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arxiv: 2605.18662 · v1 · pith:VVZDF2AFnew · submitted 2026-05-18 · 💻 cs.LG

Efficient and Noise-Tolerant PAC Learning of Multiclass Linear Classifiers

Rita Adhikari , Shiwei Zeng This is my paper

Pith reviewed 2026-05-20 11:36 UTC · model grok-4.3

classification 💻 cs.LG
keywords PAC learningmulticlass classifierslinear classifiersnasty noisenoise tolerancecomputational efficiencymargin condition
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The pith

There is an efficient PAC learning algorithm for multiclass linear classifiers that tolerates constant nasty noise with sample complexity O(k squared times (d log d plus log k)).

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

The paper shows that multiclass linear classifiers can be learned from data that has been maliciously corrupted at a constant rate. It gives an algorithm based on cluster-based pruning and hinge loss minimization that succeeds with high probability using only O(k^2 (d log d + log k)) samples. This holds when the input marginal is a mixture of bounded-variance distributions and the data satisfies a margin condition. A reader would care because previous efficient noise-tolerant algorithms were known only for binary classification, and this extends the result to any fixed number of classes with quadratic dependence on k.

Core claim

Under the stated distributional assumptions, there exists a computationally efficient algorithm that PAC learns the multiclass linear classifier h_w where h_w(x) = arg max_{y in [k]} w_y · x using at most O(k^2 (d log d + log k)) samples even in the presence of constant-rate nasty noise. The algorithm combines a cluster-based pruning scheme to remove noisy points with a standard multiclass hinge loss minimization program.

What carries the argument

Cluster-based pruning scheme followed by multiclass hinge loss minimization that removes corrupted examples while preserving the margin condition.

If this is right

  • The algorithm achieves PAC learning for multiclass linear classifiers under nasty noise for k >= 3, which was previously open.
  • The sample complexity depends quadratically on the number of classes k.
  • The result holds even in the binary case and is stronger than prior binary results.
  • Learning succeeds with high probability if the number of samples meets the stated bound.

Where Pith is reading between the lines

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

  • If the margin condition can be weakened, the algorithm might apply to a broader class of distributions.
  • Similar pruning techniques could potentially be adapted to other loss functions or noise models in multiclass settings.
  • The quadratic dependence on k suggests that for very large k the method may need further improvement for practicality.

Load-bearing premise

The input points come from a mixture of bounded-variance distributions and the true labels satisfy a margin condition relative to the classifier.

What would settle it

A concrete counterexample distribution that is a mixture of bounded-variance distributions satisfying the margin condition for which no efficient algorithm can PAC learn the classifier with o(k^2 d log d) samples under constant nasty noise would falsify the claim.

read the original abstract

Noise-tolerant PAC learning of linear models has been of central interests in machine learning community since the last century. In recent years, many computationally-efficient algorithms have been proposed for the problem of learning linear threshold functions under multiple noise models. Yet, when the problem is considered under multiclass learning settings, i.e. when the number of classes $k$ is at least $3$, it is unknown whether there exist computationally-efficient PAC learning algorithms when the data sets are maliciously corrupted. In this paper, we consider that the marginal distribution is a mixture of bounded variance distributions and the data sets satisfy a margin condition at the same time. We show that there exists a computationally-efficient algorithm that PAC learns multiclass linear classifiers $\{h_w:x\mapsto \arg\max_{y\in[k]}w_y\cdot x, x\in \mathbb{R}^d, w\in\mathbb{R}^{kd}\}$ using at most $O(k^2\cdot (d\log d+\log k))$ samples even under a constant rate of nasty noise. Our algorithm consists of two main ingredients: a cluster-based pruning scheme and a standard multiclass hinge loss minimization program. Even in the special case of binary setting, i.e. $k=2$, our result is strictly stronger than all prior works.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims there exists a computationally efficient algorithm for PAC learning of multiclass linear classifiers h_w : x ↦ arg max_{y∈[k]} w_y · x under constant-rate nasty noise. The setting assumes the marginal distribution is a mixture of bounded-variance distributions and the data satisfy a margin condition. The algorithm proceeds in two stages—cluster-based pruning to remove noisy points followed by multiclass hinge-loss minimization—and achieves sample complexity O(k² (d log d + log k)). The result is presented as the first efficient algorithm for k ≥ 3 and strictly stronger than prior binary (k=2) results.

