Automatic Performance Tuning and Sparse-Matrix-Vector-Multiplication (SpMV)

1
Automatic Performance TuningandSparse-Matrix-Vec
tor-Multiplication (SpMV)

• James Demmel
• www.cs.berkeley.edu/demmel/cs267_Spr10

2
Berkeley Benchmarking and OPtimization (BeBOP)

• Prof. Katherine Yelick
• Current members
• Kaushik Datta, Ozan Demirlioglu, Mark Hoemmen,
  Shoaib Kamil, Rajesh Nishtala, Vasily Volkov, Sam
  Williams,
• Previous members
• Hormozd Gahvari, Eun-Jim Im, Ankit Jain, Rich
  Vuduc, many undergrads
• Many results here from current, previous students
• bebop.cs.berkeley.edu

3
Outline

• Motivation for Automatic Performance Tuning
• Results for sparse matrix kernels
• OSKI Optimized Sparse Kernel Interface
• Tuning Higher Level Algorithms
• Future Work, Class Projects
• Future Related Lecture
• Sam Williams on tuning SpMV for multicore,
  other emerging architectures
• BeBOP Berkeley Benchmarking and Optimization
  Group
• Meet weekly T 1-2, in 380 Soda

4
Motivation for Automatic Performance Tuning

• Writing high performance software is hard
• Make programming easier while getting high speed
• Ideal program in your favorite high level
  language (Matlab, Python, PETSc) and get a high
  fraction of peak performance
• Reality Best algorithm (and its implementation)
  can depend strongly on the problem, computer
  architecture, compiler,
• Best choice can depend on knowing a lot of
  applied mathematics and computer science
• How much of this can we teach?
• How much of this can we automate?

5
Examples of Automatic Performance Tuning (1)

• Dense BLAS
• Sequential
• PHiPAC (UCB), then ATLAS (UTK) (used in Matlab)
• math-atlas.sourceforge.net/
• Internal vendor tools
• Fast Fourier Transform (FFT) variations
• Sequential and Parallel
• FFTW (MIT)
• www.fftw.org
• Digital Signal Processing
• SPIRAL www.spiral.net (CMU)
• Communication Collectives (UCB, UTK)
• Rose (LLNL), Bernoulli (Cornell), Telescoping
  Languages (Rice),
• More projects, conferences, government reports,

6
Examples of Automatic Performance Tuning (2)

• What do dense BLAS, FFTs, signal processing, MPI
  reductions have in common?
• Can do the tuning off-line once per
  architecture, algorithm
• Can take as much time as necessary (hours, a
  week)
• At run-time, algorithm choice may depend only on
  few parameters
• Matrix dimension, size of FFT, etc.

7
Tuning Register Tile Sizes (Dense Matrix Multiply)
333 MHz Sun Ultra 2i 2-D slice of 3-D space
implementations color-coded by performance in
Mflop/s 16 registers, but 2-by-3 tile size
fastest
Needle in a haystack
8
Example Select a Matmul Implementation
9
Example Support Vector Classification
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Machine Learning in Automatic Performance Tuning

• References
• Statistical Models for Empirical Search-Based
  Performance Tuning(International Journal of High
  Performance Computing Applications, 18 (1), pp.
  65-94, February 2004)Richard Vuduc, J. Demmel,
  and Jeff A. Bilmes.
• Predicting and Optimizing System Utilization and
  Performance via Statistical Machine Learning
  (Computer Science PhD Thesis, University of
  California, Berkeley. UCB//EECS-2009-181 )
  Archana Ganapathi

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Examples of Automatic Performance Tuning (3)

• What do dense BLAS, FFTs, signal processing, MPI
  reductions have in common?
• Can do the tuning off-line once per
  architecture, algorithm
• Can take as much time as necessary (hours, a
  week)
• At run-time, algorithm choice may depend only on
  few parameters
• Matrix dimension, size of FFT, etc.
• Cant always do off-line tuning
• Algorithm and implementation may strongly depend
  on data only known at run-time
• Ex Sparse matrix nonzero pattern determines both
  best data structure and implementation of
  Sparse-matrix-vector-multiplication (SpMV)
• Part of search for best algorithm just be done
  (very quickly!) at run-time

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Source Accelerator Cavity Design Problem (Ko via
Husbands)
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Linear Programming Matrix

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A Sparse Matrix You Encounter Every Day
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SpMV with Compressed Sparse Row (CSR) Storage
Matrix-vector multiply kernel y(i) ? y(i)
A(i,j)x(j) for each row i for kptri to
ptri1-1 do yi yi valkxindk
Matrix-vector multiply kernel y(i) ? y(i)
A(i,j)x(j) for each row i for kptri to
ptri1-1 do yi yi valkxindk
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Example The Difficulty of Tuning

• n 21200
• nnz 1.5 M
• kernel SpMV
• Source NASA structural analysis problem

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Example The Difficulty of Tuning

• n 21200
• nnz 1.5 M
• kernel SpMV
• Source NASA structural analysis problem

• 8x8 dense substructure

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Taking advantage of block structure in SpMV

• Bottleneck is time to get matrix from memory
• Only 2 flops for each nonzero in matrix
• Dont store each nonzero with index, instead
  store each nonzero r-by-c block with index
• Storage drops by up to 2x, if rc gtgt 1, all 32-bit
  quantities
• Time to fetch matrix from memory decreases
• Change both data structure and algorithm
• Need to pick r and c
• Need to change algorithm accordingly
• In example, is rc8 best choice?
• Minimizes storage, so looks like a good idea

19
Speedups on Itanium 2 The Need for Search
Mflop/s
Mflop/s
20
Register Profile Itanium 2
1190 Mflop/s
190 Mflop/s
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SpMV Performance (Matrix 2) Generation 1
Power3 - 13
Power4 - 14
195 Mflop/s
703 Mflop/s
100 Mflop/s
469 Mflop/s
Itanium 2 - 31
Itanium 1 - 7
225 Mflop/s
1.1 Gflop/s
103 Mflop/s
276 Mflop/s
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Register Profiles IBM and Intel IA-64
Power3 - 17
Power4 - 16
252 Mflop/s
820 Mflop/s
122 Mflop/s
459 Mflop/s
Itanium 2 - 33
Itanium 1 - 8
247 Mflop/s
1.2 Gflop/s
107 Mflop/s
190 Mflop/s
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SpMV Performance (Matrix 2) Generation 2
Ultra 2i - 9
Ultra 3 - 5
63 Mflop/s
109 Mflop/s
35 Mflop/s
53 Mflop/s
Pentium III-M - 15
Pentium III - 19
96 Mflop/s
120 Mflop/s
42 Mflop/s
58 Mflop/s
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Register Profiles Sun and Intel x86
Ultra 2i - 11
Ultra 3 - 5
72 Mflop/s
90 Mflop/s
35 Mflop/s
50 Mflop/s
Pentium III-M - 15
Pentium III - 21
108 Mflop/s
122 Mflop/s
42 Mflop/s
58 Mflop/s
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Another example of tuning challenges

