Belief Samples Are All You Need For Social Learning

In this paper, we consider the problem of social learning, where a group of agents embedded in a social network are interested in learning an underlying state of the world. Agents have incomplete, noisy, and heterogeneous sources of information, providing them with recurring private observations of the underlying state of the world. Agents can share their learning experience with their peers by taking actions observable to them, with values from a finite feasible set of states. Actions can be interpreted as samples from the beliefs which agents may form and update on what the true state of the world is. Sharing samples, in place of full beliefs, is motivated by the limited communication, cognitive, and information-processing resources available to agents especially in large populations. Previous work (Salhab et al.) poses the question as to whether learning with probability one is still achievable if agents are only allowed to communicate samples from their beliefs. We provide a definite positive answer to this question, assuming a strongly connected network and a ``collective distinguishability'' assumption, which are both required for learning even in full-belief-sharing settings. In our proposed belief update mechanism, each agent's belief is a normalized weighted geometric interpolation between a fully Bayesian private belief -- aggregating information from the private source -- and an ensemble of empirical distributions of the samples shared by her neighbors over time. By carefully constructing asymptotic almost-sure lower/upper bounds on the frequency of shared samples matching the true state/or not, we rigorously prove the convergence of all the beliefs to the true state, with probability one.