Fault-tolerant least squares solvers for wireless sensor networks based on gossiping

Abstract

Many applications in large loosely connected distributed networks (such as wireless sensor networks) require the distributed solution of linear least squares (dLLS) problems. Ideally, a truly distributed algorithm should require very little coordination between the nodes. This favours algorithms which do not require a fusion centre, cluster heads or any multi-hop communication.We present the novel dLLS solver GLS-IR for overdetermined linear systems. We investigate two variants of our novel solver, one of them based on the semi-normal equations, the other based on the normal equations. Both are combined with iterative refinement in mixed precision, which not only stabilises the methods but also decreases the communication cost. In GLS-IR, all communication between nodes is contained within a gossip-based algorithm for distributed aggregation, which limits the communication of each node to its immediate neighbourhood. Therefore, GLS-IR benefits directly from efficient and fault-tolerant algorithms for distributed aggregation. We use a fault-tolerant alternative to the push-sum method, the push-flow algorithm, which is able to recover from silent message loss and temporary or permanent link failures.We analytically compare the communication cost of GLS-IR to existing truly distributed algorithms. Since the theoretical analysis contains problem-dependent parameters, numerical experiments are needed in order to get a complete picture. Our simulation experiments illustrate a significantly reduced communication cost of GLS-IR compared to other existing truly distributed least squares solvers. We also illustrate that due to the properties of iterative refinement and push-flow, GLS-IR can achieve a result accurate to machine precision even if a high amount of message loss occurs.