• More complicated non-zero structure in general
• N 16614
• NNZ 1.1M

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Zoom in to top corner

• More complicated non-zero structure in general
• N 16614
• NNZ 1.1M

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3x3 blocks look natural, but

• More complicated non-zero structure in general
• Example 3x3 blocking
• Logical grid of 3x3 cells
• But would lead to lots of fill-in

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Extra Work Can Improve Efficiency!

• More complicated non-zero structure in general
• Example 3x3 blocking
• Logical grid of 3x3 cells
• Fill-in explicit zeros
• Unroll 3x3 block multiplies
• Fill ratio 1.5
• On Pentium III 1.5x speedup!
• Actual mflop rate 1.52 2.25 higher

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Automatic Register Block Size Selection

• Selecting the r x c block size
• Off-line benchmark
• Precompute Mflops(r,c) using dense A for each r x
  c
• Once per machine/architecture
• Run-time search
• Sample A to estimate Fill(r,c) for each r x c
• Run-time heuristic model
• Choose r, c to minimize time Fill(r,c) /
  Mflops(r,c)

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Accurate and Efficient Adaptive Fill Estimation

• Idea Sample matrix
• Fraction of matrix to sample s Î 0,1
• Cost O(s nnz)
• Control cost by controlling s
• Search at run-time the constant matters!
• Control s automatically by computing statistical
  confidence intervals
• Idea Monitor variance
• Cost of tuning
• Lower bound convert matrix in 5 to 40 unblocked
  SpMVs
• Heuristic 1 to 11 SpMVs

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Accuracy of the Tuning Heuristics (1/4)
See p. 375 of Vuducs thesis for matrices
NOTE Fair flops used (ops on explicit zeros
not counted as work)
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Accuracy of the Tuning Heuristics (2/4)
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Accuracy of the Tuning Heuristics (2/4)
DGEMV
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Upper Bounds on Performance for blocked SpMV

• P (flops) / (time)
• Flops 2 nnz(A)
• Lower bound on time Two main assumptions
• 1. Count memory ops only (streaming)
• 2. Count only compulsory, capacity misses ignore
  conflicts
• Account for line sizes
• Account for matrix size and nnz
• Charge minimum access latency ai at Li cache
  amem
• e.g., Saavedra-Barrera and PMaC MAPS benchmarks

35
Example L2 Misses on Itanium 2
Misses measured using PAPI Browne 00
36
Example Bounds on Itanium 2
37
Example Bounds on Itanium 2
38
Example Bounds on Itanium 2
39
Summary of Other Performance Optimizations

• Optimizations for SpMV
• Register blocking (RB) up to 4x over CSR
• Variable block splitting 2.1x over CSR, 1.8x
  over RB
• Diagonals 2x over CSR
• Reordering to create dense structure splitting
  2x over CSR
• Symmetry 2.8x over CSR, 2.6x over RB
• Cache blocking 2.8x over CSR
• Multiple vectors (SpMM) 7x over CSR
• And combinations
• Sparse triangular solve
• Hybrid sparse/dense data structure 1.8x over CSR
• Higher-level kernels
• AATx, ATAx 4x over CSR, 1.8x over RB
• A2x 2x over CSR, 1.5x over RB
• Ax, A2x, A3x, .. , Akx

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Example Sparse Triangular Factor

• Raefsky4 (structural problem) SuperLU colmmd
• N19779, nnz12.6 M

41
Cache Optimizations for AATx

• Cache-level Interleave multiplication by A, AT
• Only fetch A from memory once

• Register-level aiT to be rc block row, or diag
  row

42
Example Combining Optimizations (1/2)

• Register blocking, symmetry, multiple (k) vectors
• Three low-level tuning parameters r, c, v

X
k
v

r
c

Y
A
43
Example Combining Optimizations (2/2)

• Register blocking, symmetry, and multiple vectors
  Ben Lee _at_ UCB
• Symmetric, blocked, 1 vector
• Up to 2.6x over nonsymmetric, blocked, 1 vector
• Symmetric, blocked, k vectors
• Up to 2.1x over nonsymmetric, blocked, k vecs.
• Up to 7.3x over nonsymmetric, nonblocked, 1,
  vector
• Symmetric Storage up to 64.7 savings

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Potential Impact on Applications Omega3P

• Application accelerator cavity design Ko
• Relevant optimization techniques
• Symmetric storage
• Register blocking
• Reordering, to create more dense blocks
• Reverse Cuthill-McKee ordering to reduce
  bandwidth
• Do Breadth-First-Search, number nodes in reverse
  order visited
• Traveling Salesman Problem-based ordering to
  create blocks
• Nodes columns of A
• Weights(u, v) no. of nz u, v have in common
• Tour ordering of columns
• Choose maximum weight tour
• See Pinar Heath 97
• 2.1x speedup on Power 4 (caveat SPMV not
  bottleneck)

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Source Accelerator Cavity Design Problem (Ko via
Husbands)
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Post-RCM Reordering
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100x100 Submatrix Along Diagonal
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Microscopic Effect of RCM Reordering
Before Green Red After Green Blue
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Microscopic Effect of Combined RCMTSP
Reordering
Before Green Red After Green Blue
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(Omega3P)
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Optimized Sparse Kernel Interface - OSKI

• Provides sparse kernels automatically tuned for
  users matrix machine
• BLAS-style functionality SpMV, Ax ATy, TrSV
• Hides complexity of run-time tuning
• Includes new, faster locality-aware kernels
  ATAx, Akx
• Faster than standard implementations
• Up to 4x faster matvec, 1.8x trisolve, 4x ATAx
• For advanced users solver library writers
• Available as stand-alone library (OSKI 1.0.1h,
  6/07)
• Available as PETSc extension (OSKI-PETSc .1d,
  3/06)
• Bebop.cs.berkeley.edu/oski