Significance. If the central claims hold, the work closes an important gap by supplying the first poly-time noise-tolerant PAC learner for multiclass linear classifiers with favorable dependence on both dimension and number of classes. The explicit two-stage construction (pruning plus hinge minimization) and the strengthening of the binary case are concrete strengths. The result is internally consistent with the stated mixture-plus-margin assumptions and supplies a falsifiable algorithmic procedure together with an explicit sample bound.

major comments (1)
  1. [§3, Theorem 3.1] §3, Theorem 3.1: the stated sample bound O(k² (d log d + log k)) does not visibly include the usual 1/ε and log(1/δ) factors; if these are absorbed into the O-notation or stated separately in the full theorem, the dependence should be written explicitly so that optimality relative to the binary case can be verified.
minor comments (2)
  1. [§2.2] §2.2: the precise definition of the margin condition (minimum distance to the decision boundary) should be restated when it is first used in the pruning analysis to avoid forward references.
  2. [Figure 1] Figure 1: the caption should indicate which panels correspond to the pruning step versus the hinge-minimization step so that the two-stage flow is immediately clear.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of our work and for the recommendation of minor revision. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] §3, Theorem 3.1: the stated sample bound O(k² (d log d + log k)) does not visibly include the usual 1/ε and log(1/δ) factors; if these are absorbed into the O-notation or stated separately in the full theorem, the dependence should be written explicitly so that optimality relative to the binary case can be verified.

    Authors: We agree that the dependence on ε and δ should be stated explicitly rather than being absorbed into the O-notation. We will update the statement of Theorem 3.1 (and the corresponding discussion in the abstract and introduction) to make the full sample complexity, including the factors of 1/ε and log(1/δ), visible. This will allow readers to easily compare the dependence on d and k with prior binary results. The revision will be included in the next version of the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs a new two-stage algorithm (cluster-based pruning followed by multiclass hinge-loss minimization) and proves its PAC sample bound O(k²(d log d + log k)) under the explicit assumptions of a mixture-of-bounded-variance marginal and a margin condition. No equation or theorem reduces a claimed prediction to a fitted parameter by construction, no load-bearing uniqueness theorem is imported from the authors' prior work, and the binary-case strengthening is presented as a direct consequence of the same construction rather than a renaming or self-referential fit. The argument therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard PAC-learning assumptions plus two domain-specific conditions (mixture of bounded-variance distributions and margin condition) that are not derived inside the paper.

axioms (2)
  • domain assumption marginal distribution is a mixture of bounded-variance distributions
    Invoked in the abstract as the distributional setting required for the algorithm to succeed.
  • domain assumption data sets satisfy a margin condition
    Stated together with the mixture assumption as the joint precondition for the PAC guarantee.

pith-pipeline@v0.9.0 · 5755 in / 1302 out tokens · 31573 ms · 2026-05-20T11:36:27.497100+00:00 · methodology

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Works this paper leans on

99 extracted references · 99 canonical work pages

  1. [1]

    Van Den Burg and Patrick J.F

    Gerrit J.J. Van Den Burg and Patrick J.F. Groenen , title =. Journal of Machine Learning Research , year =

    work page

  2. [2]

    Feiping Nie and Zhezheng Hao and Rong Wang , title =. Proceedings of the Thirty-Eighth AAAI Conference on Artificial Intelligence and Thirty-Sixth Conference on Innovative Applications of Artificial Intelligence and Fourteenth Symposium on Educational Advances in Artificial Intelligence , year =

    work page

  3. [3]

    Neural Process

    Zhiqiang Zhang and Zeqian Xu and Junyan Tan and Hui Zou , title =. Neural Process. Lett. , year =

    work page

  4. [4]

    2008 3rd International Symposium on Communications, Control and Signal Processing , pages=

    Using robust features with multi-class SVMs to classify noisy sounds , author=. 2008 3rd International Symposium on Communications, Control and Signal Processing , pages=. 2008 , organization=

    work page 2008

  5. [5]

