Robust Consensus in Distributed Networks using Total Variation

TL;DR

This paper introduces a distributed optimization method using total variation regularization for consensus problems in networks, offering robustness to unreliable agents and broad applicability beyond average consensus.

Contribution

It proposes a novel distributed algorithm based on total variation regularization that achieves robust consensus and is effective for various consensus objectives.

Findings

Algorithms are efficient in simulations.

Optimal solutions are consistent with initial objectives under simple conditions.

Method is robust to unreliable agents injecting false data.

Abstract

Consider a connected network of agents endowed with local cost functions representing private objectives. Agents seek to find an agreement on some minimizer of the aggregate cost, by means of repeated communications between neighbors. Consensus on the average over the network, usually addressed by gossip algorithms, is a special instance of this problem, corresponding to quadratic private objectives. Consensus on the median, or more generally quantiles, is also a special instance, as many more consensus problems. In this paper we show that optimizing the aggregate cost function regularized by a total variation term has appealing properties. First, it can be done very naturally in a distributed way, yielding algorithms that are efficient on numerical simulations. Secondly, the optimum for the regularized cost is shown to be also the optimum for the initial aggregate cost function under assumptions that are simple to state and easily verifiable. Finally, these algorithms are robust to unreliable agents that keep injecting some false value in the network. This is remarkable enough, and is not the case, for instance, of gossip algorithms, that are entirely ruled by unreliable agents as detailed in the paper.