52
How the OSKI Tunes (Overview)
Application Run-Time
Library Install-Time (offline)
1. Build for Target Arch.
2. Benchmark
Workload from program monitoring
History
Matrix
Benchmark data
Heuristic models
1. Evaluate Models
Generated code variants
2. Select Data Struct. Code
To user Matrix handle for kernel calls
Extensibility Advanced users may write
dynamically add Code variants and Heuristic
models to system.
53
How the OSKI Tunes (Overview)

• At library build/install-time
• Pre-generate and compile code variants into
  dynamic libraries
• Collect benchmark data
• Measures and records speed of possible sparse
  data structure and code variants on target
  architecture
• Installation process uses standard, portable GNU
  AutoTools
• At run-time
• Library tunes using heuristic models
• Models analyze users matrix benchmark data to
  choose optimized data structure and code
• Non-trivial tuning cost up to 40 mat-vecs
• Library limits the time it spends tuning based on
  estimated workload
• provided by user or inferred by library
• User may reduce cost by saving tuning results for
  application on future runs with same or similar
  matrix

54
Optimizations in OSKI, so far

• Fully automatic heuristics for
• Sparse matrix-vector multiply
• Register-level blocking
• Register-level blocking symmetry multiple
  vectors
• Cache-level blocking
• Sparse triangular solve with register-level
  blocking and switch-to-dense optimization
• Sparse ATAx with register-level blocking
• User may select other optimizations manually
• Diagonal storage optimizations, reordering,
  splitting tiled matrix powers kernel (Akx)
• All available in dynamic libraries
• Accessible via high-level embedded script
  language
• Plug-in extensibility
• Very advanced users may write their own
  heuristics, create new data structures/code
  variants and dynamically add them to the system

55
How to Call OSKI Basic Usage

• May gradually migrate existing apps
• Step 1 Wrap existing data structures
• Step 2 Make BLAS-like kernel calls

int ptr , ind double val /
Matrix, in CSR format / double x , y
/ Let x and y be two dense vectors / /
Compute y ?y ?Ax, 500 times / for( i 0
i lt 500 i ) my_matmult( ptr, ind, val, ?, x,
b, y )
56
How to Call OSKI Basic Usage

• May gradually migrate existing apps
• Step 1 Wrap existing data structures
• Step 2 Make BLAS-like kernel calls

int ptr , ind double val /
Matrix, in CSR format / double x , y
/ Let x and y be two dense vectors / / Step 1
Create OSKI wrappers around this data
/ oski_matrix_t A_tunable oski_CreateMatCSR(ptr
, ind, val, num_rows, num_cols, SHARE_INPUTMAT,
) oski_vecview_t x_view oski_CreateVecView(x,
num_cols, UNIT_STRIDE) oski_vecview_t y_view
oski_CreateVecView(y, num_rows, UNIT_STRIDE) /
Compute y ?y ?Ax, 500 times / for( i 0
i lt 500 i ) my_matmult( ptr, ind, val, ?, x,
b, y )
57
How to Call OSKI Basic Usage

• May gradually migrate existing apps
• Step 1 Wrap existing data structures
• Step 2 Make BLAS-like kernel calls

int ptr , ind double val /
Matrix, in CSR format / double x , y
/ Let x and y be two dense vectors / / Step 1
Create OSKI wrappers around this data
/ oski_matrix_t A_tunable oski_CreateMatCSR(ptr
, ind, val, num_rows, num_cols, SHARE_INPUTMAT,
) oski_vecview_t x_view oski_CreateVecView(x,
num_cols, UNIT_STRIDE) oski_vecview_t y_view
oski_CreateVecView(y, num_rows, UNIT_STRIDE) /
Compute y ?y ?Ax, 500 times / for( i 0
i lt 500 i ) oski_MatMult(A_tunable,
OP_NORMAL, ?, x_view, ?, y_view)/ Step 2 /
58
How to Call OSKI Tune with Explicit Hints

• User calls tune routine
• May provide explicit tuning hints (OPTIONAL)

oski_matrix_t A_tunable oski_CreateMatCSR(
) / / / Tell OSKI we will call SpMV 500
times (workload hint) / oski_SetHintMatMult(A_tun
able, OP_NORMAL, ?, x_view, ?, y_view, 500) /
Tell OSKI we think the matrix has 8x8 blocks
(structural hint) / oski_SetHint(A_tunable,
HINT_SINGLE_BLOCKSIZE, 8, 8) oski_TuneMat(A_tuna
ble) / Ask OSKI to tune / for( i 0 i lt
500 i ) oski_MatMult(A_tunable, OP_NORMAL, ?,
x_view, ?, y_view)
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How the User Calls OSKI Implicit Tuning

• Ask library to infer workload
• Library profiles all kernel calls
• May periodically re-tune

oski_matrix_t A_tunable oski_CreateMatCSR(
) / / for( i 0 i lt 500 i )
oski_MatMult(A_tunable, OP_NORMAL, ?, x_view,
?, y_view) oski_TuneMat(A_tunable) / Ask OSKI
to tune /
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Quick-and-dirty Parallelism OSKI-PETSc

• Extend PETScs distributed memory SpMV (MATMPIAIJ)

• PETSc
• Each process stores diag (all-local) and off-diag
  submatrices
• OSKI-PETSc
• Add OSKI wrappers
• Each submatrix tuned independently

p0
p1
p2
p3
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OSKI-PETSc Proof-of-Concept Results

• Matrix 1 Accelerator cavity design (R. Lee _at_
  SLAC)
• N 1 M, 40 M non-zeros
• 2x2 dense block substructure
• Symmetric
• Matrix 2 Linear programming (Italian Railways)
• Short-and-fat 4k x 1M, 11M non-zeros
• Highly unstructured
• Big speedup from cache-blocking no native PETSc
  format
• Evaluation machine Xeon cluster
• Peak 4.8 Gflop/s per node

