Improved worst-case spectral-norm bound

Determine whether the worst-case upper bound on the spectral norm of the FedAPM information-mixing matrix can be improved under the stated client-availability assumptions.

Background

The paper bounds the spectral norm governing information mixing over a sliding availability window. This quantity controls the decay of consensus errors and therefore enters the convergence-rate guarantees.

The authors derive a worst-case bound using graph conductance and Cheeger’s inequality, but explicitly identify possible looseness in its dependence on the number of clients and the window length.

References

It remains an open question whether the worst-case bound can be improved, and we would like to leave this as future work.

— Resilience Beyond Stationary Client Unavailability: Unlocking Efficient and Unbiased Federated Learning  (2609.04763 - Xiang et al., 4 Sep 2026) in Remark 4.4, Section 4.3, subsection “Main Convergence Results”