Structure, Analysis, and Synthesis of First-Order Algorithms

TL;DR

This paper presents a control-theoretic framework for analyzing and synthesizing first-order optimization algorithms, enabling their design over networks with dynamical phenomena like delays and crosstalk.

Contribution

It introduces a novel factorization approach using the internal model principle to synthesize robust optimization algorithms for networked systems.

Findings

Factorization of existing algorithms into network-dependent models

Automated synthesis of new algorithms with robustness to network effects

Achievement of exponential convergence rate through optimization techniques

Abstract

Optimization algorithms can be interpreted through the lens of dynamical systems as the interconnection of linear systems and a set of subgradient nonlinearities. This dynamical systems formulation allows for the analysis and synthesis of optimization algorithms by solving robust control problems. In this work, we use the celebrated internal model principle in control theory to structurally factorize convergent composite optimization algorithms into suitable network-dependent internal models and core subcontrollers. As the key benefit, we reveal that this permits us to synthesize optimization algorithms even if information is transmitted over networks featuring dynamical phenomena such as time delays, channel memory, or crosstalk. Design of these algorithms is achieved under bisection in the exponential convergence rate either through a nonconvex local search or by alternation of convex semidefinite programs. We demonstrate factorization of existing optimization algorithms and the automated synthesis of new optimization algorithms in the networked setting.