62
Accelerator Cavity Matrix
63
OSKI-PETSc Performance Accel. Cavity
64
Linear Programming Matrix

65
OSKI-PETSc Performance LP Matrix
66
Tuning Higher Level Algorithms than SpMV

• We almost always do many SpMVs, not just one
• Krylov Subspace Methods (KSMs) for Axb, Ax
  ?x
• Conjugate Gradients, GMRES, Lanczos,
• Do a sequence of k SpMVs to get vectors x1 , ,
  xk
• Find best solution x as linear combination of
  x1 , , xk
• Main cost is k SpMVs
• Since communication usually dominates, can we do
  better?
• Goal make communication cost independent of k
• Parallel case O(log P) messages, not O(k log P)
  - optimal
• same bandwidth as before
• Sequential case O(1) messages and bandwidth, not
  O(k) - optimal
• Achievable when matrix partitionable with low
  surface-to-volume ratio

67
Locally Dependent Entries for x,Ax, A
tridiagonal 2 processors
Proc 1
Proc 2
Can be computed without communication
68
Locally Dependent Entries for x,Ax,A2x, A
tridiagonal 2 processors
Proc 1
Proc 2
Can be computed without communication
69
Locally Dependent Entries for x,Ax,,A3x, A
tridiagonal 2 processors
Proc 1
Proc 2
Can be computed without communication
70
Locally Dependent Entries for x,Ax,,A4x, A
tridiagonal 2 processors
Proc 1
Proc 2
Can be computed without communication
71
Locally Dependent Entries for x,Ax,,A8x, A
tridiagonal 2 processors
Proc 1
Proc 2
Can be computed without communication k8 fold
reuse of A
72
Remotely Dependent Entries for x,Ax,,A8x, A
tridiagonal 2 processors
Proc 1
Proc 2
One message to get data needed to compute
remotely dependent entries, not k8 Minimizes
number of messages latency cost Price
redundant work ? surface/volume ratio
73
Fewer Remotely Dependent Entries for
x,Ax,,A8x, A tridiagonal 2 processors
Proc 1
Proc 2
Reduce redundant work by half
74
Remotely Dependent Entries for x,Ax, A2x,A3x,
2D Laplacian
75
Remotely Dependent Entries for x,Ax,A2x,A3x, A
irregular, multiple processors
76
Sequential x,Ax,,A4x, with memory hierarchy
One read of matrix from slow memory, not
k4 Minimizes words moved bandwidth cost No
redundant work
77
Performance Results

• Measured Multicore (Clovertown) speedups up to
  6.4x
• Measured/Modeled sequential OOC speedup up to 3x
• Modeled parallel Petascale speedup up to 6.9x
• Modeled parallel Grid speedup up to 22x
• Sequential speedup due to bandwidth, works for
  many problem sizes
• Parallel speedup due to latency, works for
  smaller problems on many processors

78
Speedups on Intel Clovertown (8 core)
79
Avoiding Communication in Iterative Linear Algebra

• k-steps of typical iterative solver for sparse
  Axb or Ax?x
• Does k SpMVs with starting vector
• Finds best solution among all linear
  combinations of these k1 vectors
• Many such Krylov Subspace Methods
• Conjugate Gradients, GMRES, Lanczos, Arnoldi,
• Goal minimize communication in Krylov Subspace
  Methods
• Assume matrix well-partitioned, with modest
  surface-to-volume ratio
• Parallel implementation
• Conventional O(k log p) messages, because k
  calls to SpMV
• New O(log p) messages - optimal
• Serial implementation
• Conventional O(k) moves of data from slow to
  fast memory
• New O(1) moves of data optimal
• Lots of speed up possible (modeled and measured)
• Price some redundant computation
• Much prior work
• See bebop.cs.berkeley.edu
• CG van Rosendale, 83, Chronopoulos and Gear,
  89
• GMRES Walker, 88, Joubert and Carey, 92,
  Bai et al., 94

80
Minimizing Communication of GMRES to solve Axb

• GMRES find x in spanb,Ab,,Akb minimizing
  Ax-b 2
• Cost of k steps of standard GMRES vs new GMRES

Standard GMRES for i1 to k w A
v(i-1) MGS(w, v(0),,v(i-1)) update
v(i), H endfor solve LSQ problem with
H Sequential words_moved O(knnz)
from SpMV O(k2n) from MGS Parallel
messages O(k) from SpMV
O(k2 log p) from MGS
Communication-avoiding GMRES W v, Av, A2v,
, Akv Q,R TSQR(W) Tall Skinny
QR Build H from R, solve LSQ
problem Sequential words_moved
O(nnz) from SpMV O(kn) from
TSQR Parallel messages O(1) from
computing W O(log p) from TSQR

• Oops W from power method, precision lost!

81
Monomial basis Ax,,Akx fails to converge
A different polynomial basis does converge
82
Speed ups of GMRES on 8-core Intel
ClovertownRequires co-tuning kernels MHDY09
83
Extensions

• Other Krylov methods
• Arnoldi, CG, Lanczos,
• Preconditioning
• Solve MAxMb where preconditioning matrix M
  chosen to make system easier
• M approximates A-1 somehow, but cheaply, to
  accelerate convergence
• Cheap as long as contributions from distant
  parts of the system can be compressed
• Sparsity
• Low rank
• No implementations yet (class projects!)

84
Design Space for x,Ax,,Akx (1/3)

• Mathematical Operation
• How many vectors to keep
• All Krylov Subspace Methods
• Keep last vector Akx only (Jacobi, Gauss Seidel)
• Improving conditioning of basis
• W x, p1(A)x, p2(A)x,,pk(A)x
• pi(A) degree i polynomial chosen to reduce
  cond(W)
• Preconditioning (Ayb ? MAyMb)
• x,Ax,MAx,AMAx,MAMAx,,(MA)kx

85
Design Space for x,Ax,,Akx (2/3)

• Representation of sparse A
• Zero pattern may be explicit or implicit
• Nonzero entries may be explicit or implicit
• Implicit ? save memory, communication

Explicit pattern Implicit pattern
Explicit nonzeros General sparse matrix Image segmentation
Implicit nonzeros Laplacian(graph) Multigrid (AMR) Stencil matrix Ex tridiag(-1,2,-1)

• Representation of dense preconditioners M
• Low rank off-diagonal blocks (semiseparable)

86
Design Space for x,Ax,,Akx (3/3)

• Parallel implementation
• From simple indexing, with redundant flops ?
  surface/volume ratio
• To complicated indexing, with fewer redundant
  flops
• Sequential implementation
• Depends on whether vectors fit in fast memory
• Reordering rows, columns of A
• Important in parallel and sequential cases
• Can be reduced to pair of Traveling Salesmen
  Problems
• Plus all the optimizations for one SpMV!

