Distributed optimization in an energy-constrained network

We consider a distributed optimization problem whereby a network of N nodes, S&ell, &ell &isin {1, ..., N }, wish to minimize a common strongly convex function f(x), x = [x1, ..., x N]T, under the constraint that node S&ell controls variable x&ell only. The nodes locally update their respective variables and periodically exchange their values over a set of pre-defined communication channels. Previous studies of this problem have focused mainly on the convergence issue and the analysis of convergence rate. In this work, we consider noisy communication channels and study the impact of communication energy on convergence. In particular, we study the minimum amount of communication energy required for nodes to obtain an epsilon-minimizer of f(x) in the mean square sense. For linear analog communication schemes, we prove that the communication energy to obtain an epsilon-minimizer of f(x) must grow at least at the rate of O(1/epsilon), and this bound is tight when f is convex quadratic. Furthermore, we show that the same energy requirement can be reduced to O (log2 1/epsilon) if suitably designed digital communication schemes are used.