Asynchronous Convex Consensus in the Presence of Crash Faults

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].

💡 Research Summary

The paper introduces a novel distributed agreement problem called convex consensus, which generalizes the well‑studied vector consensus. In convex consensus each process starts with a d‑dimensional real vector (a point in ℝ^d) and the goal is not a single point but a convex polytope that lies inside the convex hull of the inputs supplied by all fault‑free processes. The output polytope should be as large as possible, i.e., it should contain as much of the true convex hull as the fault model permits.

The authors work in an asynchronous complete‑graph model where up to f processes may crash and, in addition, may provide arbitrarily incorrect inputs. This “crash‑with‑incorrect‑inputs” model is weaker than the Byzantine model but stronger than the classic crash‑with‑correct‑inputs model, because a faulty process can corrupt the geometric information even though it follows the algorithm faithfully until it crashes. As in the classic FLP result, exact consensus is impossible under these conditions, so the paper targets approximate consensus: all correct processes must output polytopes whose Hausdorff distance is at most a pre‑specified ε > 0.

The main contribution is Algorithm CC, an asynchronous round‑based protocol that achieves ε‑approximate convex consensus with optimal fault tolerance. The algorithm works as follows:

• Round 0 (initialization) – Each process broadcasts its input vector (x_i, i, 0). Using a stable‑vector primitive (originally designed for Byzantine settings) each correct process obtains a set R_i containing at least n − f distinct input tuples. From R_i the process builds a multiset X_i of the received points. It then computes the intersection of the convex hulls of all size‑|X_i| − f subsets of X_i. This intersection, denoted h_i