Continuous interpolation coefficients in FedAPM

Develop a theoretical analysis of FedAPM with a continuous interpolation coefficient k, rather than restricting k to discrete values, to determine whether the discreteness assumption is merely technical.

Background

FedAPM uses the interpolation coefficient k in its global moving-average update. The theoretical analysis models k through an augmented system containing k virtual clients, which requires k to be an integer.

The paper argues empirically that k=Theta(m) is generally reasonable, but leaves open whether the same analysis can be extended to continuous interpolation coefficients.

References

We conjecture that discrete $k$'s are only necessary for the technical convenience of our analysis, and leave it as a future work on how to analyze continuous $k$ theoretically.

— Resilience Beyond Stationary Client Unavailability: Unlocking Efficient and Unbiased Federated Learning  (2609.04763 - Xiang et al., 4 Sep 2026) in Section 3.2, subsection “Global moving average”

We hypothesize that the spectral norm $\rho_{k}$ would decrease w.r.t.\,$k$, which is numerically demonstrated in Example~\ref{example: rho monoto} by explicit realizations of~\prettyref{ass: prob lower bound}.

— Resilience Beyond Stationary Client Unavailability: Unlocking Efficient and Unbiased Federated Learning  (2609.04763 - Xiang et al., 4 Sep 2026) in Section 5.1, subsection “On the Scaling of Convergence Results”

However, which term will ultimately dominate the monotonicity of~eq: x bar rate as $k$ increases remains unclear.

— Resilience Beyond Stationary Client Unavailability: Unlocking Efficient and Unbiased Federated Learning  (2609.04763 - Xiang et al., 4 Sep 2026) in Section 5.1, subsection “On convergence upper bound in \eqref{eq: x bar rate}”