-based regularization actually ensure global convergence and consensus.

Editorial extensions

If this is right

  • Cluster formation via one-time D2D initialization removes the need for persistent server infrastructure in decentralized FL.
  • The Cheeger-inequality-based clustering algorithm produces partitions that respect both data heterogeneity and optimizer differences.
  • Experimental results show faster convergence and reduced communication volume compared with three categories of existing decentralized FL approaches.
  • The heterogeneity scoring metric quantifies the combined effect of data and optimizer variation across devices.

Reading between the lines

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

  • The same segmentation idea might extend to dynamic reclustering when device participation changes over time.
  • The approach could reduce the communication burden in edge networks where devices already form natural geographic clusters.
  • If the effective loss construction generalizes, similar regularization terms might improve other distributed optimization settings that mix local solvers.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 2 minor

Summary. The paper proposes SSD-FL, a serverless semi-decentralized federated learning method for decentralized settings with heterogeneous ML optimizers. Cluster formation occurs via a one-time D2D initialization phase; training then proceeds serverlessly by segmenting global rounds into intra-cluster and inter-cluster regimes. Novel effective loss functions integrate device-specific optimizers with network graph regularization to ensure global convergence and consensus. An iterative clustering algorithm is derived using the consensus gap and Cheeger inequality, evaluated against convergence/consensus bounds that incorporate a heterogeneity scoring metric. Experiments across network graphs, datasets, and optimizer regimes claim improvements in convergence speed and communication efficiency over three categories of decentralized FL baselines.

Significance. If the effective loss functions and derived bounds are valid, the work would offer a practical serverless alternative to centralized or fully decentralized FL under optimizer heterogeneity, with potential gains in scalability and efficiency. The use of Cheeger inequality for clustering and the heterogeneity metric are technically interesting extensions, and the experimental scope (multiple graphs/datasets/optimizers) strengthens the practical case if the theory holds.

major comments (3)
  1. [Abstract (and the section deriving effective loss functions)] The central claim depends on the novel effective loss functions (combining per-device optimizer dynamics with graph regularization) producing the stated global convergence and consensus. The abstract and description provide no explicit derivation or proof sketch showing that the composite losses yield the claimed behavior for arbitrary optimizer heterogeneity; this coupling is load-bearing for both the bounds and the experimental improvements.
  2. [Abstract (and the section on bounds and clustering algorithm)] The convergence and consensus bounds incorporate a unique scoring metric for data/optimizer heterogeneity and are used to evaluate the clustering algorithm. Without the explicit form of the bounds or the metric (and confirmation that they are not circular with the clustering objective), it is unclear whether the experimental validation against the bounds is independent or tautological.
  3. [Abstract (experimental evaluation)] The experimental claims of improved convergence speeds and communication efficiency rest on the theoretical guarantees. If the effective loss functions do not ensure the bounds under heterogeneous optimizers, the reported gains versus the three categories of baselines cannot be attributed to the proposed segmentation into intra- and inter-cluster regimes.
minor comments (2)
  1. [Abstract] The abstract states that bounds are derived but contains no equations, proof sketches, dataset details, or error bars; the full manuscript should include these for reproducibility.
  2. [Throughout] Notation for the effective loss functions, heterogeneity score, and intra-/inter-cluster regimes should be defined clearly with explicit equations before the experimental section.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the constructive feedback on our manuscript. We address each major comment below with clarifications from the full paper and commit to revisions that enhance the explicitness of our theoretical derivations without altering the core contributions.

read point-by-point responses

  1. Referee: [Abstract (and the section deriving effective loss functions)] The central claim depends on the novel effective loss functions (combining per-device optimizer dynamics with graph regularization) producing the stated global convergence and consensus. The abstract and description provide no explicit derivation or proof sketch showing that the composite losses yield the claimed behavior for arbitrary optimizer heterogeneity; this coupling is load-bearing for both the bounds and the experimental improvements.

    Authors: Section 3.2 of the full manuscript derives the effective loss functions by integrating device-specific optimizer dynamics with graph-based regularization. A proof sketch is included using a Lyapunov function that accounts for heterogeneity to establish global convergence and consensus. To improve accessibility, we will expand this section with a more detailed proof sketch and explicit steps in the revised manuscript. revision: yes

  2. Referee: [Abstract (and the section on bounds and clustering algorithm)] The convergence and consensus bounds incorporate a unique scoring metric for data/optimizer heterogeneity and are used to evaluate the clustering algorithm. Without the explicit form of the bounds or the metric (and confirmation that they are not circular with the clustering objective), it is unclear whether the experimental validation against the bounds is independent or tautological.

