Stability conditions for hierarchical attribution

Determine the conditions under which the hierarchical attribution among the shared global process \(f_g\), the client-specific deviation processes \(f_{\delta,i}\), and the local residual processes \(f_i\) remains stable, particularly as the number of clients, client sample-size balance, and degree of cross-client heterogeneity vary.

Background

Because the deviation kernel is constrained to be a scaled copy of the global kernel, the split between the global process and a client-specific deviation is not uniquely identifiable from a single client's marginal process. The paper treats the resulting decomposition as a regularized attribution supported by multiple clients rather than as a classically point-identified decomposition.

The paper further states that attribution may become less stable when there are few clients, strongly unbalanced client sample sizes, or weak cross-client heterogeneity. It identifies determining the conditions for stable attribution as a future research problem, alongside other extensions such as privacy, robust aggregation, and scalable inducing-budget selection.

References

Future work will study the conditions under which this hierarchical attribution is stable, including the number of clients, sample-size balance, and degree of cross-client heterogeneity, as well as differentially private updates, online updating, robust aggregation under adversarial participation, and scalable inducing-budget selection in multi-output federated settings.

— Personalized Federated Hierarchical Gaussian Processes for Privacy-Preserving Modeling of Heterogeneous Distributed Systems  (2609.19337 - Xie et al., 16 Sep 2026) in Section 6, "Limitations and Future Work"