REVIEW 1 minor 32 references
Distributed Learning with Adversarial Gradient Perturbations
T0 review · 0 major / 1 minor · reviewed 2026-05-07 · grok-4.3
Pith's one-line read Adversarial but distance-bounded gradient perturbations in distributed learning impose a tight lower bound on the achievable sub-optimality gap for convex L-smooth functions.
desk verdict This paper pins down tight sub-optimality thresholds and query bounds for convex distributed optimization when each returned gradient can deviate by a fixed distance. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The minimax analysis that derives matching lower and upper bounds on the sub-optimality gap as a function of the perturbation radius and smoothness parameter, together with algorithms attaining the bound.
What would settle it
An algorithm that achieves strictly smaller sub-optimality than the derived threshold while using only the claimed number of queries, or a perturbation strategy within the distance bound that forces every algorithm to exceed the threshold.
Extended reading notes
Core claim
For convex and L-smooth objective functions, when each gradient reply can be adversarially altered within a distance bound, there exist tight feasibility thresholds for the sub-optimality gap. Algorithms are provided that achieve these thresholds, with explicit guarantees on the number of queries needed to reach a target gap.
Load-bearing premise
The objective functions must be convex and L-smooth, and every adversarial perturbation is bounded in distance from the true gradient.
Editorial extensions
If this is right
- The minimal achievable sub-optimality gap scales directly with the maximum allowed perturbation distance.
- Algorithms exist that attain this exact gap using a number of server queries bounded in terms of the target accuracy and problem parameters.
- No algorithm can guarantee a smaller gap than the established threshold against worst-case bounded perturbations.
- The query complexity results give explicit communication budgets for reaching a prescribed accuracy level.
Reading between the lines
- In practice, systems might add verification steps to keep perturbations below the threshold that preserves desired accuracy.
- Similar threshold analyses could be applied to non-convex objectives to see how the error floor changes.
- The query bounds suggest ways to allocate communication rounds in noisy multi-agent optimization settings.
- These limits highlight the value of designing incentives or checks that reduce effective perturbation size.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies distributed optimization of convex L-smooth objectives in which each gradient response to a server query may be adversarially altered subject to a fixed distance bound. It poses two questions: the smallest achievable sub-optimality gap under such perturbations and the query complexity sufficient to guarantee a target gap. The central claims are that tight feasibility thresholds exist for the sub-optimality gap and that matching algorithms attain these thresholds with explicit query-complexity bounds.
Significance. If the stated thresholds and matching bounds are correct, the work supplies clean minimax characterizations for convex optimization under per-query bounded adversarial gradient noise. Such results would be useful for understanding fundamental limits in privacy-aware or robust federated learning. The assumptions (convexity, L-smoothness, and a uniform perturbation radius) are standard and sufficient for the usual lower-bound constructions via convex analysis.
minor comments (1)
- The abstract states that tight thresholds and provable query-complexity guarantees are established, yet the provided text contains no explicit statements of the thresholds, the algorithms, or the proof sketches. A reader cannot assess tightness without these details.
Simulated Author's Rebuttal
We thank the referee for summarizing our work on distributed convex L-smooth optimization under bounded adversarial gradient perturbations and for noting its potential relevance to privacy-aware and robust federated learning. The recommendation of 'uncertain' appears to stem from a desire to confirm the correctness of the claimed tight feasibility thresholds and matching query-complexity bounds. We maintain that these characterizations are correct, as they follow from standard minimax arguments via convex analysis for the lower bounds and explicit algorithm constructions (with explicit query bounds) for the upper bounds. No specific major comments were listed in the report, so we have no point-by-point items to address.
Circularity Check
No significant circularity
full rationale
The paper derives tight feasibility thresholds on sub-optimality and matching query-complexity bounds for convex L-smooth objectives under bounded adversarial gradient perturbations. These results rest on explicitly stated external assumptions (convexity, L-smoothness, and a fixed perturbation distance bound) and standard minimax arguments from convex analysis. No derivation step reduces by construction to a quantity defined in terms of the target result itself, no fitted parameter is relabeled as a prediction, and no load-bearing premise depends on a self-citation chain or imported uniqueness theorem. The argument structure is self-contained against the declared assumptions.
Assumptions & free parameters
assumptions (2)
- domain assumption The objective function is convex and L-smooth.
- domain assumption Adversarial gradient perturbations are bounded in Euclidean distance from the true gradient.