    Authors: The explicit forms of the bounds and heterogeneity scoring metric appear in Section 4. The clustering algorithm is derived independently via the consensus gap and Cheeger inequality; the bounds are applied afterward for evaluation. We will revise to include the explicit mathematical expressions and a dedicated paragraph confirming the independence to eliminate any ambiguity regarding circularity. revision: yes

  3. Referee: [Abstract (experimental evaluation)] The experimental claims of improved convergence speeds and communication efficiency rest on the theoretical guarantees. If the effective loss functions do not ensure the bounds under heterogeneous optimizers, the reported gains versus the three categories of baselines cannot be attributed to the proposed segmentation into intra- and inter-cluster regimes.

    Authors: The experiments validate the framework under the derived effective loss functions. We will add an explicit discussion in the experimental evaluation section linking the observed gains in convergence and communication efficiency to the intra-/inter-cluster segmentation enabled by the loss functions and clustering, thereby strengthening the attribution to the proposed regimes. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation relies on external inequalities and independent experiments

full rationale

The abstract describes novel effective loss functions integrating optimizers with graph regularization, then applies the Cheeger inequality (a standard external result) to derive clustering and bounds that include a heterogeneity scoring metric. No equations or self-citations are provided that reduce the bounds, metric, or convergence claims to a fit or definition from the clustering outputs themselves. Experimental validation is against three external categories of decentralized FL methods across graphs/datasets/optimizers, providing independent content. The central claims do not reduce by construction to inputs; this is the common honest non-finding.

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

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted. The effective loss functions and scoring metric appear to be introduced by the paper but their definitions and grounding are not provided.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Towards Serverless Semi-Decentralized Federated Learning with Heterogeneous Optimizers." pith.science (2026). https://pith.science/paper/XCSTWLKR

@misc{pith2026260606687,
  author       = {Pith},
  title        = {Pith review of: Towards Serverless Semi-Decentralized Federated Learning with Heterogeneous Optimizers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XCSTWLKR}},
  note         = {Machine review of arXiv:2606.06687}
}
read the original abstract

We investigate cluster formation, involving the number and composition of clusters, in decentralized federated learning (FL) with heterogeneous machine learning (ML) optimizers. While clustering in centralized FL has enabled scalability and resource savings, its value and development in fully decentralized environments have yet to be explored. Optimizing cluster formation in such environments is challenging, especially due to the complex coupling between network graph structures, local data heterogeneity, and different local ML model optimizers. To address these challenges, we propose serverless semi-decentralized FL (SSD-FL), a methodology requiring no persistent server infrastructure. In SSD-FL, cluster formation occurs via a lightweight, one-time device-to-device (D2D) initialization phase, after which actual ML model training (alongside consensus and convergence processes) is fully serverless. Functionally, SSD-FL segments global rounds into intra-cluster and inter-cluster regimes, ensuring global convergence and consensus through novel "effective loss functions" that integrate device-specific ML optimizers with network graph-based regularization. Next, SSD-FL leverages the consensus gap via the Cheeger inequality to develop an iterative clustering algorithm evaluated against our derived convergence and consensus bounds, which incorporate a unique scoring metric to quantify data and optimizer heterogeneity across devices. Finally, experimental evaluation against three categories of decentralized FL methodologies validate that SSD-FL improves both convergence speeds and communication efficiency across various network graphs, datasets, and local optimizer regimes.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

55 extracted references · 3 canonical work pages

  1. [1]

    A robust privacy-preserving federated learning model against model poisoning attacks,

    A. Yazdinejad, A. Dehghantanha, H. Karimipour, G. Srivastava, and R. M. Parizi, “A robust privacy-preserving federated learning model against model poisoning attacks,”IEEE Transactions on Information Forensics and Security, vol. 19, no. 1, pp. 6693–6708, 2024

    2024

  2. [2]

    Decentralized federated learning: A survey on security and privacy,

    E. Hallaji, R. Razavi-Far, M. Saif, B. Wang, and Q. Yang, “Decentralized federated learning: A survey on security and privacy,”IEEE Transactions on Big Data, vol. 10, no. 2, pp. 194–213, 2024