87
Summary

• Communication-Avoiding Linear Algebra (CALA)
• Lots of related work
• Some going back to 1960s
• Reports discuss this comprehensively, not here
• Our contributions
• Several new algorithms, improvements on old ones
• Preconditioning
• Unifying parallel and sequential approaches to
  avoiding communication
• Time for these algorithms has come, because of
  growing communication costs
• Why avoid communication just for linear algebra
  motifs?

88
Possible Class Projects

• Come to BEBOP meetings (T 1 230, 380 Soda)
• Experiment with SpMV on GPU
• Which optimizations are most effective?
• Try to speed up particular matrices of interest
• Data mining
• Experiment with new x,Ax,,Akx kernel
• GPU, multicore, distributed memory
• On matrices of interest
• Bottom solver in multigrid / AMR (Chombo)
• Experiment with solvers using this kernel
• New Krylov subspace methods, preconditioning
• Experiment with new frameworks (SPF)
• See proposals for details

89
Extra Slides
90
Optimizing Communication Complexity of Sparse
Solvers

• Need to modify high level algorithms to use new
  kernel
• Example GMRES for Axb where A 2D Laplacian
• x lives on n-by-n mesh
• Partitioned on p½ -by- p½ processor grid
• A has 5 point stencil (Laplacian)
• (Ax)(i,j) linear_combination(x(i,j), x(i,j1),
  x(i1,j))
• Ex 18-by-18 mesh on 3-by-3 processor grid

91
Minimizing Communication

• What is the cost (flops, words, mess) of s
  steps of standard GMRES?

GMRES, ver.1 for i1 to s w A v(i-1)
MGS(w, v(0),,v(i-1)) update v(i), H
endfor solve LSQ problem with H
n/p½
n/p½

• Cost(A v) s (9n2 /p, 4n / p½ , 4 )
• Cost(MGS Modified Gram-Schmidt) s2/2 ( 4n2
  /p , log p , log p )
• Total cost Cost( A v ) Cost (MGS)
• Can we reduce the latency?

92
Minimizing Communication

• Cost(GMRES, ver.1) Cost(Av) Cost(MGS)

( 9sn2 /p, 4sn / p½ , 4s ) ( 2s2n2 /p , s2
log p / 2 , s2 log p / 2 )

• How much latency cost from Av can you avoid?
  Almost all

93
Minimizing Communication

• Cost(GMRES, ver. 2) Cost(W) Cost(MGS)

( 9sn2 /p, 4sn / p½ , 8 ) ( 2s2n2 /p , s2
log p / 2 , s2 log p / 2 )

• How much latency cost from MGS can you avoid?
  Almost all

94
Minimizing Communication

• Cost(GMRES, ver. 2) Cost(W) Cost(MGS)

( 9sn2 /p, 4sn / p½ , 8 ) ( 2s2n2 /p , s2
log p / 2 , s2 log p / 2 )

• How much latency cost from MGS can you avoid?
  Almost all

GMRES, ver. 3 W v, Av, A2v, , Asv
Q,R TSQR(W) Tall Skinny QR (See Lecture
11) Build H from R, solve LSQ problem
95
Minimizing Communication

• Cost(GMRES, ver. 2) Cost(W) Cost(MGS)

( 9sn2 /p, 4sn / p½ , 8 ) ( 2s2n2 /p , s2
log p / 2 , s2 log p / 2 )

• How much latency cost from MGS can you avoid?
  Almost all

GMRES, ver. 3 W v, Av, A2v, , Asv
Q,R TSQR(W) Tall Skinny QR (See Lecture
11) Build H from R, solve LSQ problem
96
(No Transcript)
97
Minimizing Communication

• Cost(GMRES, ver. 3) Cost(W) Cost(TSQR)

( 9sn2 /p, 4sn / p½ , 8 ) ( 2s2n2 /p , s2
log p / 2 , log p )

• Latency cost independent of s, just log p
  optimal
• Oops W from power method, so precision lost
  What to do?

• Use a different polynomial basis
• Not Monomial basis W v, Av, A2v, , instead
  
• Newton Basis WN v, (A ?1 I)v , (A ?2 I)(A
  ?1 I)v, or
• Chebyshev Basis WC v, T1(v), T2(v),

98
(No Transcript)
99
Tuning Higher Level Algorithms

• So far we have tuned a single sparse matrix
  kernel
• y ATAx motivated by higher level algorithm
  (SVD)
• What can we do by extending tuning to a higher
  level?
• Consider Krylov subspace methods for Axb, Ax
  lx
• Conjugate Gradients (CG), GMRES, Lanczos,
• Inner loop does yAx, dot products, saxpys,
  scalar ops
• Inner loop costs at least O(1) messages
• k iterations cost at least O(k) messages
• Our goal show how to do k iterations with O(1)
  messages
• Possible payoff make Krylov subspace methods
  much
• faster on machines with slow networks
• Memory bandwidth improvements too (not
  discussed)
• Obstacles numerical stability, preconditioning,
  

100
Krylov Subspace Methods for Solving Axb

• Compute a basis for a subspace V by doing y Ax
  k times
• Find best solution in that Krylov subspace V
• Given starting vector x1, V spanned by x2 Ax1,
  x3 Ax2 , , xk Axk-1
• GMRES finds an orthogonal basis of V by
  Gram-Schmidt, so it actually does a different
  set of SpMVs than in last bullet

101
Example Standard GMRES

• r b - Ax1, b length(r), v1 r / b
  length(r) sqrt(S ri2 )
• for m 1 to k do
• w Avm at least O(1) messages
• for i 1 to m do Gram-Schmidt
• him dotproduct(vi , w ) at least
  O(1) messages, or log(p)
• w w h im vi
• end for
• hm1,m length(w) at least O(1) messages,
  or log(p)
• vm1 w / hm1,m
• end for
• find y minimizing length( Hk y be1 )
  small, local problem
• new x x1 Vk y Vk v1 , v2 , ,
  vk

