Distributed Algorithms for Optimization in Networks
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We will overview the distributed optimization algorithms starting with the basic underlying idea illustrated on a prototype problem in machine learning. In particular, we will focus on convex minimization problem where the objective function is given as the sum of convex functions, each of which is known by an agent in a network. The agents communicate over the network with a task to jointly determine a minimum of the sum of their objective functions. The communication network can vary over time, which is modeled through a sequence of graphs over a static set of nodes (representing the agents in a system). In this setting, the distributed first-order methods will be discussed that make use of an agreement protocol, which is a mechanism replacing the role of a coordinator. We will discuss some refinements of the basic method and conclude with more recent developments of fast methods that can match the performance of centralized methods (up to a logarithmic factor).