Accelerated Gradient Methods for Networked Optimization

TL;DR

This paper introduces multi-step gradient methods tailored for network-constrained optimization of strongly convex functions, achieving faster convergence by leveraging network topology and problem bounds, with applications in engineering.

Contribution

It develops accelerated gradient algorithms for networked optimization, providing parameter tuning strategies and analyzing performance under data uncertainty, outperforming standard methods.

Findings

Proposed methods converge faster than traditional gradient descent.

Performance gains are significant under network constraints.

Applications demonstrate rapid convergence in engineering problems.

Abstract

We develop multi-step gradient methods for network-constrained optimization of strongly convex functions with Lipschitz-continuous gradients. Given the topology of the underlying network and bounds on the Hessian of the objective function, we determine the algorithm parameters that guarantee the fastest convergence and characterize situations when significant speed-ups can be obtained over the standard gradient method. Furthermore, we quantify how the performance of the gradient method and its accelerated counterpart are affected by uncertainty in the problem data, and conclude that in most cases our proposed method outperforms gradient descent. Finally, we apply the proposed technique to three engineering problems: resource allocation under network-wide budget constraints, distributed averaging, and Internet congestion control. In all cases, we demonstrate that our algorithm converges more rapidly than alternative algorithms reported in the literature.