  In this paper, we showcase the interplay between discrete and continuous
optimization in network-structured settings. We propose the first fully
decentralized optimization method for a wide class of non-convex objective
functions that possess a diminishing returns property. More specifically, given
an arbitrary connected network and a global continuous submodular function,
formed by a sum of local functions, we develop Decentralized Continuous Greedy
(DCG), a message passing algorithm that converges to the tight (1-1/e)
approximation factor of the optimum global solution using only local
computation and communication. We also provide strong convergence bounds as a
function of network size and spectral characteristics of the underlying
topology. Interestingly, DCG readily provides a simple recipe for decentralized
discrete submodular maximization through the means of continuous relaxations.
Formally, we demonstrate that by lifting the local discrete functions to
continuous domains and using DCG as an interface we can develop a consensus
algorithm that also achieves the tight (1-1/e) approximation guarantee of the
global discrete solution once a proper rounding scheme is applied.
