  Semi-decentralized federated learning blends the conventional device
to-server (D2S) interaction structure of federated model training with
localized device-to-device (D2D) communications. We study this architecture
over practical edge networks with multiple D2D clusters modeled as time-varying
and directed communication graphs. Our investigation results in an algorithm
that controls the fundamental trade-off between (a) the rate of convergence of
the model training process towards the global optimizer, and (b) the number of
D2S transmissions required for global aggregation. Specifically, in our
semi-decentralized methodology, D2D consensus updates are injected into the
federated averaging framework based on column-stochastic weight matrices that
encapsulate the connectivity within the clusters. To arrive at our algorithm,
we show how the expected optimality gap in the current global model depends on
the greatest two singular values of the weighted adjacency matrices (and hence
on the densities) of the D2D clusters. We then derive tight bounds on these
singular values in terms of the node degrees of the D2D clusters, and we use
the resulting expressions to design a threshold on the number of clients
required to participate in any given global aggregation round so as to ensure a
desired convergence rate. Simulations performed on real-world datasets reveal
that our connectivity-aware algorithm reduces the total communication cost
required to reach a target accuracy significantly compared with baselines
depending on the connectivity structure and the learning task.