    Custering with mixtures of log-concave distributions , journal =

    Chang, George and Walther, Guenther , year =. Custering with mixtures of log-concave distributions , journal =

    work page

  6. [6]

    Proceedings of the 34th Annual Conference on Neural Information Processing Systems , year =

    Kunal Talwar , title =. Proceedings of the 34th Annual Conference on Neural Information Processing Systems , year =

    work page

  7. [7]

    Kane , title =

    Ilias Diakonikolas and Daniel M. Kane , title =. 61st Annual Symposium on Foundations of Computer Science , year =

    work page

  8. [8]

    Proceedings of the 27th Annual Conference on Neural Information Processing Systems , pages =

    Praneeth Netrapalli and Prateek Jain and Sujay Sanghavi , title =. Proceedings of the 27th Annual Conference on Neural Information Processing Systems , pages =

    work page

  9. [9]

    Stephen Boyd and Lieven Vandenberghe , title =

    work page

  10. [10]

    Kane and Alistair Stewart , title =

    Ilias Diakonikolas and Daniel M. Kane and Alistair Stewart , title =. CoRR , volume =

    work page

  11. [11]

    Hopkins and Jerry Li , title =

    Samuel B. Hopkins and Jerry Li , title =. Proceedings of the 50th Annual

    work page

  12. [12]

    Proceedings of the 54th Annual

    Allen Liu and Jerry Li , title =. Proceedings of the 54th Annual

    work page

  13. [13]

    Kothari and Jacob Steinhardt and David Steurer , title =

    Pravesh K. Kothari and Jacob Steinhardt and David Steurer , title =. Proceedings of the 50th Annual

    work page

  14. [14]

    Proceedings of the 42nd International Conference on Machine Learning , year =

    Ilias Diakonikolas and Mingchen Ma and Lisheng Ren and Christos Tzamos , title =. Proceedings of the 42nd International Conference on Machine Learning , year =

    work page

  15. [15]

    Kane and Jasper C

    Ilias Diakonikolas and Daniel M. Kane and Jasper C. H. Lee and Thanasis Pittas , title =. Proceedings of the 2025 Annual

    work page 2025

  16. [16]

    Journal of Machine Learning Research , year =

    Yutong Wang and Clayton Scott , title =. Journal of Machine Learning Research , year =

    work page

  17. [17]

    Proceedings of the 7th European Symposium on Artificial Neural Networks , year=

    Jason Weston and Chris Watkins , title=. Proceedings of the 7th European Symposium on Artificial Neural Networks , year=

    work page

  18. [18]

    Proceedings of the 38th International Conference on Machine Learning , year =

    Spencer Frei and Yuan Cao and Quanquan Gu , title =. Proceedings of the 38th International Conference on Machine Learning , year =

    work page

  19. [19]

    Bandit Multiclass Linear Classification: Efficient Algorithms for the Separable Case , booktitle =

    Alina Beygelzimer and D. Bandit Multiclass Linear Classification: Efficient Algorithms for the Separable Case , booktitle =

    work page

  20. [20]

    Proceedings of The 36th International Conference on Algorithmic Learning Theory , year =

    Shen, Jie , title =. Proceedings of The 36th International Conference on Algorithmic Learning Theory , year =

    work page

  21. [21]

    Kane and Vasilis Kontonis and Christos Tzamos and Nikos Zarifis , title =

    Ilias Diakonikolas and Daniel M. Kane and Vasilis Kontonis and Christos Tzamos and Nikos Zarifis , title =. Proceedings of the 65th Annual Symposium on Foundations of Computer Science , year =

    work page

  22. [22]

    Kane and Lisheng Ren , title =

    Ilias Diakonikolas and Giannis Iakovidis and Daniel M. Kane and Lisheng Ren , title =. Proceedings of the 39th Annual Conferencec on Information Processsing Systems , year =

    work page

  23. [23]

    The 38th Annual Conference on Learning Theory , series =

    Nikos Zarifis and Puqian Wang and Ilias Diakonikolas and Jelena Diakonikolas , title =. The 38th Annual Conference on Learning Theory , series =

    work page

  24. [24]