    2024

  3. [3]

    Device sampling and resource optimization for federated learning in cooperative edge networks,

    S. Wang, R. Morabito, S. Hosseinalipour, M. Chiang, and C. G. Brinton, “Device sampling and resource optimization for federated learning in cooperative edge networks,”IEEE/ACM Transactions on Networking, vol. 32, no. 5, pp. 4365 – 4381, 2024

    2024

  4. [4]

    A review of federated learning methods in heterogeneous scenarios,

    J. Pei, W. Liu, J. Li, L. Wang, and C. Liu, “A review of federated learning methods in heterogeneous scenarios,”IEEE Transactions on Consumer Electronics, vol. 70, no. 3, pp. 5983–5999, 2024

    2024

  5. [5]

    Toward cooperative federated learning over heterogeneous edge/fog networks,

    S. Wang, S. Hosseinalipour, V . Aggarwal, C. G. Brinton, D. J. Love, W. Su, and M. Chiang, “Toward cooperative federated learning over heterogeneous edge/fog networks,”IEEE Communications Magazine, vol. 61, no. 12, pp. 54–60, 2023

    2023

  6. [6]

    Adaptive federated learning in resource constrained edge com- puting systems,

    S. Wang, T. Tuor, T. Salonidis, K. K. Leung, C. Makaya, T. He, and K. Chan, “Adaptive federated learning in resource constrained edge com- puting systems,”IEEE Journal on Selected Areas in Communications, vol. 37, no. 6, pp. 1205–1221, 2019

    2019

  7. [7]

    Decentralized federated learning: A survey and perspective,

    L. Yuan, Z. Wang, L. Sun, S. Y . Philip, and C. G. Brinton, “Decentralized federated learning: A survey and perspective,”IEEE Internet of Things Journal, vol. 11, no. 21, pp. 34 617 – 34 638, 2024

    2024

  8. [8]

    Semi-decentralized federated learning with collaborative relaying,

    M. Yemini, R. Saha, E. Ozfatura, D. G ¨und¨uz, and A. J. Goldsmith, “Semi-decentralized federated learning with collaborative relaying,” in Proceedings of the 2022 IEEE International Symposium on Information Theory. IEEE, 2022, pp. 1471–1476

    2022

Show all 55 references
  1. [9]

    Semi-decentralized federated learning with cooperative d2d local model aggregations,

    F. P.-C. Lin, S. Hosseinalipour, S. S. Azam, C. G. Brinton, and N. Michelusi, “Semi-decentralized federated learning with cooperative d2d local model aggregations,”IEEE Journal on Selected Areas in Communications, vol. 39, no. 12, pp. 3851–3869, 2021

    2021

  2. [10]

    Sdflmq: A semi-decentralized federated learning framework over mqtt,

    A. Ali-Pour and J. Gascon-Samson, “Sdflmq: A semi-decentralized federated learning framework over mqtt,” inProceedings of the 2025 IEEE International Parallel and Distributed Processing Symposium Workshops. IEEE, 2025, pp. 1100–1107

    2025

  3. [11]

    A novel framework for the analysis and design of heterogeneous federated learning,

    J. Wang, Q. Liu, H. Liang, G. Joshi, and H. V . Poor, “A novel framework for the analysis and design of heterogeneous federated learning,”IEEE Transactions on Signal Processing, vol. 69, pp. 5234–5249, 2021

    2021

  4. [12]

    Accelerating decentralized federated learning with probabilistic com- munication in heterogeneous edge computing,

    J. Liu, J. Yan, H. Xu, L. Wang, Z. Wang, J. Huang, and C. Qiao, “Accelerating decentralized federated learning with probabilistic com- munication in heterogeneous edge computing,”IEEE Transactions on Networking, 2025, to appear

    2025

  5. [13]

    Gossipfl: A decentralized federated learning framework with sparsified and adaptive communication,

    Z. Tang, S. Shi, B. Li, and X. Chu, “Gossipfl: A decentralized federated learning framework with sparsified and adaptive communication,”IEEE Transactions on Parallel and Distributed Systems, vol. 34, no. 3, pp. 909–922, 2022

    2022

  6. [14]

    Decentralized federated learning with gradient tracking over time- varying directed networks,

