# Communication-Efficient Algorithms For Distributed Optimization

This thesis is concerned with the design of distributed algorithms for solving optimization problems. The particular scenario we consider is a network with P compute nodes, where each node p has exclusive access to a cost function *f*_{p}. We design algorithms in which all the nodes cooperate to find the minimum of the sum of all the cost functions,* f*_{1} + · · · + *f*_{P} . Several problems in signal processing, control, and machine learning can be posed as such optimization problems. Given that communication is often the most energy-consuming operation in networks and, many times, also the slowest one, it is important to design distributed algorithms with low communication requirements, that is, communication-efficient algorithms. The two main contributions of this thesis are a classification scheme for distributed optimization problems of the kind explained above and a set of corresponding communication-efficient algorithms.

The class of optimization problems we consider is quite general, since we allow that each function may depend on arbitrary components of the optimization variable, and not necessarily on all of them. In doing so, we go beyond the commonly used assumption in distributed optimization and create additional structure that can be explored to reduce the total number of communications. This structure is captured by our classification scheme, which identifies particular instances of the problem that are easier to solve. One example is the standard distributed optimization problem, in which all the functions depend on all the components of the variable.

All our algorithms are distributed in the sense that no central node coordinates the network, all the communications occur exclusively between neighboring nodes, and the data associated with each node is always processed locally. We show several applications of our algorithms, including average consensus, support vector machines, network flows, and several distributed scenarios for compressed sensing. We also propose a new framework for distributed model predictive control, which can be solved with our algorithms. Through extensive numerical experiments, we show that our algorithms outperform prior distributed algorithms in terms of communication-efficiency, even some that were specifically designed for a particular application.