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Byzantine-Robust Optimization under $(L_0, L_1)$-Smoothness

arXiv:2603.12512v1 Announce Type: new Abstract: We consider distributed optimization under Byzantine attacks in the presence of $(L_0,L_1)$-smoothness, a generalization of standard $L$-smoothness that captures functions with state-dependent gradient Lipschitz constants. We propose Byz-NSGDM, a normalized stochastic gradient descent method with momentum that achieves robustness against Byzantine workers while maintaining convergence guarantees. Our algorithm combines momentum normalization with Byzantine-robust aggregation enhanced by Nearest Neighbor Mixing (NNM) to handle both the challenges posed by $(L_0,L_1)$-smoothness and Byzantine adversaries. We prove that Byz-NSGDM achieves a convergence rate of $O(K^{-1/4})$ up to a Byzantine bias floor proportional to the robustness coefficient and gradient heterogeneity. Experimental validation on heterogeneous MNIST classification, synthetic $(L_0,L_1)$-smooth optimization, and character-level language modeling with a small GPT model demo

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Arman Bolatov, Samuel Horv\'ath, Martin Tak\'a\v{c}, Eduard Gorbunov · · 1 min read · 3 views
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arXiv:2603.12512v1 Announce Type: new Abstract: We consider distributed optimization under Byzantine attacks in the presence of $(L_0,L_1)$-smoothness, a generalization of standard $L$-smoothness that captures functions with state-dependent gradient Lipschitz constants. We propose Byz-NSGDM, a normalized stochastic gradient descent method with momentum that achieves robustness against Byzantine workers while maintaining convergence guarantees. Our algorithm combines momentum normalization with Byzantine-robust aggregation enhanced by Nearest Neighbor Mixing (NNM) to handle both the challenges posed by $(L_0,L_1)$-smoothness and Byzantine adversaries. We prove that Byz-NSGDM achieves a convergence rate of $O(K^{-1/4})$ up to a Byzantine bias floor proportional to the robustness coefficient and gradient heterogeneity. Experimental validation on heterogeneous MNIST classification, synthetic $(L_0,L_1)$-smooth optimization, and character-level language modeling with a small GPT model demonstrates the effectiveness of our approach against various Byzantine attack strategies. An ablation study further shows that Byz-NSGDM is robust across a wide range of momentum and learning rate choices.

Executive Summary

This article presents Byz-NSGDM, a novel optimization algorithm designed to tackle Byzantine attacks in distributed optimization under $(L_0,L_1)$-smoothness. Byz-NSGDM combines momentum normalization with Byzantine-robust aggregation enhanced by Nearest Neighbor Mixing (NNM) to achieve robustness against Byzantine workers while maintaining convergence guarantees. The proposed algorithm is experimentally validated on various tasks, demonstrating its effectiveness against Byzantine attacks. The study also provides an ablation analysis, showcasing the robustness of Byz-NSGDM across a range of momentum and learning rate choices. The findings contribute significantly to the field of distributed optimization, offering a promising solution to tackle the challenges posed by $(L_0,L_1)$-smoothness and Byzantine adversaries.

Key Points

  • ▸ Byz-NSGDM is the first optimization algorithm to achieve robustness against Byzantine workers under $(L_0,L_1)$-smoothness.
  • ▸ The proposed algorithm combines momentum normalization with Byzantine-robust aggregation enhanced by Nearest Neighbor Mixing (NNM).
  • ▸ Byz-NSGDM achieves a convergence rate of $O(K^{-1/4})$ up to a Byzantine bias floor proportional to the robustness coefficient and gradient heterogeneity.

Merits

Strength

Byz-NSGDM's ability to achieve robustness against Byzantine workers under $(L_0,L_1)$-smoothness is a significant merit of the proposed algorithm. This capability addresses a critical challenge in distributed optimization and makes the algorithm more applicable to real-world scenarios.

Robustness

The ablation study demonstrating Byz-NSGDM's robustness across a wide range of momentum and learning rate choices is a notable strength of the algorithm. This characteristic makes the algorithm more adaptable and user-friendly.

Convergence Rate

The $O(K^{-1/4})$ convergence rate of Byz-NSGDM is a notable merit, as it provides a guaranteed rate of convergence despite the presence of Byzantine workers and $(L_0,L_1)$-smoothness.

Demerits

Limitation

The dependence of Byz-NSGDM's convergence rate on the robustness coefficient and gradient heterogeneity may limit its applicability in scenarios with high Byzantine bias or gradient heterogeneity.

Scalability

The experimental validation of Byz-NSGDM is limited to relatively small-scale tasks, and its scalability to larger-scale distributed optimization problems remains an open question.

Expert Commentary

The proposed Byz-NSGDM algorithm is a notable contribution to the field of distributed optimization, addressing the critical challenge of Byzantine attacks under $(L_0,L_1)$-smoothness. The algorithm's ability to achieve robustness against Byzantine workers while maintaining convergence guarantees is a significant strength. However, the dependence of its convergence rate on the robustness coefficient and gradient heterogeneity may limit its applicability in scenarios with high Byzantine bias or gradient heterogeneity. Overall, the study's findings have significant implications for the development of secure and robust distributed optimization algorithms and highlight the need for further research in this area.

Recommendations

  • ✓ Further research is needed to investigate the scalability of Byz-NSGDM to larger-scale distributed optimization problems.
  • ✓ The dependence of Byz-NSGDM's convergence rate on the robustness coefficient and gradient heterogeneity should be explored further to improve its applicability in scenarios with high Byzantine bias or gradient heterogeneity.

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