    D. T. A. Nguyen, S. Wang, D. T. Nguyen, A. Nedich, and H. V . Poor, “Decentralized federated learning with gradient tracking over time- varying directed networks,” arXiv:2409.17189, 2024

    2024

  7. [15]

    Peer-to- peer trading in electricity networks: An overview,

    W. Tushar, T. K. Saha, C. Yuen, D. Smith, and H. V . Poor, “Peer-to- peer trading in electricity networks: An overview,”IEEE Transactions on Smart Grid, vol. 11, no. 4, pp. 3185–3200, 2020

    2020

  8. [16]

    Peer to peer distributed solar energy trading,

    Q. Li and D. Chen, “Peer to peer distributed solar energy trading,”ACM SIGMETRICS Performance Evaluation Review, vol. 50, no. 4, pp. 44– 46, 2023. 14

    2023

  9. [17]

    An ad-hoc sensor network for disaster relief operations,

    N. Pogkas, G. Karastergios, C. Antonopoulos, S. Koubias, and G. Pa- padopoulos, “An ad-hoc sensor network for disaster relief operations,” inProceedings of the 2005 IEEE Conference on Emerging Technologies and Factory Automation. IEEE, 2005, pp. 131–139

    2005

  10. [18]

    Information-centric robotic ad hoc networking based continuous data routing and delivery for disaster scenes,

    X. Wang and Y . Lu, “Information-centric robotic ad hoc networking based continuous data routing and delivery for disaster scenes,”IEEE Transactions on Green Communications and Networking, vol. 9, no. 3, pp. 768–777, 2024

    2024

  11. [19]

    Joint multi-domain resource allocation and trajectory optimization in uav-assisted maritime iot networks,

    L. P. Qian, H. Zhang, Q. Wang, Y . Wu, and B. Lin, “Joint multi-domain resource allocation and trajectory optimization in uav-assisted maritime iot networks,”IEEE Internet of Things Journal, vol. 10, no. 1, pp. 539– 552, 2022

    2022

  12. [20]

    Fmd-iov: Security and robust enhancement for federated multi-domain learning–based iov,

    C. Zhang, G. Shan, and B.-H. Roh, “Fmd-iov: Security and robust enhancement for federated multi-domain learning–based iov,”IEEE Transactions on Intelligent Transportation Systems, vol. 26, no. 9, pp. 14 225–14 236, 2025

    2025

  13. [21]

    Coded cooperative networks for semi-decentralized federated learning,

    S. Weng, M. Xiao, C. Ren, and M. Skoglund, “Coded cooperative networks for semi-decentralized federated learning,”IEEE Wireless Communications Letters, vol. 14, no. 3, pp. 626–630, 2024

    2024

  14. [22]

    Decentralized federated averaging,

    T. Sun, D. Li, and B. Wang, “Decentralized federated averaging,”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 45, no. 4, pp. 4289–4301, 2022

    2022

  15. [23]

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

    A. Koloskova, N. Loizou, S. Boreiri, M. Jaggi, and S. Stich, “A unified theory of decentralized sgd with changing topology and local updates,” inProceedings of the 2020 International conference on machine learn- ing. PMLR, 2020, pp. 5381–5393

    2020

  16. [24]

    Hierarchical federated learning with quantization: Convergence analysis and system design,

    L. Liu, J. Zhang, S. Song, and K. B. Letaief, “Hierarchical federated learning with quantization: Convergence analysis and system design,” IEEE Transactions on Wireless Communications, vol. 22, no. 1, pp. 2– 18, 2022

    2022

  17. [25]

    Hierarchical federated learning for social network with mobil- ity,

    Z. Chen, W. Chen, J. Li, Q. Wu, M. Ding, X. Han, X. Deng, and L. Wang, “Hierarchical federated learning for social network with mobil- ity,”IEEE Transactions on Cognitive Communications and Networking, 2025, to appear

    2025

  18. [26]

    Uav-assisted online machine learning over multi-tiered networks: A hierarchical nested personalized federated learning approach,

    S. Wang, S. Hosseinalipour, M. Gorlatova, C. G. Brinton, and M. Chi- ang, “Uav-assisted online machine learning over multi-tiered networks: A hierarchical nested personalized federated learning approach,”IEEE Transactions on Network and Service Management, vol. 20, no. 2, pp. ...