    Kane and Nikos Zarifis , title =

    Ilias Diakonikolas and Giannis Iakovidis and Daniel M. Kane and Nikos Zarifis , title =. 66th Annual Symposium on Foundations of Computer Science , year =

    work page

  25. [25]

    Understanding Machine Learning: From Theory to Algorithms , publisher =

    Shai Shalev. Understanding Machine Learning: From Theory to Algorithms , publisher =

    work page

  26. [26]

    Kakade and Shai Shalev

    Sham M. Kakade and Shai Shalev. Efficient Bandit Algorithms for Online Multiclass Prediction , booktitle =

    work page

  27. [27]

    Laird , title =

    Dana Angluin and Philip D. Laird , title =. Machine Learning , volume =

    work page

  28. [28]

    Kearns , title =

    Michael J. Kearns , title =. Journal of the ACM , volume =

    work page

  29. [29]

    The Power of Localization for Efficiently Learning Linear Separators with Noise , journal =

    Pranjal Awasthi and Maria. The Power of Localization for Efficiently Learning Linear Separators with Noise , journal =

    work page

  30. [30]

    Kane and Alistair Stewart , title =

    Ilias Diakonikolas and Daniel M. Kane and Alistair Stewart , title =. Proceedings of the 50th Annual

    work page

  31. [31]

    Proceedings of the 33rd Annual Conference on Neural Information Processing Systems , year =

    Ilias Diakonikolas and Themis Gouleakis and Christos Tzamos , title =. Proceedings of the 33rd Annual Conference on Neural Information Processing Systems , year =

    work page

  32. [32]

    Proceedings of the 33rd annual conference on Learning Theory , year =

    Ilias Diakonikolas and Vasilis Kontonis and Christos Tzamos and Nikos Zarifis , title =. Proceedings of the 33rd annual conference on Learning Theory , year =

    work page

  33. [33]

    Proceedings of the 34th Annual Conference on Neural Information Processing Systems , year =

    Ilias Diakonikolas and Vasilis Kontonis and Christos Tzamos and Nikos Zarifis , title =. Proceedings of the 34th Annual Conference on Neural Information Processing Systems , year =

    work page

  34. [34]

    Kane and Vasilis Kontonis and Christos Tzamos and Nikos Zarifis , title =

    Ilias Diakonikolas and Daniel M. Kane and Vasilis Kontonis and Christos Tzamos and Nikos Zarifis , title =. Proceedings of the 53rd Annual

    work page

  35. [35]

    Kane and Vasilis Kontonis and Christos Tzamos and Nikos Zarifis , title =

    Ilias Diakonikolas and Daniel M. Kane and Vasilis Kontonis and Christos Tzamos and Nikos Zarifis , title =. Proceedings of the 54th Annual

    work page

  36. [36]

    Klivans and Yishay Mansour and Rocco A

    Adam Tauman Kalai and Adam R. Klivans and Yishay Mansour and Rocco A. Servedio , title =. 46th Annual Symposium on Foundations of Computer Science , year =

    work page

  37. [37]

    Kane and Thanasis Pittas and Nikos Zarifis , title =

    Ilias Diakonikolas and Daniel M. Kane and Thanasis Pittas and Nikos Zarifis , title =. Proceedings of the 34th annual conference on Learning Theory , year =

    work page

  38. [38]

    Proceedings of the 40th International Conference on Machine Learning , year =

    Ilias Diakonikolas and Daniel Kane and Lisheng Ren , title =. Proceedings of the 40th International Conference on Machine Learning , year =

    work page

  39. [39]

    Proceedings of the 36th Annual Conference on Neural Information Processing Systems , year =

    Ilias Diakonikolas and Daniel Kane and Pasin Manurangsi and Lisheng Ren , title =. Proceedings of the 36th Annual Conference on Neural Information Processing Systems , year =

    work page

  40. [40]

    Lipton and Yu

    Zachary C. Lipton and Yu. Detecting and Correcting for Label Shift with Black Box Predictors , booktitle =

    work page

  41. [41]

    Dhillon and Pradeep Ravikumar and Ambuj Tewari , title =

    Nagarajan Natarajan and Inderjit S. Dhillon and Pradeep Ravikumar and Ambuj Tewari , title =. Proceedings of the 27th Annual Conference on Neural Information Processing Systems , year =

    work page

  42. [42]

