Unveiling the Non-Monotonic Effect of Privacy on Generalization under Byzantine Robustness

arXiv cs.LG Papers

Summary

This paper reveals a non-monotonic effect of privacy on generalization error in Byzantine-robust distributed learning: in high-noise (strong privacy) regimes, increasing privacy reduces generalization error, while in low-noise (weaker privacy) regimes, increasing privacy degrades generalization.

arXiv:2607.01492v1 Announce Type: new Abstract: Recent work has established a fundamental trilemma between Byzantine robustness, local differential privacy (LDP), and optimization error in distributed learning. We show that this trilemma does not universally extend to generalization error, but instead depends critically on the privacy regime. Specifically, in the high-noise regime (strong privacy), we prove that increasing privacy reduces the generalization error, i.e., there is no tension between robustness and privacy. In the low-noise regime (weaker privacy), however, the tension between robustness and privacy reappears and increasing privacy indeed degrades generalization. Our theory explains this surprising non-monotonic behavior of the generalization error via matching lower and upper bounds on the algorithmic stability of Byzantine-robust distributed learning under LDP constraints. We corroborate and further analyze these theoretical findings with empirical evaluations.
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# Unveiling the Non-Monotonic Effect of Privacy on Generalization under Byzantine Robustness
Source: [https://arxiv.org/abs/2607.01492](https://arxiv.org/abs/2607.01492)
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> Abstract:Recent work has established a fundamental trilemma between Byzantine robustness, local differential privacy \(LDP\), and optimization error in distributed learning\. We show that this trilemma does not universally extend to generalization error, but instead depends critically on the privacy regime\. Specifically, in the high\-noise regime \(strong privacy\), we prove that increasing privacy reduces the generalization error, i\.e\., there is no tension between robustness and privacy\. In the low\-noise regime \(weaker privacy\), however, the tension between robustness and privacy reappears and increasing privacy indeed degrades generalization\. Our theory explains this surprising non\-monotonic behavior of the generalization error via matching lower and upper bounds on the algorithmic stability of Byzantine\-robust distributed learning under LDP constraints\. We corroborate and further analyze these theoretical findings with empirical evaluations\.

## Submission history

From: Thomas Boudou \[[view email](https://arxiv.org/show-email/020f0b6f/2607.01492)\] **\[v1\]**Wed, 1 Jul 2026 21:42:47 UTC \(201 KB\)

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