Gossip Algorithms for Convex Consensus Optimization over Networks

TL;DR

This paper introduces novel gossip algorithms, PE and PB, for convex consensus optimization over networks, offering advantages over traditional subgradient methods by ensuring convergence and flexibility in implementation.

Contribution

It presents two new non-gradient gossip algorithms, PE and PB, that improve convergence and flexibility for convex consensus problems in networked systems.

Findings

PE generalizes Pairwise Averaging and Gossip Algorithm.

Both algorithms are easy to implement and converge asymptotically.

PB relaxes sharing requirements, enhancing privacy and robustness.

Abstract

In many applications, nodes in a network desire not only a consensus, but an optimal one. To date, a family of subgradient algorithms have been proposed to solve this problem under general convexity assumptions. This paper shows that, for the scalar case and by assuming a bit more, novel non-gradient-based algorithms with appealing features can be constructed. Specifically, we develop Pairwise Equalizing (PE) and Pairwise Bisectioning (PB), two gossip algorithms that solve unconstrained, separable, convex consensus optimization problems over undirected networks with time-varying topologies, where each local function is strictly convex, continuously differentiable, and has a minimizer. We show that PE and PB are easy to implement, bypass limitations of the subgradient algorithms, and produce switched, nonlinear, networked dynamical systems that admit a common Lyapunov function and asymptotically converge. Moreover, PE generalizes the well-known Pairwise Averaging and Randomized Gossip Algorithm, while PB relaxes a requirement of PE, allowing nodes to never share their local functions.