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Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

PortabilityandScalabilityofSparseTensorDecompositionsonCPU/MIC/GPUArchitectures

ChristopherForster,KeitaTeranishi,GregMackey,DanielDunlavy andTamaraKolda

SAND2017-6575C

SparseTensorDecomposition

§ DevelopproductionqualitylibrarysoftwaretoperformCPfactorizationwithAlternatingPoissonRegression onHPCplatforms§ SparTen

§ SupportseveralHPCplatforms§ Nodeparallelism(Multicore,Manycore andGPUs)

§ MajorQuestions§ SoftwareDesign§ PerformanceTuning

§ Thistalk§ Weareinterestedintwomajorvariants

§ MultiplicativeUpdates§ ProjectedDampedNewtonforRow-subproblems

2

CPTensorDecomposition

3

§ Expresstheimportantfeatureofdatausingasmallnumberofvectorouterproducts

Key references: Hitchcock (1927), Harshman (1970), Carroll and Chang (1970)

CANDECOMP/PARAFAC (CP) Model

Model:

PoissonforSparseCountData

4

Gaussian (typical) PoissonThe random variable x is a

continuous real-valued number.The random variable x is a

discrete nonnegative integer.

Model: Poisson distribution (nonnegative factorization)

SparsePoissonTensorFactorization

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§ Nonconvexproblem!§ AssumeRisgiven

§ Minimizationproblemwithconstraint§ Thedecomposedvectorsmustbenon-negative

§ AlternatingPoissonRegression(ChiandKolda,2011)§ Assume(d-1)factormatricesareknownandsolvefortheremainingone

NewMethod:AlternatingPoissonRegression(CP-APR)

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Repeat until converged…

Fix B,C;solveforA

Fix A,C;solvefor B

Fix A,B;solvefor C

Theorem: The CP-APR algorithm will converge to a constrained stationary point if the subproblems are strictly convex and solved exactly at each iteration. (Chi and Kolda, 2011)

Convergence Theory

CP-APR

7

Minimization problem is expressed as:

CP-APR

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Minimization problem is expressed as:

• 2 major approaches• Multiplicative Updates like Lee & Seung

(2000) for matrices, but extended by Chi and Kolda (2011) for tensors

• Newton and Quasi-Newton method for Row-subpblems by Hansen, Plantenga and Kolda(2014)

KeyElementsofMUandPDNRmethods

§ Keycomputations§ Khatri-RaoProduct§ Modifier(10+iterations)

§ Keyfeatures§ Factormatrixisupdatedallatonce

§ Exploitstheconvexityofrowsubproblems forglobalconvergence

§ Keycomputations§ Khatri-RaoProduct§ ConstrainedNon-linearNewton-basedoptimizationforeachrow

§ Keyfeatures§ Factormatrixcanbe

updatedbyrows§ Exploitstheconvexityof

row-subproblems

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Multiplicative Update (MU) Projected Damped Newton for Row-subproblems (PDNR)

CP-APR-MU

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Key Computations

CP-APR-PDNR

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Key Computations

Algorithm 1: CPAPR-PDNR algorithm

1 CPAPR PDNR (X ,M);

Input : Sparse N -mode Tensor X of size I

1

⇥ I

2

⇥ . . . IN and the

number of components R

Output: Kruskal Tensor M = [�;A

(1)

. . . A

(N)

]

2 Initialize3 repeat4 for n = 1, . . . , N do5 Let ⇧

(n)= (A

(N) � · · ·�A

(n+1) �A

(n�1) � . . . A

(1)

)

T

6 for i = 1, . . . , In do

7 Find b

(n)i s.t. min

b(n)i �0

f

row

(b

(n)i , x

(n)i ,⇧

(n))

8 end

9 � = e

TB

(n)where B

(n)= [b

(n)1

. . . b

(n)In

]

T

10 A

(n) B

(n)⇤

�1

, where ⇤ = diag(�)

11 end

12 until all mode subproblems converged ;

