Fundamentalness of the robustness-coefficient restriction

Determine whether the restriction imposed on the robustness coefficient \(\kappa\) in Equation (\ref{eq:main-kappa-condition}) is fundamental for Byzantine-robust nonlinear representation learning with personalized client heads and a shared representation; in particular, establish whether this restriction can be replaced by a sufficiently large sample burn-in condition.

Background

The paper analyzes federated nonlinear representation learning under Byzantine attacks by allowing each honest client to learn a personalized linear head while robustly aggregating updates for a shared nonlinear representation. Its convergence and recovery guarantees require the robustness coefficient κ\kappa, which measures the deviation of the robust aggregation rule from the honest average, to be sufficiently small relative to problem-dependent curvature, feature, and Jacobian parameters.

The authors note that prior work on single-model Byzantine-robust linear and nonlinear regression showed that a restriction on κ\kappa may instead be replaced by a sufficiently large sample burn-in condition. It remains unresolved whether an analogous result holds in the personalized representation-learning setting, where client-specific heads absorb heterogeneity and the server aggregates only shared-representation updates. Resolving this would clarify the statistical limits of personalization under adversarial clients.

References

We leave for future work to determine whether the restriction on the robustness coefficient $\kappa$ (i.e., Eq. eq:main-kappa-condition) is fundamental. In particular, , for single-model Byzantine-robust linear and nonlinear regression demonstrated that a restriction on $\kappa$ can instead be replaced by a sufficiently large sample burn-in condition. Establishing whether a similar guarantee also holds for our setting would clarify the statistical limits of personalization under adversarial clients.

— Byzantine-Robust Federated Representation Learning  (2609.36660 - Toso et al., 29 Sep 2026) in Section 5, Conclusion and Future Work

Whether this condition is fundamental remains a direction for future work.

— Byzantine-Robust Federated Representation Learning  (2609.36660 - Toso et al., 29 Sep 2026) in Appendix, Section 8, remark following Lemma \ref{lem:G-empirical-curvature-learned-heads}