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

Robust distributed learning algorithms aim to maintain reliable performance despite the presence of misbehaving workers. Such misbehaviors are commonly modeled as \textit{Byzantine failures}, allowing arbitrarily corrupted communication, or as \textit{data poisoning}, a weaker form of corruption restricted to local training data. While prior work shows similar optimization guarantees for both models, an important question remains: \textit{How do these threat models impact generalization?} We show, for the first time, a fundamental gap in generalization guarantees between the two threat models: Byzantine failures yield strictly worse rates than those achievable under data poisoning. Our findings are based upon a tight algorithmic stability analysis of robust distributed learning. Specifically, with $f$ out of $n$ workers misbehaving, we prove that: \textit{(i)} under data poisoning, the uniform algorithmic stability of a robust distributed learning algorithm

Understanding the relationship between generalization and privacy remains a central challenge in modern machine learning theory, particularly for deep networks trained by variants of differentially private stochastic gradient descent (DP-SGD). In this work we make progress on this persistent open problem by proving a finite-sample bound on the approximate max-information of DP-SGD that exhibits scaling properties comparable with (Dwork et al, 2015)'s classic result for $ε$-differentially private algorithms, namely at most linear in the dataset size. From our result we obtain a general-purpose PAC-Bayes generalization bound in which the necessary prior distribution can be learned by DP-SGD, as well as a generalization bound for DP-SGD-trained models themselves, with a complexity term that is fully explicit and controlled by the optimization hyperparameters.

Generalization is a critical property of data-driven models, particularly deep learning models deployed in safety-critical applications. Robustness-based generalization bounds have gained attention as a principled way to link robustness properties to generalization performance, often in a data-dependent manner. However, most existing bounds suffer from vacuousness in practical settings, yielding loose upper bounds that greatly exceed the actual error rates and limiting their usefulness for real-world evaluation. While this issue is often attributed to the uncertainty term, a substantial part of the problem originates from the robustness term itself, particularly for the 0-1 loss. Existing approaches typically treat the robustness term as a global measure, ignoring its variation across different sub-regions of the input space. In this work, we propose a generalization bound that addresses this limitation by scaling the robustness term according to the number of stable and unstable samples within each sub-region. Our bounds incorporate both data- and model-dependent factors while maintaining practical relevance (yielding tighter upper bounds on true error). Experiments on models trained on the ImageNet dataset show that our bounds remain consistently non-vacuous and achieve the tightest estimates among existing methods, closely aligning with empirical performance across a range of robust deep neural networks.