PARALLELCP-APRALGORITHMS

12

ParallelizingCP-APR§ Focusonon-nodeparallelismformultiplearchitectures

§ Multiplechoicesforprogramming§ OpenMP,OpenACC, CUDA,Pthread …§ Managedifferentlow-levelhardwarefeatures(cache,devicememory,NUMA…)

§ OurSolution:UseKokkos forproductivityandperformanceportability§ Abstractionofparallelloops§ AbstractionDatalayout(row-major,columnmajor,programmablememory)§ Samecodetosupportmultiplearchitectures

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Kokkos

Intel Multicore Intel Manycore NVIDIA GPU IBM PowerAMD Multicore/APU ARM

Support multiple Architectures

WhatisKokkos?

§ TemplatedC++LibrarybySandiaNationalLabs(Edwards,etal)§ Serveassubstratelayerofsparsematrixandvectorkernels§ Supportanymachineprecisions

§ Float§ Double§ QuadandHalffloatifneeded.

§ Kokkos::View()accommodatesperformance-awaremultidimensionalarraydataobjects§ Light-weightC++classto

§ ParallelizingloopsusingC++languagestandard§ Lambda§ Functors

§ Extensivesupportofatomics

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ParallelProgramingwithKokkos

§ ProvideparallelloopoperationsusingC++languagefeatures§ Conceptually,theusageisnomoredifficultthanOpenMP.

Theannotationsjustgoindifferentplaces. 15

for (size_t i = 0; i < N; ++i) {

/* loop body */}

#pragma omp parallel forfor (size_t i = 0; i < N; ++i) {

/* loop body */}

parallel_for (( N, [=], (const size_t i) {

/* loop body */});

Seria

lO

penM

PKo

kkos

Kokkos information courtesy of Carter Edwards

WhyKokkos?

§ ComplyC++languagestandard!§ Supportmultipleback-ends

§ Pthread,OpenMP,CUDA,IntelTBBandQthread

§ Supportmultipledatalayoutoptions§ ColumnvsRowMajor§ Device/CPUmemory

§ Supportdifferentparallelism§ Nestingsupport§ Vector,threads,Warp,etc.§ Taskparallelism(underdevelopment)

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ArrayAccessbyKokkos

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Row-majorThread 0 reads

Thread 1 reads

Column-major

Thread 0 reads

Thread 1 reads

Kokkos::View<double **, Layout, Space>

View<double **, Right, Space> View<double **, Left, Space>

ArrayAccessbyKokkos

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Row-major

Thread 0 reads

Thread 1 reads

Contiguous reads per thread

Column-major

Thread 0 reads

Thread 1 reads

Coalesced reads w

ithin warp

View<double **, Right, Host> View<double **, Left, CUDA>

Kokkos::View<double **, Layout, Space>

ParallelCP-APR-MU

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Data Parallel

ParallelCP-APR-PDNR

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Data Parallel

Task Parallel

NotesonDataStructure

§ UseKokkos::View§ SparseTensor

§ SimilartotheCoordinate(COO)FormatinSparseMatrixrepresentation

§ Kruskal Tensor&KhatriRaoProduct§ Providesoptionsforroworcolumnmajor

§ Kokkos::Viewprovidesanoptiontodefinetheleadingdimension.§ Determinedduringcompileorruntime

§ AvoidAtomics§ ExpensiveinCPUsandManycore§ Useextraindexingdatastructure

§ CP-APR-PDNR§ Createsapooloftasks§ Adedicatedbufferspace(Kokkos::View)isassignedtoindividualtask

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PERFORMANCE

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PerformanceTest§ StrongScalability

§ Problemsizeisfixed§ RandomTensor

§ 3Kx4Kx5K,10Mnonzeroentries§ 100outeriterations

§ RealisticProblems§ CountData(Non-negative)§ Availableathttp://frostt.io/§ 10outeriterations

§ DoublePrecision

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Data Dimensions Nonzeros RankLBNL 2K x 4K x 2K x 4K x 866K 1.7M 10NELL-2 12K x 9K x 29K 77M 10NELL-1 3M x 2M x 25M 144M 10Delicious 500K x 17M x 3M x 1K 140M 10