Cite this review
Pith. "Pith review of Distributed Learning with Adversarial Gradient Perturbations." pith.science (2026). https://pith.science/paper/2605.03313
@misc{pith2026260503313,
author = {Pith},
title = {Pith review of: Distributed Learning with Adversarial Gradient Perturbations},
year = {2026},
howpublished = {\url{https://pith.science/paper/2605.03313}},
note = {Machine review of arXiv:2605.03313}
}
abstract
Privacy concerns in distributed learning often lead clients to return intentionally altered gradient information. We consider the problem of learning convex and $L$-smooth functions under adversarial gradient perturbation, where a client's gradient reply to a server query can deviate arbitrarily from the true gradient subject to a distance bound. Our study focuses on two fundamental questions: (i) what is the smallest achievable sub-optimality gap (i.e., excess error in optimization) under such responses, and (ii) how many queries are sufficient to guarantee a given sub-optimality gap? We establish tight feasibility thresholds on the sub-optimality gap and provide algorithms that achieve these thresholds with provable query complexity guarantees.
Figures
Reference graph
Works this paper leans on
-
[1]
Bartlett, Pradeep Ravikumar, and Martin J
Alekh Agarwal, Peter L. Bartlett, Pradeep Ravikumar, and Martin J. Wainwright. Information-theoretic lower bounds on the oracle complexity of stochastic convex opti- mization.IEEE Transactions on Information Theory, 58(5):3235–3249, 2012
work page 2012
-
[2]
Byzantine stochastic gradient descent
Dan Alistarh, Zeyuan Allen-Zhu, and Jerry Li. Byzantine stochastic gradient descent. In Proceedings of Neural Information Processing Systems (NIPS), pages 4618–4628, 2018
work page 2018
-
[3]
Machine learning with adversaries: Byzantine tolerant gradient descent
Peva Blanchard, El Mahdi El Mhamdi, Rachid Guerraoui, and Julien Stainer. Machine learning with adversaries: Byzantine tolerant gradient descent. InProceedings of Neural Information Processing Systems (NIPS), pages 119–129, 2017
work page 2017
-
[4]
Digvijay Boob, Qi Deng, and Guanghui Lan. Stochastic first-order methods for convex and nonconvex functional constrained optimization.Mathematical Programming, 197(1):215– 279, 2023
work page 2023
-
[5]
Sebastien Bubeck. Convex optimization: Algorithms and complexity.Foundations and Trends in Machine Learning, 8(3-4):231–357, 2015
work page 2015
-
[6]
Efficient coreset selection with cluster-based methods
Chengliang Chai, Jiayi Wang, Nan Tang, Ye Yuan, Jiabin Liu, Yuhao Deng, and Guoren Wang. Efficient coreset selection with cluster-based methods. InProceedings of ACM Knowledge Discovery and Data Mining (SIGKDD), pages 167–178, 2023
work page 2023
-
[7]
Gradient descent: Robustness to adversarial corruption
Fu-Chieh Chang, Farhang Nabiei, Pei-Yuan Wu, Alexandru Cioba, Sattar Vakili, and Al- berto Bernacchia. Gradient descent: Robustness to adversarial corruption. InProceedings of International OPT Workshop on Optimization for Machine Learning, 2022
work page 2022
-
[8]
Smooth optimization with approximate gradient.SIAM Journal on Optimization, 19(3):1171–1183, 2008
Alexandre d’Aspremont. Smooth optimization with approximate gradient.SIAM Journal on Optimization, 19(3):1171–1183, 2008
work page 2008
Show all 32 references
-
[9]
Optimal distributed online prediction using mini-batches.Journal of Machine Learning Research (JMLR), 13:165–202, 2012
Ofer Dekel, Ran Gilad-Bachrach, Ohad Shamir, and Lin Xiao. Optimal distributed online prediction using mini-batches.Journal of Machine Learning Research (JMLR), 13:165–202, 2012
2012
-
[10]
Nesterov
Olivier Devolder, Francois Glineur, and Yurii E. Nesterov. First-order methods of smooth convex optimization with inexact oracle.Mathematical Programming, 146(1-2):37–75, 2014
2014
-
[11]
Guillaume Garrigos and Robert M. Gower. Handbook of convergence theorems for (stochas- tic) gradient methods.ArXiv, abs/2301.11235, 2024
2024
-
[12]
A study of first-order methods with a determinis- tic relative-error gradient oracle
Nadav Hallak and Kfir Yehuda Levy. A study of first-order methods with a determinis- tic relative-error gradient oracle. InProceedings of International Conference on Machine Learning (ICML), 2024
2024
-
[13]
Freris, and Hu Ding
Jiawei Huang, Ruomin Huang, Wenjie Liu, Nikolaos M. Freris, and Hu Ding. A novel sequential coreset method for gradient descent algorithms. InProceedings of International Conference on Machine Learning (ICML), pages 4412–4422, 2021. 12
2021
-
[14]
How to learn when data gradually reacts to your model
