  This work proposes a novel strategy for social learning by introducing the
critical feature of adaptation. In social learning, several distributed agents
update continually their belief about a phenomenon of interest through: i)
direct observation of streaming data that they gather locally; and ii)
diffusion of their beliefs through local cooperation with their neighbors.
Traditional social learning implementations are known to learn well the
underlying hypothesis (which means that the belief of every individual agent
peaks at the true hypothesis), achieving steady improvement in the learning
accuracy under stationary conditions. However, these algorithms do not perform
well under nonstationary conditions commonly encountered in online learning,
exhibiting a significant inertia to track drifts in the streaming data. In
order to address this gap, we propose an Adaptive Social Learning (ASL)
strategy, which relies on a small step-size parameter to tune the adaptation
degree. First, we provide a detailed characterization of the learning
performance by means of a steady-state analysis. Focusing on the small
step-size regime, we establish that the ASL strategy achieves consistent
learning under standard global identifiability assumptions. We derive reliable
Gaussian approximations for the probability of error (i.e., of choosing a wrong
hypothesis) at each individual agent. We carry out a large deviations analysis
revealing the universal behavior of adaptive social learning: the error
probabilities decrease exponentially fast with the inverse of the step-size,
and we characterize the resulting exponential learning rate. Second, we
characterize the adaptation performance by means of a detailed transient
analysis, which allows us to obtain useful analytical formulas relating the
adaptation time to the step-size.