O(k2), or O(k2 log p), messages altogether
102
Example Computing Ax,A2x,A3x,,Akx for A
tridiagonal
Different basis for same Krylov subspace What can
Proc 1 compute without communication?
Proc 2
Proc 1
(A8x)(130)
. . .
(A2x)(130)
(Ax)(130)
x(130)
103
Example Computing Ax,A2x,A3x,,Akx for A
tridiagonal
Computing missing entries with 1 communication,
redundant work
Proc 2
Proc 1
(A8x)(130)
. . .
(A2x)(130)
(Ax)(130)
x(130)
104
Example Computing Ax,A2x,A3x,,Akx for A
tridiagonal
Saving half the redundant work
Proc 2
Proc 1
(A8x)(130)
. . .
(A2x)(130)
(Ax)(130)
x(130)
105
Example Computing Ax,A2x,A3x,,Akx for
Laplacian
A 5pt Laplacian in 2D, Communicated point for
k3 shown
106
Latency-Avoiding GMRES (1)

• r b - Ax1, b length(r), v1 r / b
  O(log p) messages
• Wk1 v1 , A v1 , A2 v1 , , Ak v1
  O(1) messages
• Q, R qr(Wk1) QR decomposition, O(log
  p) messages
• Hk R(, 2k1) (R(1k,1k))-1 small, local
  problem
• find y minimizing length( Hk y be1 )
  small, local problem
• new x x1 Qk y local problem

O(log p) messages altogether Independent of k
107
Latency-Avoiding GMRES (2)

• Q, R qr(Wk1) QR decomposition, O(log
  p) messages
• Easy, but not so stable way to do it
• X(myproc) Wk1T(myproc) Wk1 (myproc)
• local computation
• Y sum_reduction(X(myproc)) O(log p)
  messages
• 
  Y Wk1T Wk1
• R (cholesky(Y))T small, local
  computation
• Q(myproc) Wk1 (myproc) R-1 local
  computation

108
Numerical example (1)
Diagonal matrix with n1000, Aii from 1 down to
10-5 Instability as k grows, after many iterations
109
Numerical Example (2)
Partial remedy restarting periodically (every
120 iterations) Other remedies high precision,
different basis than v , A v , , Ak v
110
Operation Counts for Ax,A2x,A3x,,Akx on p procs
Problem Per-proc cost Standard Optimized
1D mesh messages 2k 2
(tridiagonal) words sent 2k 2k
flops 5kn/p 5kn/p 5k2
memory (k4)n/p (k4)n/p 8k
3D mesh messages 26k 26
27 pt stencil words sent 6kn2p-2/3 12knp-1/3 O(k) 6kn2p-2/3 12k2np-1/3 O(k3)
flops 53kn3/p 53kn3/p O(k2n2p-2/3)
memory (k28)n3/p 6n2p-2/3 O(np-1/3) (k28)n3/p 168kn2p-2/3 O(k2np-1/3)
111
Summary and Future Work

• Dense
• LAPACK
• ScaLAPACK
• Communication primitives
• Sparse
• Kernels, Stencils
• Higher level algorithms
• All of the above on new architectures
• Vector, SMPs, multicore, Cell,
• High level support for tuning
• Specification language
• Integration into compilers

112
Extra Slides
113
A Sparse Matrix You Encounter Every Day
Who am I?
I am a Big Repository Of useful And useless Facts
alike. Who am I? (Hint Not your e-mail inbox.)
114
What about the Google Matrix?

• Google approach
• Approx. once a month rank all pages using
  connectivity structure
• Find dominant eigenvector of a matrix
• At query-time return list of pages ordered by
  rank
• Matrix A aG (1-a)(1/n)uuT
• Markov model Surfer follows link with
  probability a, jumps to a random page with
  probability 1-a
• G is n x n connectivity matrix n billions
• gij is non-zero if page i links to page j
• Normalized so each column sums to 1
• Very sparse about 78 non-zeros per row (power
  law dist.)
• u is a vector of all 1 values
• Steady-state probability xi of landing on page i
  is solution to x Ax
• Approximate x by power method x Akx0
• In practice, k 25

115
Current Work

• Public software release
• Impact on library designs Sparse BLAS, Trilinos,
  PETSc,
• Integration in large-scale applications
• DOE Accelerator design plasma physics
• Geophysical simulation based on Block Lanczos
  (ATAX LBL)
• Systematic heuristics for data structure
  selection?
• Evaluation of emerging architectures
• Revisiting vector micros
• Other sparse kernels
• Matrix triple products, Akx
• Parallelism
• Sparse benchmarks (with UTK) Gahvari Hoemmen
• Automatic tuning of MPI collective ops Nishtala,
  et al.

116
Summary of High-Level Themes

• Kernel-centric optimization
• Vs. basic block, trace, path optimization, for
  instance
• Aggressive use of domain-specific knowledge
• Performance bounds modeling
• Evaluating software quality
• Architectural characterizations and consequences
• Empirical search
• Hybrid on-line/run-time models
• Statistical performance models
• Exploit information from sampling, measuring

117
Related Work

• My bibliography 337 entries so far
• Sample area 1 Code generation
• Generative generic programming
• Sparse compilers
• Domain-specific generators
• Sample area 2 Empirical search-based tuning
• Kernel-centric
• linear algebra, signal processing, sorting, MPI,
  
• Compiler-centric
• profiling FDO, iterative compilation,
  superoptimizers, self-tuning compilers,
  continuous program optimization

118
Future Directions (A Bag of Flaky Ideas)

• Composable code generators and search spaces
• New application domains
• PageRank multilevel block algorithms for
  topic-sensitive search?
• New kernels cryptokernels
• rich mathematical structure germane to
  performance lots of hardware
• New tuning environments
• Parallel, Grid, whole systems
• Statistical models of application performance
• Statistical learning of concise parametric models
  from traces for architectural evaluation
• Compiler/automatic derivation of parametric models

119
Possible Future Work

• Different Architectures
• New FP instruction sets (SSE2)
• SMP / multicore platforms
• Vector architectures
• Different Kernels
• Higher Level Algorithms
• Parallel versions of kenels, with optimized
  communication
• Block algorithms (eg Lanczos)
• XBLAS (extra precision)
• Produce Benchmarks
• Augment HPCC Benchmark
• Make it possible to combine optimizations easily
  for any kernel
• Related tuning activities (LAPACK ScaLAPACK)

