  In vertical federated learning (FL), the features of a data sample are
distributed across multiple agents. As such, inter-agent collaboration can be
beneficial not only during the learning phase, as is the case for standard
horizontal FL, but also during the inference phase. A fundamental theoretical
question in this setting is how to quantify the cost, or performance loss, of
decentralization for learning and/or inference. In this paper, we consider
general supervised learning problems with any number of agents, and provide a
novel information-theoretic quantification of the cost of decentralization in
the presence of privacy constraints on inter-agent communication within a
Bayesian framework. The cost of decentralization for learning and/or inference
is shown to be quantified in terms of conditional mutual information terms
involving features and label variables.