    2022

  19. [27]

    Hierarchical federated learning in multi-hop cluster-based vanets,

    M. S. HaghighiFard and S. Coleri, “Hierarchical federated learning in multi-hop cluster-based vanets,”IEEE Transactions on Vehicular Technology, 2025, to appear

    2025

  20. [28]

    Semi-decentralized federated edge learning for fast convergence on non-iid data,

    Y . Sun, J. Shao, Y . Mao, J. H. Wang, and J. Zhang, “Semi-decentralized federated edge learning for fast convergence on non-iid data,” inPro- ceedings of the 2022 IEEE Wireless Communications and Networking Conference (WCNC). IEEE, 2022, pp. 1898–1903

    2022

  21. [29]

    Accelerating federated learning with cluster construction and hierarchical aggrega- tion,

    Z. Wang, H. Xu, J. Liu, Y . Xu, H. Huang, and Y . Zhao, “Accelerating federated learning with cluster construction and hierarchical aggrega- tion,”IEEE Transactions on Mobile Computing, vol. 22, no. 7, pp. 3805–3822, 2022

    2022

  22. [30]

    Towards hierarchical clustered federated learning with model stability on mobile devices,

    B. Gong, T. Xing, Z. Liu, W. Xi, and X. Chen, “Towards hierarchical clustered federated learning with model stability on mobile devices,” IEEE Transactions on Mobile Computing, vol. 23, no. 6, pp. 7148–7164, 2023

    2023

  23. [31]

    Distributed optimization over time-varying directed graphs,

    A. Nedi ´c and A. Olshevsky, “Distributed optimization over time-varying directed graphs,”IEEE Transactions on Automatic Control, vol. 60, no. 3, pp. 601–615, 2014

    2014

  24. [32]

    Quantized decentralized stochastic learning over directed graphs,

    H. Taheri, A. Mokhtari, H. Hassani, and R. Pedarsani, “Quantized decentralized stochastic learning over directed graphs,” inProceedings of the 2020 International Conference on Machine Learning. PMLR, 2020, pp. 9324–9333

    2020

  25. [33]

    Matcha: A matching-based link scheduling strategy to speed up distributed optimization,

    J. Wang, A. K. Sahu, G. Joshi, and S. Kar, “Matcha: A matching-based link scheduling strategy to speed up distributed optimization,”IEEE Transactions on Signal Processing, vol. 70, pp. 5208–5221, 2022

    2022

  26. [34]

    Decentralized federated learning: Balancing communication and computing costs,

    W. Liu, L. Chen, and W. Zhang, “Decentralized federated learning: Balancing communication and computing costs,”IEEE Transactions on Signal and Information Processing over Networks, vol. 8, pp. 131–143, 2022

    2022

  27. [35]

    On the benefits of multiple gossip steps in communication-constrained decentralized federated learning,

    A. Hashemi, A. Acharya, R. Das, H. Vikalo, S. Sanghavi, and I. Dhillon, “On the benefits of multiple gossip steps in communication-constrained decentralized federated learning,”IEEE Transactions on Parallel and Distributed Systems, vol. 33, no. 11, pp. 2727–2739, 2021

    2021

  28. [36]

    Distributed subgradient methods for multi- agent optimization,

    A. Nedic and A. Ozdaglar, “Distributed subgradient methods for multi- agent optimization,”IEEE Transactions on Automatic Control, vol. 54, no. 1, pp. 48–61, 2009

    2009

  29. [37]

    Decentralized sporadic federated learning: A unified algorithmic framework with convergence guarantees,

    S. Zehtabi, D.-J. Han, R. Parasnis, S. Hosseinalipour, and C. G. Brin- ton, “Decentralized sporadic federated learning: A unified algorithmic framework with convergence guarantees,” arXiv:2402.03448, 2024

    2024

  30. [38]

    Extra: An exact first-order algorithm for decentralized consensus optimization,

    W. Shi, Q. Ling, G. Wu, and W. Yin, “Extra: An exact first-order algorithm for decentralized consensus optimization,”SIAM Journal on Optimization, vol. 25, no. 2, pp. 944–966, 2015

    2015

  31. [39]

    Swift: Rapid decentralized federated learning via wait-free model communi- cation,