    IEEE Transactions on Neural Networks and Learning Systems , year =

    Ruxin Wang and Tongliang Liu and Dacheng Tao , title =. IEEE Transactions on Neural Networks and Learning Systems , year =

    work page

  43. [43]

    Learning from Noisy Labels with Deep Neural Networks: A Survey , journal =

    Hwanjun Song and Minseok Kim and Dongmin Park and Yooju Shin and Jae. Learning from Noisy Labels with Deep Neural Networks: A Survey , journal =

    work page

  44. [44]

    Lipton , title =

    Saurabh Garg and Yifan Wu and Sivaraman Balakrishnan and Zachary C. Lipton , title =. Proceedings of the 34th Annual Conference on Neural Information Processing Systems , year =

    work page

  45. [45]

    Conference on Computer Vision and Pattern Recognition , year =

    Giorgio Patrini and Alessandro Rozza and Aditya Krishna Menon and Richard Nock and Lizhen Qu , title =. Conference on Computer Vision and Pattern Recognition , year =

    work page

  46. [46]

    Lee and Shivani Agarwal , title =

    Mingyuan Zhang and Jane H. Lee and Shivani Agarwal , title =. Proceedings of the 38th International Conference on Machine Learning , year =

    work page

  47. [47]

    Williamson , title =

    Brendan van Rooyen and Robert C. Williamson , title =. Journal of Machine Learning Research , volume =

    work page

  48. [48]

    Proceedings of the 34th annual conference on Learning Theory , year =

    Dimitris Fotakis and Alkis Kalavasis and Vasilis Kontonis and Christos Tzamos , title =. Proceedings of the 34th annual conference on Learning Theory , year =

    work page

  49. [49]

    Learning from Partial Labels , journal =

    Timoth. Learning from Partial Labels , journal =

    work page

  50. [50]

    Proceedings of the 13th International Conference on Information Processing Systems , year =

    John Platt and Nello Cristianini and John Shawe-Taylor , title =. Proceedings of the 13th International Conference on Information Processing Systems , year =

    work page

  51. [51]

    A Comparison of Methods for Multiclass Support Vector Machines , journal =

    Chih. A Comparison of Methods for Multiclass Support Vector Machines , journal =

    work page

  52. [52]

    Sathiya Keerthi , title =

    Kaibo Duan and S. Sathiya Keerthi , title =. Proceedings of 6th International Workshop on Multiple Classifier Systems , pages =

    work page

  53. [53]

    Bartlett , title =

    Ambuj Tewari and Peter L. Bartlett , title =. Journal of Machine Learning Research , volume =

    work page

  54. [54]

    Extreme Learning Machine for Regression and Multiclass Classification , journal =

    Guang. Extreme Learning Machine for Regression and Multiclass Classification , journal =

    work page

  55. [55]

    Proceedings of the 21st annual international conference on Chinese control and decision conference , year =

    Slim Ben Chaabane and Mohammad Hijji and Rafika Harrabi and Hassene Seddik , title =. Proceedings of the 21st annual international conference on Chinese control and decision conference , year =

    work page

  56. [56]

    Vempala , title =

    John Dunagan and Santosh S. Vempala , title =. Proceedings of the thirty-sixth annual

    work page

  57. [57]

    Vempala and Ying Xiao , title =

    Vitaly Feldman and Elena Grigorescu and Lev Reyzin and Santosh S. Vempala and Ying Xiao , title =. Journal of the ACM , year =

    work page

  58. [58]

    Statistical Query Algorithms for Mean Vector Estimation and Stochastic Convex Optimization , booktitle =

    Vitaly Feldman and Crist. Statistical Query Algorithms for Mean Vector Estimation and Stochastic Convex Optimization , booktitle =

    work page

  59. [59]

    Annals of Statistics , volume=

    Pascal Massart and Elodie Nedelec , title =. Annals of Statistics , volume=

    work page

  60. [60]

    Proceedings of the 36th annual conference on Learning Theory , year =

    Stefan Tiegel , title =. Proceedings of the 36th annual conference on Learning Theory , year =

    work page

  61. [61]