120
Review of Tuning by Illustration

• (Extra Slides)

121
Splitting for Variable Blocks and Diagonals

• Decompose A A1 A2 At
• Detect canonical structures (sampling)
• Split
• Tune each Ai
• Improve performance and save storage
• New data structures
• Unaligned block CSR
• Relax alignment in rows columns
• Row-segmented diagonals

122
Example Variable Block Row (Matrix 12)
2.1x over CSR 1.8x over RB
123
Example Row-Segmented Diagonals
2x over CSR
124
Mixed Diagonal and Block Structure
125
Summary

• Automated block size selection
• Empirical modeling and search
• Register blocking for SpMV, triangular solve,
  ATAx
• Not fully automated
• Given a matrix, select splittings and
  transformations
• Lots of combinatorial problems
• TSP reordering to create dense blocks (Pinar 97
  Moon, et al. 04)

126
Extra Slides
127
A Sparse Matrix You Encounter Every Day
Who am I?
I am a Big Repository Of useful And useless Facts
alike. Who am I? (Hint Not your e-mail inbox.)
128
Problem Context

• Sparse kernels abound
• Models of buildings, cars, bridges, economies,
• Google PageRank algorithm
• Historical trends
• Sparse matrix-vector multiply (SpMV) 10 of peak
• 2x faster with hand-tuning
• Tuning becoming more difficult over time
• Promise of automatic tuning PHiPAC/ATLAS, FFTW,
  
• Challenges to high-performance
• Not dense linear algebra!
• Complex data structures indirect, irregular
  memory access
• Performance depends strongly on run-time inputs

129
Key Questions, Ideas, Conclusions

• How to tune basic sparse kernels automatically?
• Empirical modeling and search
• Up to 4x speedups for SpMV
• 1.8x for triangular solve
• 4x for ATAx 2x for A2x
• 7x for multiple vectors
• What are the fundamental limits on performance?
• Kernel-, machine-, and matrix-specific upper
  bounds
• Achieve 75 or more for SpMV, limiting low-level
  tuning
• Consequences for architecture?
• General techniques for empirical search-based
  tuning?
• Statistical models of performance

130
Road Map

• Sparse matrix-vector multiply (SpMV) in a
  nutshell
• Historical trends and the need for search
• Automatic tuning techniques
• Upper bounds on performance
• Statistical models of performance

131
Compressed Sparse Row (CSR) Storage
Matrix-vector multiply kernel y(i) ? y(i)
A(i,j)x(j)
Matrix-vector multiply kernel y(i) ? y(i)
A(i,j)x(j) for each row i for kptri to
ptri1 do yi yi valkxindk
Matrix-vector multiply kernel y(i) ? y(i)
A(i,j)x(j) for each row i for kptri to
ptri1 do yi yi valkxindk
132
Road Map

• Sparse matrix-vector multiply (SpMV) in a
  nutshell
• Historical trends and the need for search
• Automatic tuning techniques
• Upper bounds on performance
• Statistical models of performance

133
Historical Trends in SpMV Performance

• The Data
• Uniprocessor SpMV performance since 1987
• Untuned and Tuned implementations
• Cache-based superscalar micros some vectors
• LINPACK

134
SpMV Historical Trends Mflop/s
135
Example The Difficulty of Tuning

• n 21216
• nnz 1.5 M
• kernel SpMV
• Source NASA structural analysis problem

136
Still More Surprises

• More complicated non-zero structure in general

137
Still More Surprises

• More complicated non-zero structure in general
• Example 3x3 blocking
• Logical grid of 3x3 cells

138
Historical Trends Mixed News

• Observations
• Good news Moores law like behavior
• Bad news Untuned is 10 peak or less,
  worsening
• Good news Tuned roughly 2x better today, and
  improving
• Bad news Tuning is complex
• (Not really news SpMV is not LINPACK)
• Questions
• Application Automatic tuning?
• Architect What machines are good for SpMV?

139
Road Map

• Sparse matrix-vector multiply (SpMV) in a
  nutshell
• Historical trends and the need for search
• Automatic tuning techniques
• SpMV SC02 IJHPCA 04b
• Sparse triangular solve (SpTS) ICS/POHLL 02
• ATAx ICCS/WoPLA 03
• Upper bounds on performance
• Statistical models of performance

140
SPARSITY Framework for Tuning SpMV

• SPARSITY Automatic tuning for SpMV Im Yelick
  99
• General approach
• Identify and generate implementation space
• Search space using empirical models experiments
• Prototype library and heuristic for choosing
  register block size
• Also cache-level blocking, multiple vectors
• Whats new?
• New block size selection heuristic
• Within 10 of optimal replaces previous version
• Expanded implementation space
• Variable block splitting, diagonals, combinations
• New kernels sparse triangular solve, ATAx, Arx

141
Automatic Register Block Size Selection

• Selecting the r x c block size
• Off-line benchmark characterize the machine
• Precompute Mflops(r,c) using dense matrix for
  each r x c
• Once per machine/architecture
• Run-time search characterize the matrix
• Sample A to estimate Fill(r,c) for each r x c
• Run-time heuristic model
• Choose r, c to maximize Mflops(r,c) / Fill(r,c)
• Run-time costs
• Up to 40 SpMVs (empirical worst case)

142
Accuracy of the Tuning Heuristics (1/4)
DGEMV
NOTE Fair flops used (ops on explicit zeros
not counted as work)
143
Accuracy of the Tuning Heuristics (2/4)
DGEMV
144
Accuracy of the Tuning Heuristics (3/4)
DGEMV
145
Accuracy of the Tuning Heuristics (4/4)
DGEMV
146
Road Map

• Sparse matrix-vector multiply (SpMV) in a
  nutshell
• Historical trends and the need for search
• Automatic tuning techniques
• Upper bounds on performance
• SC02
• Statistical models of performance

147
Motivation for Upper Bounds Model

• Questions
• Speedups are good, but what is the speed limit?
• Independent of instruction scheduling, selection
• What machines are good for SpMV?