    M. Bornstein, T. Rabbani, E. Z. Wang, A. Bedi, and F. Huang, “Swift: Rapid decentralized federated learning via wait-free model communi- cation,” inProceedings of the Eleventh International Conference on Learning Representations, 2023, pp. 1–30

    2023

  32. [40]

    Collaborative deep learning in fixed topology networks,

    Z. Jiang, A. Balu, C. Hegde, and S. Sarkar, “Collaborative deep learning in fixed topology networks,”Advances in Neural Information Processing Systems, vol. 30, pp. 3322–3330, 2017

    2017

  33. [41]

    On the complexity of finding stationary points of smooth functions in one dimension,

    S. Chewi, S. Bubeck, and A. Salim, “On the complexity of finding stationary points of smooth functions in one dimension,” inProceedings of the 34th International Conference on Algorithmic Learning Theory. PMLR, 2023, pp. 358–374

    2023

  34. [42]

    A proximal stochastic gradient method with progressive variance reduction,

    L. Xiao and T. Zhang, “A proximal stochastic gradient method with progressive variance reduction,”SIAM Journal on Optimization, vol. 24, no. 4, pp. 2057–2075, 2014

  35. [43]

    On the importance of initialization and momentum in deep learning,

    I. Sutskever, J. Martens, G. Dahl, and G. Hinton, “On the importance of initialization and momentum in deep learning,” inProceedings of the 30th International Conference on Machine Learning. PMLR, 2013, pp. 1139–1147

    2013

  36. [44]

    The perron-frobenius theorem: some of its applications,

    S. U. Pillai, T. Suel, and S. Cha, “The perron-frobenius theorem: some of its applications,”IEEE Signal Processing Magazine, vol. 22, no. 2, pp. 62–75, 2005

    2005

  37. [45]

    F. R. K. Chung,Spectral Graph Theory, ser. CBMS Regional Conference Series in Mathematics. Providence, RI: American Mathematical Society, 1997, vol. 92

    1997

  38. [46]

    Jensen–shannon divergence and hilbert space embedding,

    B. Fuglede and F. Topsøe, “Jensen–shannon divergence and hilbert space embedding,” inProceedings of the 2004 IEEE International Symposium on Information Theory. IEEE, 2004, p. 31

    2004

  39. [47]

    Energy distance,

    M. L. Rizzo and G. J. Sz ´ekely, “Energy distance,”Wiley Interdisciplinary Reviews: Computational Statistics, vol. 8, no. 1, pp. 27–38, 2016

    2016

  40. [48]

    Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms,

    H. Xiao, K. Rasul, and R. V ollgraf, “Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms,” arXiv:1708.07747, 2017

    2017 arXiv

  41. [49]

    Learning multiple layers of features from tiny images,

    A. Krizhevsky, “Learning multiple layers of features from tiny images,” University of Toronto, Tech. Rep., 2009, technical Report

    2009

  42. [50]

    Random graphs,

    E. N. Gilbert, “Random graphs,”The Annals of Mathematical Statistics, vol. 30, no. 4, pp. 1141–1144, 1959

    1959

  43. [51]

    Distributed average consensus with time-varying metropolis weights,

    L. Xiao, S. Boyd, and S. Lall, “Distributed average consensus with time-varying metropolis weights,”Automatica, vol. 41, no. 12, pp. 1895– 1906, 2005

    1906

  44. [52]

    Emergence of scaling in random net- works,

    A.-L. Barab ´asi and R. Albert, “Emergence of scaling in random net- works,”Science, vol. 286, no. 5439, pp. 509–512, 1999

    1999

  45. [53]

    M. D. Penrose,Random Geometric Graphs, ser. Oxford Studies in Probability. Oxford, UK: Oxford University Press, 2003, vol. 5

    2003

  46. [54]

    Collective dynamics of ‘small- world’networks,

    D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small- world’networks,”Nature, vol. 393, no. 6684, pp. 440–442, 1998

    1998

  47. [55]

    Some remarks on the theory of graphs,

    P. Erd ˝os, “Some remarks on the theory of graphs,”Bulletin of the American Mathematical Society, vol. 53, no. 4, pp. 292–294, 1947. 15 APPENDIX TABLE OFCONTENTS Appendix A: Proof of Proposition 1 16 Appendix B: Proof of Corollary 1 18 Appendix C: Proof of Theorem 1 19 Appendi...

    1947

Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.