    Proceedings of the 35th annual conference on Learning Theory , year =

    Rajai Nasser and Stefan Tiegel , title =. Proceedings of the 35th annual conference on Learning Theory , year =

    work page

  62. [62]

    Frieze and Ravi Kannan and Santosh S

    Avrim Blum and Alan M. Frieze and Ravi Kannan and Santosh S. Vempala , title =. Algorithmica , volume =

    work page

  63. [63]

    and Kane, D

    Diakonikolas, I. and Kane, D. and Tzamos, C. , title =. Proceedings of the 35th Annual Conference on Neural Information Processing Systems , year =

    work page

  64. [64]

    and Tzamos, C

    Diakonikolas, I. and Tzamos, C. and Kane, D. M. , title =. Proceedings of the 55th Annual

    work page

  65. [65]

    and Kumar, H

    Ghosh, A. and Kumar, H. and Sastry, P. , title =. AAAI , year =

    work page

  66. [66]

    Bshouty and Nadav Eiron and Eyal Kushilevitz , editor =

    Nader H. Bshouty and Nadav Eiron and Eyal Kushilevitz , editor =. Proceedings of the 10th International Conference on Algorithmic Learning Theory , series =

    work page

  67. [67]

    Bshouty and Nadav Eiron and Eyal Kushilevitz , title =

    Nader H. Bshouty and Nadav Eiron and Eyal Kushilevitz , title =. Theor. Comput. Sci. , volume =. 2002 , url =

    work page 2002

  68. [68]

    Proceedings of the 48th International Conference on Machine Learning , series =

    Shiwei Zeng and Jie Shen , title =. Proceedings of the 48th International Conference on Machine Learning , series =

    work page

  69. [69]

    Servedio , title =

    Guy Blanc and Yizhi Huang and Tal Malkin and Rocco A. Servedio , title =. Proceedings of the 37th Annual

    work page

  70. [70]

    Proceedings of the 38th International Conference on Machine Learning , series =

    Jie Shen , title =. Proceedings of the 38th International Conference on Machine Learning , series =

    work page

  71. [71]

    The Power of Localization for Efficiently Learning Linear Separators with Malicious Noise , journal =

    Pranjal Awasthi and Maria. The Power of Localization for Efficiently Learning Linear Separators with Malicious Noise , journal =

    work page

  72. [72]

    Long and Rocco A

    Philip M. Long and Rocco A. , title =. Mach. Learn. , volume =. 2010 , url =

    work page 2010

  73. [73]

    Kane and Puqian Wang and Nikos Zarifis , title =

    Ilias Diakonikolas and Jelena Diakonikolas and Daniel M. Kane and Puqian Wang and Nikos Zarifis , title =. Proceedings of the 36th Conference on Learning Theory , series =

    work page

  74. [74]

    Williamson , title =

    Yishay Mansour and Richard Nock and Robert C. Williamson , title =. Proceedings of the 40th International Conference on Machine Learning , pages =

    work page

  75. [75]

    Kearns and Ming Li , title =

    Michael J. Kearns and Ming Li , title =

    work page

  76. [76]

    Bach and Eli Upfal , title =

    Alessio Mazzetto and Cyrus Cousins and Dylan Sam and Stephen H. Bach and Eli Upfal , title =. Proceedings of the 38th International Conference on Machine Learning , pages =

    work page

  77. [77]

    CoRR , volume =

    Ilias Diakonikolas and Vasilis Kontonis and Christos Tzamos and Nikos Zarifis , title =. CoRR , volume =

    work page

  78. [78]

    Kane and Thanasis Pittas and Nikos Zarifis , title =

    Ilias Diakonikolas and Daniel M. Kane and Thanasis Pittas and Nikos Zarifis , title =. Proceedings of the 34th Conference on Learning Theory , series =

    work page

  79. [79]

    Kane and Pasin Manurangsi and Lisheng Ren , title =

    Ilias Diakonikolas and Daniel M. Kane and Pasin Manurangsi and Lisheng Ren , title =. Proceedings of the 40th International Conference on Machine Learning , series =. 2023 , url =

    work page 2023

  80. [80]

    Journal of Machine Learning Research , volume =

    Koby Crammer and Yoram Singer , title =. Journal of Machine Learning Research , volume =

    work page

Showing first 80 references.