Implementing Randomized Matrix Algorithms in Parallel and Distributed Environments.

In this era of large-scale data, distributed systems built on top of clusters of commodity hardware provide cheap and reliable storage and scalable processing of massive data. With cheap storage, instead of storing only currently relevant data, it is common to store as much data as possible, hoping that its value can be extracted later. In this way, exabytes (10 $^{18} $ bytes) of data are being created on a daily basis. Extracting value from these data, however, requires scalable implementations of advanced analytical algorithms beyond simple data processing, e.g., statistical regression methods, linear algebra, and optimization algorithms. Most such traditional methods are designed to minimize floating-point operations, which is the dominant cost of in-memory computation on a single machine. In parallel and distributed environments, however, load balancing and communication, including disk and network input/output (I/O) , can easily dominate computation. These factors greatly increase the complexity of algorithm design and challenge traditional ways of thinking about the design of parallel and distributed algorithms. Here, we review recent work on developing and implementing randomized matrix algorithms in large-scale parallel and distributed environments. Randomized algorithms for matrix problems have received a great deal of attention in recent years, thus far typically either in theory or in machine learning applications or with implementations on a single machine. Our main focus is on the underlying theory and practical implementation of random projection and random sampling algorithms for very large very overdetermined (i.e., overconstrained) $ell_{1}$- and $ell_{2} $-regression problems. Randomization can be used in one of two related ways: ei- her to construct subsampled problems that can be solved, exactly or approximately, with traditional numerical methods; or to construct preconditioned versions of the original full problem that are easier to solve with traditional iterative algorithms. Theoretical results demonstrate that in near input-sparsity time and with only a few passes through the data one can obtain very strong relative-error approximate solutions, with high probability. Empirical results highlight the importance of various tradeoffs (e.g., between the time to construct an embedding and the conditioning quality of the embedding, between the relative importance of computation versus communication, etc.) and demonstrate that $ell_{1}$- and $ell_{2} $-regression problems can be solved to low, medium, or high precision in existing distributed systems on up to terabyte-sized data.