Zachary Izzo, James Zou, and Lexing Ying. How to learn when data gradually reacts to your model. InProceedings of International Conference on Artificial Intelligence and Statistics (AISTATS), pages 3998–4035, 2022
2022
-
[15]
Revisiting frank-wolfe: Projection-free sparse convex optimization
Martin Jaggi. Revisiting frank-wolfe: Projection-free sparse convex optimization. InPro- ceedings of International Conference on Machine Learning (ICML), pages 427–435, 2013
2013
-
[16]
Learning from history for byzantine robust optimization
Sai Praneeth Karimireddy, Lie He, and Martin Jaggi. Learning from history for byzantine robust optimization. InProceedings of International Conference on Machine Learning (ICML), pages 5311–5319, 2021
2021
-
[17]
Kerger, Marco Molinaro, Hongyi Jiang, and Amitabh Basu
Phillip A. Kerger, Marco Molinaro, Hongyi Jiang, and Amitabh Basu. A universal transfer theorem for convex optimization algorithms using inexact first-order oracles. InProceedings of International Conference on Machine Learning (ICML), 2024
2024
-
[18]
Sub-sampled cubic regularization for non-convex optimization
Jonas Moritz Kohler and Aurelien Lucchi. Sub-sampled cubic regularization for non-convex optimization. InProceedings of International Conference on Machine Learning (ICML), pages 1895–1904, 2017
1904
-
[19]
Springer, 2020
Guanghui Lan.First-order and Stochastic Optimization Methods for Machine Learning. Springer, 2020
2020
-
[20]
Learning rates for stochastic gradient descent with nonconvex objectives.IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), 43(12):4505–4511, 2021
Yunwen Lei and Ke Tang. Learning rates for stochastic gradient descent with nonconvex objectives.IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), 43(12):4505–4511, 2021
2021
-
[21]
Federated learning: Challenges, methods, and future directions.IEEE Signal Processing Magazine, 37(3):50– 60, 2020
Tian Li, Anit Kumar Sahu, Ameet Talwalkar, and Virginia Smith. Federated learning: Challenges, methods, and future directions.IEEE Signal Processing Magazine, 37(3):50– 60, 2020
2020
-
[22]
Bilmes, and Jure Leskovec
Baharan Mirzasoleiman, Jeff A. Bilmes, and Jure Leskovec. Coresets for data-efficient train- ing of machine learning models. InProceedings of International Conference on Machine Learning (ICML), pages 6950–6960, 2020
2020
-
[23]
MIT Press, 2018
Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar.Foundations of machine learning. MIT Press, 2018
2018
-
[24]
Juditsky, Guanghui Lan, and Alexander Shapiro
Arkadi Nemirovski, Anatoli B. Juditsky, Guanghui Lan, and Alexander Shapiro. Robust stochastic approximation approach to stochastic programming.SIAM Journal on Opti- mization, 19(4):1574–1609, 2009
2009
-
[25]
John Wiley & Sons, 1983
Arkadi Nemirovsky and David Berkovich Yudin.Problem Complexity and Method Effi- ciency in Optimization. John Wiley & Sons, 1983
1983
-
[26]
Nesterov
Yurii E. Nesterov. Efficiency of coordinate descent methods on huge-scale optimization problems.SIAM Journal on Optimization, 22(2):341–362, 2012
2012
-
[27]
Tsybakov
Alexandre B. Tsybakov. Optimal rates of aggregation. In Bernhard Scholkopf and Man- fred K. Warmuth, editors,Proceedings of Conference on Learning Theory (COLT), volume 2777, pages 303–313, 2003
2003
-
[28]
Shuche Wang and Vincent Y. F. Tan. Robust distributed gradient descent to corruption over noisy channels. InIEEE International Symposium on Information Theory, pages 2520–2525, 2024
2024
-
[29]
Shuche Wang and Vincent Y. F. Tan. A mirror descent-based algorithm for corruption- tolerant distributed gradient descent.IEEE Transactions on Signal Processing, 73:827–842, 2025. 13
2025
-
[30]
if-branch
Dong Yin, Yudong Chen, Kannan Ramchandran, and Peter L. Bartlett. Byzantine-robust distributed learning: Towards optimal statistical rates. InProceedings of International Conference on Machine Learning (ICML), pages 5636–5645, 2018. Appendix A Completing the Proofs of Theorems...
2018
-
[31]
by magic
studied the behavior of gradient descent (GD) when it runs under anϵ-AGP, as described in the pseudocode below: algorithm GD(K) 1.w 0 ←0,k←0 2.whilek < Kdo 3.g k ←oracle(w k) /* output of theϵ-AGP oracle */ 4.w k+1 ←w k − 1 2L gk 5.k←k+ 1 6.returnw k For anyK≥1, [14, Lemma 5] ...
-
[32]
9”, “13”, “17
did not analyze the sub-optimality gap ofw +. Nevertheless, we can do an analysis for them by resorting to our theory. For this purpose, let us augment their algorithm by requiring it to return the currentw k as soon as∥g k∥falls below 4ϵ(as in Line 4 ofAGP-opt). If the algori...
Reviewed May 7, 2026 · model on record in the stance chip above.
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