DISH: A Distributed Hybrid Primal-Dual Optimization Framework to Utilize System Heterogeneity

TL;DR

DISH is a novel distributed optimization framework that leverages system heterogeneity by combining Newton and gradient updates, achieving linear convergence for strongly convex problems.

Contribution

DISH introduces a hybrid primal-dual approach that handles agent heterogeneity and unifies existing methods, with proven convergence guarantees.

Findings

DISH achieves linear convergence rate for strongly convex functions.

DISH generalizes and includes methods like EXTRA, DIGing, and ESOM-0.

Numerical results confirm practical effectiveness.

Abstract

We consider solving distributed consensus optimization problems over multi-agent networks. Current distributed methods fail to capture the heterogeneity among agents' local computation capacities. We propose DISH as a distributed hybrid primal-dual algorithmic framework to handle and utilize system heterogeneity. Specifically, DISH allows those agents with higher computational capabilities or cheaper computational costs to implement Newton-type updates locally, while other agents can adopt the much simpler gradient-type updates. We show that DISH is a general framework and includes EXTRA, DIGing, and ESOM-0 as special cases. Moreover, when all agents take both primal and dual Newton-type updates, DISH approximates Newton's method by estimating both primal and dual Hessians. Theoretically, we show that DISH achieves a linear (Q-linear) convergence rate to the exact optimal solution for strongly convex functions, regardless of agents' choices of gradient-type and Newton-type updates. Finally, we perform numerical studies to demonstrate the efficacy of DISH in practice. To the best of our knowledge, DISH is the first hybrid method allowing heterogeneous local updates for distributed consensus optimization under general network topology with provable convergence and rate guarantees.