Distributed min-max optimization in networks

We consider a setup where we are given a network of agents with their local objective functions which are coupled through a common decision variable. We provide a distributed stochastic gradient algorithm for the agents to compute an optimal decision variable that minimizes the worst case loss incurred by any agent. We establish almost sure convergence of the agent's estimates to a common optimal point. We demonstrate the use of our algorithm to a problem of min-max fair power allocation in a cellular network.