QR decomposition of a matrix is one of the essential operations that is used for solving linear equations and finding least-squares solutions. We propose a coded computing strategy for parallel QR decomposition with applications to the high- performance computing (HPC) setup. Our strategy is based on the parallel Gram-Schmidt algorithm, which is one of the three commonly used algorithms for QR decomposition. Protecting QR decomposition from failures consists of two parts: protecting the left factor Q (orthogonal) and the right factor R (upper- triangular). For protecting Q, we prove a condition for a generator matrix to preserve the orthogonality of the matrix Q and construct a generator matrix for single-node failures. For protecting R, we derive a lower bound on the amount of checksums required under the in-node checksum storage setting, where checksums are stored in original nodes, and propose an encoding scheme that meets the lower bound.
This work considers the distributed multivariate polynomial evaluation (DMPE) problem using a master-worker framework, which was originally considered by Yu et al., where Lagrange Coded Computing (LCC) was proposed as a coded computation scheme to provide resilience against stragglers for the DMPE problem. In this work, we propose a variant of the LCC scheme, termed Product Lagrange Coded Computing (PLCC), by combining ideas from classical product codes and LCC. The main advantage of PLCC is that they are more numerically stable than LCC; however, their resilience to stragglers is sub-optimal.
Today's blockchain designs suffer from a trilemma claiming that no blockchain system can simultaneously achieve decentralization, security, and performance scalability. For current blockchain systems, as more nodes join the network, the efficiency of the system (computation, communication, and storage) stays constant at best. A leading idea for enabling blockchains to scale efficiency is the notion of sharding: different subsets of nodes handle different portions of the blockchain, thereby reducing the load for each individual node. However, existing sharding proposals achieve efficiency scaling by compromising on trust - corrupting the nodes in a given shard will lead to the permanent loss of the corresponding portion of data. In this paper, we settle the trilemma by demonstrating a new protocol for coded storage and computation in blockchains. In particular, we propose PolyShard: ``polynomially coded sharding'' scheme that achieves information-theoretic upper bounds on the efficiency of the storage, system throughput, as well as on trust, thus enabling a truly scalable system.
The alternating direction method of multipliers (ADMM) has recently been recognized as a promising approach for large-scale machine learning models. However, very few results study ADMM from the aspect of communication costs, especially jointly with privacy preservation. We investigate the communication efficiency and privacy of ADMM in solving the consensus optimization problem over decentralized networks. We first propose incremental ADMM (I-ADMM), the updating order of which follows a Hamiltonian cycle. To protect privacy for agents against external eavesdroppers, we investigate I-ADMM with privacy preservation, where randomized initialization and step size perturbation are adopted. Using numerical results from simulations, we demonstrate that the proposed I-ADMM with step size perturbation can be both communication efficient and privacy preserving.