A Generalized Framework for Social Learning Over State-Dependent Networks

Many social and distributed learning applications often exhibit highly correlated observations. In particular, knowledge of the underlying parameters is often insufficient to decouple observations statistically. This feature challenges the analysis of these learning systems and the design of learning rules. In many cases, this coupling can be effectively captured by an additional state variable. To this end, a new framework for social learning, based on the notion of state, is derived. The framework allows for extensions of several classical results for the conditionally independent case to the conditionally dependent case, considerably simplifying the design and analysis of decision rules. A numerical example focusing on Byzantine attacks in sensor networks is explored. Specifically, it is shown that the problem of learning under Byzantine attacks belongs to the proposed framework for some widely used attack models, highlighting the utility of the framework. Furthermore, the optimal decision rules for large networks are derived and shown to be superior to those computed under the assumption of conditionally independent observations.