  This paper addresses the problem of distributed detection in multi-agent
networks. Agents receive private signals about an unknown state of the world.
The underlying state is globally identifiable, yet informative signals may be
dispersed throughout the network. Using an optimization-based framework, we
develop an iterative local strategy for updating individual beliefs. In
contrast to the existing literature which focuses on asymptotic learning, we
provide a finite-time analysis. Furthermore, we introduce a Kullback-Leibler
cost to compare the efficiency of the algorithm to its centralized counterpart.
Our bounds on the cost are expressed in terms of network size, spectral gap,
centrality of each agent and relative entropy of agents' signal structures. A
key observation is that distributing more informative signals to central agents
results in a faster learning rate. Furthermore, optimizing the weights, we can
speed up learning by improving the spectral gap. We also quantify the effect of
link failures on learning speed in symmetric networks. We finally provide
numerical simulations which verify our theoretical results.