CPAPR-MUonCPU(Random)

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0

200

400

600

800

1000

1200

1400

1600

1800

1 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Exec

utio

n Ti

me

(s)

Core Count

CP-APR-MU method, 100 outer-iterations, (3000 x 4000 x 5000, 10M nonzero entries), R=10, PC cluster, 2 Haswell (14 core) CPUs

per node, MKL-11.3.3, HyperThreading disabled

Pi Phi+ Update

Results:CPAPR-MUScalability

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DataHaswell CPU

1-core

2 Haswell CPUs

14-cores

2 Haswell CPUs

28-coresKNL

68-core CPUNVIDIA

P100 GPUTime(s) Speedup Time(s) Speedup Time(s) Speedup Time(s) Speedup Time(s) Speedup

Random 1715* 1 279 6.14 165 10.39 20 85.74 10 171.5LBNL 131 1 32 4.09 32 4.09 103 1.27NELL-2 1226 1 159 7.77 92 13.32 873 1.40NELL-1 5410 1 569 9.51 349 15.50 1690 3.20Delicious 5761 1 2542 2.26 2524 2.28

100 outer iterations for the random problem10 outer iterations for realistic problems* Pre-Kokkos C++ code on 2 Haswell CPUs:

1-core, 2136 sec14-cores, 762 sec28-cores, 538 sec

CPAPR-PDNRonCPU(Random)

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0100200300400500600700800900

1 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Exec

utio

n Ti

me

(s)

Core Count

CPAPR-PDNR method, 100 outer-iterations, 1831221 inner iterations total, (3000 x 4000 x 5000, 10M nonzero entries), R=10,

PC cluster, 2 Haswell (14 core) CPUs per node, MKL-11.3.3, HyperThreading disabled

Pi RowSub

Results:CPAPR-PDNRScalability

27

DataHaswell CPU

1 core2 Haswell CPUs

14 cores2 Haswell CPUs

28 coresTime(s) Speedup Time(s) Speedup Time(s) Speedup

Random 817* 1 73 11.19 44 18.58

LBNL 441 1 187 2.35 191 2.30

NELL-2 2162 1 326 6.63 319 6.77

NELL-1 17212 1 4241 4.05 3974 4.33

Delicious 18992 1 3684 5.15 3138 6.05

100 outer iterations for the random problem10 outer iterations for realistic problems* Pre-Kokkos C++ code spends 3270 sec on 1 core

PerformanceIssues

§ Ourimplementationexhibitsverygoodscalabilitywiththerandomtensor.§ Similarmodesizes§ Regulardistributionofnonzeroentries

§ Somecacheeffects§ Kokkos isNUMA-awareforcontiguousmemoryaccess(first-touch)

§ Somescalabilityissueswiththerealistictensorproblems.§ Irregularnonzerodistributionanddisparityinmodesizes§ Task-parallelcodemayhavesomememorylocalityissuestoaccess

sparsetensor,Kruskal Tensor,andKhatori-Raoproduct§ Preprocessingcouldimprovethelocality

§ ExplicitDatapartitioning(SmithandKarypis)§ PossibletoimplementusingKokkos

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MemoryBandwidth(StreamBenchmark)

§ Allcoresdeliverapproximately8xperformanceimprovementfromsinglethread

§ Hardtoscaleusingallcoreswithmemory-boundcode.

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0

20000

40000

60000

80000

100000

120000

0 5 10 15 20 25 30

MBy

tes/

Sec

# of Cores

Stream Benchmark on 2x 14 core Intel Haswell CPUs

Conclusion

§ DevelopmentofPortableon-nodeParallelCP-APRSolvers§ DataparallelismforMUmethod§ MixedData/TaskparallelismforPDNRmethod§ MultipleArchitectureSupportusingKokkos

§ ScalablePerformanceforrandomsparsetensor

§ FutureWork§ ProjectedQuasi-NewtonforRow-subproblems (PQNR)§ GPUandManycore supportforPDNRandPQNR§ Performancetuningtohandleirregularnonzerodistributionsand

disparityinmodesizes

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