  This work examines a social learning problem, where dispersed agents
connected through a network topology interact locally to form their opinions
(beliefs) as regards certain hypotheses of interest. These opinions evolve over
time, since the agents collect observations from the environment, and update
their current beliefs by accounting for: their past beliefs, the innovation
contained in the new data, and the beliefs received from the neighbors. The
distinguishing feature of the present work is that agents are constrained to
share opinions regarding only a single hypothesis. We devise a novel learning
strategy where each agent forms a valid belief by completing the partial
beliefs received from its neighbors. This completion is performed by exploiting
the knowledge accumulated in the past beliefs, thanks to a principled
memory-aware rule inspired by a Bayesian criterion. The analysis allows us to
characterize the role of memory in social learning under partial information
sharing, revealing novel and nontrivial learning dynamics. Surprisingly, we
establish that the standard classification rule based on selecting the maximum
belief is not optimal under partial information sharing, while there exists a
consistent threshold-based decision rule that allows each agent to classify
correctly the hypothesis of interest. We also show that the proposed strategy
outperforms previously considered schemes, highlighting that the introduction
of memory in the social learning algorithm is critical to overcome the
limitations arising from sharing partial information.