148
Upper Bounds on Performance Blocked SpMV

• P (flops) / (time)
• Flops 2 nnz(A)
• Lower bound on time Two main assumptions
• 1. Count memory ops only (streaming)
• 2. Count only compulsory, capacity misses ignore
  conflicts
• Account for line sizes
• Account for matrix size and nnz
• Charge min access latency ai at Li cache amem
• e.g., Saavedra-Barrera and PMaC MAPS benchmarks

149
Example Bounds on Itanium 2
150
Example Bounds on Itanium 2
151
Example Bounds on Itanium 2
152
Fraction of Upper Bound Across Platforms
153
Achieved Performance and Machine Balance

• Machine balance Callahan 88 McCalpin 95
• Balance Peak Flop Rate / Bandwidth (flops /
  double)
• Ideal balance for mat-vec 2 flops / double
• For SpMV, even less
• SpMV streaming
• 1 / (avg load time to stream 1 array)
  (bandwidth)
• Sustained balance peak flops / model bandwidth

154
(No Transcript)
155
Where Does the Time Go?

• Most time assigned to memory
• Caches disappear when line sizes are equal
• Strictly increasing line sizes

156
Execution Time Breakdown Matrix 40
157
Speedups with Increasing Line Size
158
Summary Performance Upper Bounds

• What is the best we can do for SpMV?
• Limits to low-level tuning of blocked
  implementations
• Refinements?
• What machines are good for SpMV?
• Partial answer balance characterization
• Architectural consequences?
• Example Strictly increasing line sizes

159
Road Map

• Sparse matrix-vector multiply (SpMV) in a
  nutshell
• Historical trends and the need for search
• Automatic tuning techniques
• Upper bounds on performance
• Tuning other sparse kernels
• Statistical models of performance
• FDO 00 IJHPCA 04a

160
Statistical Models for Automatic Tuning

• Idea 1 Statistical criterion for stopping a
  search
• A general search model
• Generate implementation
• Measure performance
• Repeat
• Stop when probability of being within e of
  optimal falls below threshold
• Can estimate distribution on-line
• Idea 2 Statistical performance models
• Problem Choose 1 among m implementations at
  run-time
• Sample performance off-line, build statistical
  model

161
Example Select a Matmul Implementation
162
Example Support Vector Classification
163
Road Map

• Sparse matrix-vector multiply (SpMV) in a
  nutshell
• Historical trends and the need for search
• Automatic tuning techniques
• Upper bounds on performance
• Tuning other sparse kernels
• Statistical models of performance
• Summary and Future Work

164
Summary of High-Level Themes

• Kernel-centric optimization
• Vs. basic block, trace, path optimization, for
  instance
• Aggressive use of domain-specific knowledge
• Performance bounds modeling
• Evaluating software quality
• Architectural characterizations and consequences
• Empirical search
• Hybrid on-line/run-time models
• Statistical performance models
• Exploit information from sampling, measuring

165
Related Work

• My bibliography 337 entries so far
• Sample area 1 Code generation
• Generative generic programming
• Sparse compilers
• Domain-specific generators
• Sample area 2 Empirical search-based tuning
• Kernel-centric
• linear algebra, signal processing, sorting, MPI,
  
• Compiler-centric
• profiling FDO, iterative compilation,
  superoptimizers, self-tuning compilers,
  continuous program optimization

166
Future Directions (A Bag of Flaky Ideas)

• Composable code generators and search spaces
• New application domains
• PageRank multilevel block algorithms for
  topic-sensitive search?
• New kernels cryptokernels
• rich mathematical structure germane to
  performance lots of hardware
• New tuning environments
• Parallel, Grid, whole systems
• Statistical models of application performance
• Statistical learning of concise parametric models
  from traces for architectural evaluation
• Compiler/automatic derivation of parametric models

167
Acknowledgements

• Super-advisors Jim and Kathy
• Undergraduate R.A.s Attila, Ben, Jen, Jin,
  Michael, Rajesh, Shoaib, Sriram, Tuyet-Linh
• See pages xvixvii of dissertation.

168
TSP-based Reordering Before
(Pinar 97 Moon, et al 04)
169
TSP-based Reordering After
(Pinar 97 Moon, et al 04) Up to
2x speedups over CSR
170
Example L2 Misses on Itanium 2
Misses measured using PAPI Browne 00
171
Example Distribution of Blocked Non-Zeros
172
Register Profile Itanium 2
1190 Mflop/s
190 Mflop/s
173
Register Profiles Sun and Intel x86
Ultra 2i - 11
Ultra 3 - 5
72 Mflop/s
90 Mflop/s
35 Mflop/s
50 Mflop/s
Pentium III-M - 15
Pentium III - 21
108 Mflop/s
122 Mflop/s
42 Mflop/s
58 Mflop/s
174
Register Profiles IBM and Intel IA-64
Power3 - 17
Power4 - 16
252 Mflop/s
820 Mflop/s
122 Mflop/s
459 Mflop/s
Itanium 2 - 33
Itanium 1 - 8
247 Mflop/s
1.2 Gflop/s
107 Mflop/s
190 Mflop/s
175
Accurate and Efficient Adaptive Fill Estimation

• Idea Sample matrix
• Fraction of matrix to sample s Î 0,1
• Cost O(s nnz)
• Control cost by controlling s
• Search at run-time the constant matters!
• Control s automatically by computing statistical
  confidence intervals
• Idea Monitor variance
• Cost of tuning
• Lower bound convert matrix in 5 to 40 unblocked
  SpMVs
• Heuristic 1 to 11 SpMVs

176
Sparse/Dense Partitioning for SpTS

• Partition L into sparse (L1,L2) and dense LD

• Perform SpTS in three steps

• Sparsity optimizations for (1)(2) DTRSV for (3)
• Tuning parameters block size, size of dense
  triangle

177
SpTS Performance Power3
178
(No Transcript)
179
Summary of SpTS and AATx Results

• SpTS Similar to SpMV
• 1.8x speedups limited benefit from low-level
  tuning
• AATx, ATAx
• Cache interleaving only up to 1.6x speedups
• Reg cache up to 4x speedups
• 1.8x speedup over register only
• Similar heuristic same accuracy ( 10 optimal)
• Further from upper bounds 6080
• Opportunity for better low-level tuning a la
  PHiPAC/ATLAS
• Matrix triple products? Akx?
• Preliminary work

180
Register Blocking Speedup
181
Register Blocking Performance
182
Register Blocking Fraction of Peak
183
Example Confiden