A Chebyshev-Accelerated Primal-Dual Method for Distributed Optimization

TL;DR

This paper introduces a Chebyshev-accelerated primal-dual algorithm for distributed optimization, significantly improving convergence speed over networks with challenging spectral properties by leveraging Chebyshev polynomials.

Contribution

It presents a novel distributed primal-dual method that uses Chebyshev polynomials to accelerate convergence, especially in networks with poor spectral properties.

Findings

Faster ergodic convergence rates achieved.

Requires fewer gradient updates with more communication rounds.

Effective in distributed signal recovery tasks.

Abstract

We consider a distributed optimization problem over a network of agents aiming to minimize a global objective function that is the sum of local convex and composite cost functions. To this end, we propose a distributed Chebyshev-accelerated primal-dual algorithm to achieve faster ergodic convergence rates. In standard distributed primal-dual algorithms, the speed of convergence towards a global optimum (i.e., a saddle point in the corresponding Lagrangian function) is directly influenced by the eigenvalues of the Laplacian matrix representing the communication graph. In this paper, we use Chebyshev matrix polynomials to generate gossip matrices whose spectral properties result in faster convergence speeds, while allowing for a fully distributed implementation. As a result, the proposed algorithm requires fewer gradient updates at the cost of additional rounds of communications between agents. We illustrate the performance of the proposed algorithm in a distributed signal recovery problem. Our simulations show how the use of Chebyshev matrix polynomials can be used to improve the convergence speed of a primal-dual algorithm over communication networks, especially in networks with poor spectral properties, by trading local computation by communication rounds.