Provably Efficient Algorithms for Numerical Tensor Algebra

Edgar Solomonik

EECS Department
University of California, Berkeley
Technical Report No. UCB/EECS-2014-170
September 30, 2014

http://www2.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-170.pdf

This thesis targets the design of parallelizable algorithms and communication-efficient parallel schedules for numerical linear algebra as well as computations with higher-order tensors. Communication is a growing bottleneck in the execution of most algorithms on parallel computers, which manifests itself as data movement both through the network connecting different processors and through the memory hierarchy of each processor as well as synchronization between processors. We provide a rigorous theoretical model of communication and derive lower bounds as well as algorithms in this model. Our analysis concerns two broad areas of linear algebra and of tensor contractions. We demonstrate the practical quality of the new theoretically-improved algorithms by presenting results which show that our implementations outperform standard libraries and traditional algorithms.

We model the costs associated with local computation, communication, and synchronization. We introduce a new technique for deriving lower bounds on tradeoffs between these costs and apply them to algorithms in both dense and sparse linear algebra as well as graph algorithms. These lower bounds are attained by what we refer to as 2.5D algorithms, which we give for matrix multiplication, Gaussian elimination, QR factorization, the symmetric eigenvalue problem, and the Floyd-Warshall all-pairs shortest-paths algorithm. 2.5D algorithms achieve lower interprocessor bandwidth cost by exploiting auxiliary memory. Algorithms employing this technique are well known for matrix multiplication, and have been derived in the BSP model for LU and QR factorization, as well as the Floyd-Warshall algorithm. We introduce alternate versions of LU and QR algorithms which have measurable performance improvements over their BSP counterparts, and we give the first evaluations of their performance. For the symmetric eigenvalue problem, we give the first 2.5D algorithms, additionally solving challenges with memory-bandwidth efficiency that arise for this problem. We also give a new memory-bandwidth efficient algorithm for Krylov subspace methods (repeated multiplication of a vector by a sparse-matrix).

The latter half of the thesis contains algorithms for higher-order tensors, in particular tensor contractions. We introduce Cyclops Tensor Framework, which provides an automated mechanism for network-topology-aware decomposition and redistribution of tensor data. It leverages 2.5D matrix multiplication to perform tensor contractions communication-efficiently. The framework is capable of exploiting symmetry and antisymmetry in tensors and utilizes a distributed packed-symmetric storage format. Finally, we consider a theoretically novel technique for exploiting tensor symmetry to lower the number of multiplications necessary to perform a contraction via computing some redundant terms that allow preservation of symmetry and then cancelling them out with low-order cost. We analyze the numerical stability and communication efficiency of this technique and give adaptations to antisymmetric and Hermitian matrices. This technique has promising potential for accelerating coupled-cluster (electronic structure) methods both in terms of computation and communication cost.

Advisor: James Demmel


BibTeX citation:

@phdthesis{Solomonik:EECS-2014-170,
    Author = {Solomonik, Edgar},
    Title = {Provably Efficient Algorithms for Numerical Tensor Algebra},
    School = {EECS Department, University of California, Berkeley},
    Year = {2014},
    Month = {Sep},
    URL = {http://www2.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-170.html},
    Number = {UCB/EECS-2014-170},
    Abstract = {This thesis targets the design of parallelizable algorithms and communication-efficient parallel schedules for numerical linear algebra as well as computations with higher-order tensors. Communication is a growing bottleneck in the execution of most algorithms on parallel computers, which manifests itself as data movement both through the network connecting different processors and through the memory hierarchy of each processor as well as synchronization between processors. We provide a rigorous theoretical model of communication and derive lower bounds as well as algorithms in this model. Our analysis concerns two broad areas of linear algebra and of tensor contractions. We demonstrate the practical quality of the new theoretically-improved algorithms by presenting results which show that our implementations outperform standard libraries and traditional algorithms.

We model the costs associated with local computation, communication, and synchronization. We introduce a new technique for deriving lower bounds on tradeoffs between these costs and apply them to algorithms in both dense and sparse linear algebra as well as graph algorithms. These lower bounds are attained by what we refer to as 2.5D algorithms, which we give for matrix multiplication, Gaussian elimination, QR factorization, the symmetric eigenvalue problem, and the Floyd-Warshall all-pairs shortest-paths algorithm. 2.5D algorithms achieve lower interprocessor bandwidth cost by exploiting auxiliary memory. Algorithms employing this technique are well known for matrix multiplication, and have been derived in the BSP model for LU and QR factorization, as well as the Floyd-Warshall algorithm. We introduce alternate versions of LU and QR algorithms which have measurable performance improvements over their BSP counterparts, and we give the first evaluations of their performance. For the symmetric eigenvalue problem, we give the first 2.5D algorithms, additionally solving challenges with memory-bandwidth efficiency that arise for this problem. We also give a new memory-bandwidth efficient algorithm for Krylov subspace methods (repeated multiplication of a vector by a sparse-matrix).

The latter half of the thesis contains algorithms for higher-order tensors, in particular tensor contractions. We introduce Cyclops Tensor Framework, which provides an automated mechanism for network-topology-aware decomposition and redistribution of tensor data. It leverages 2.5D matrix multiplication to perform tensor contractions communication-efficiently. The framework is capable of exploiting symmetry and antisymmetry in tensors and utilizes a distributed packed-symmetric storage format. Finally, we consider a theoretically novel technique for exploiting tensor symmetry to lower the number of multiplications necessary to perform a contraction via computing some redundant terms that allow preservation of symmetry and then cancelling them out with low-order cost. We analyze the numerical stability and communication efficiency of this technique and give adaptations to antisymmetric and Hermitian matrices. This technique has promising potential for accelerating coupled-cluster (electronic structure) methods both in terms of computation and communication cost.}
}

EndNote citation:

%0 Thesis
%A Solomonik, Edgar
%T Provably Efficient Algorithms for Numerical Tensor Algebra
%I EECS Department, University of California, Berkeley
%D 2014
%8 September 30
%@ UCB/EECS-2014-170
%U http://www2.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-170.html
%F Solomonik:EECS-2014-170