Asynchronous Convex Consensus in the Presence of Crash Faults

TL;DR

This paper introduces convex consensus, a new problem where processes agree on a convex polytope within the convex hull of fault-free inputs, and provides an asynchronous algorithm with optimal fault tolerance.

Contribution

It defines convex consensus, explores its properties under crash faults, and presents an optimal fault-tolerant asynchronous algorithm for approximate convex consensus.

Findings

Achieves consensus on an optimal output polytope.

Handles crash faults with incorrect inputs.

Provides a foundation for solving convex function optimization.

Abstract

This paper defines a new consensus problem, convex consensus. Similar to vector consensus [13, 20, 19], the input at each process is a d-dimensional vector of reals (or, equivalently, a point in the d-dimensional Euclidean space). However, for convex consensus, the output at each process is a convex polytope contained within the convex hull of the inputs at the fault-free processes. We explore the convex consensus problem under crash faults with incorrect inputs, and present an asynchronous approximate convex consensus algorithm with optimal fault tolerance that reaches consensus on an optimal output polytope. Convex consensus can be used to solve other related problems. For instance, a solution for convex consensus trivially yields a solution for vector consensus. More importantly, convex consensus can potentially be used to solve other more interesting problems, such as convex function optimization [5, 4].