Essentially Decentralized Conjugate Gradients

Abstract: Solving structured systems of linear equations in a non-centralized fashion is an important step in many distributed optimization and control algorithms. Fast convergence is required in manifold applications. Known decentralized algorithms, however, typically exhibit asymptotic convergence at a linear rate. This note proposes an essentially decentralized variant of the Conjugate Gradient algorithm (d-CG). The proposed method exhibits a practical superlinear convergence rate and comes with a priori computable finite-step convergence guarantees. In contrast to previous works, we consider sum-wise decomposition instead of row-wise decomposition which enables application in multi-agent settings. We illustrate the performance of d-CG on problems from sensor fusion and compare the results to the widely-used Alternating Direction Method of Multipliers.