Automatic Performance Tuning Sparse Matrix Algorithms

1
Automatic Performance TuningSparse Matrix
Algorithms

• James Demmel
• www.cs.berkeley.edu/demmel/cs267_Spr06

2
Berkeley Benchmarking and OPtimization (BeBOP)

• Prof. Katherine Yelick
• Rich Vuduc
• Some results in this talk are from Vuducs PhD
  thesis, www.cs.berkeley.edu/richie
• Rajesh Nishtala, Mark Hoemmen, Hormozd Gahvari,
  Ankit Jain, Sam Williams, A. Gyulassy S. Kamil,
  
• Eun-Jim Im, Ben Lee, many other earlier
  contributors
• 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

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)
• Now in Matlab, many other releases
• math-atlas.sourceforge.net/
• Fast Fourier Transform (FFT) variations
• Sequential and Parallel
• FFTW (MIT)
• Widely used
• 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.
• 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)
• BEBOP project addresses this

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Tuning Dense BLAS PHiPAC
8
Tuning Dense BLAS ATLAS
Extends applicability of PHIPAC Incorporated in
Matlab (with rest of LAPACK)
9
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
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(No Transcript)
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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

12
Example Select a Matmul Implementation
13
Example Support Vector Classification
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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 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
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Motivation for Automatic Performance Tuning of
SpMV

• Historical trends
• Sparse matrix-vector multiply (SpMV) 10 of peak
  or less
• Performance depends on machine, kernel, matrix
• Matrix known at run-time
• Best data structure implementation can be
  surprising
• Our approach empirical performance modeling and
  algorithm search

17
SpMV Historical Trends Fraction of Peak
18
Example The Difficulty of Tuning

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

19
Example The Difficulty of Tuning

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

• 8x8 dense substructure

20
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

21
Speedups on Itanium 2 The Need for Search
Mflop/s
Mflop/s
22
Register Profile Itanium 2
1190 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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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
26
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
27
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 (3/4)
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Accuracy of the Tuning Heuristics (3/4)
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

38
Example L2 Misses on Itanium 2
Misses measured using PAPI Browne 00
39
Example Bounds on Itanium 2
40
Example Bounds on Itanium 2
41
Example Bounds on Itanium 2
42
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

43
Example Sparse Triangular Factor

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

44
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

45
Example Combining Optimizations

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

X
k
v

r
c

Y
A
46
Example Combining Optimizations

• 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

47
Potential Impact on Applications T3P

• Application accelerator design Ko
• 80 of time spent in SpMV
• Relevant optimization techniques
• Symmetric storage
• Register blocking
• On Single Processor Itanium 2
• 1.68x speedup
• 532 Mflops, or 15 of 3.6 GFlop peak
• 4.4x speedup with multiple (8) vectors
• 1380 Mflops, or 38 of peak

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Potential Impact on Applications Omega3P

• Application accelerator cavity design Ko
• Relevant optimization techniques
• Symmetric storage
• Register blocking
• Reordering
• Reverse Cuthill-McKee ordering to reduce
  bandwidth
• 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, but SPMV not dominant

49
Source Accelerator Cavity Design Problem (Ko via
Husbands)
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100x100 Submatrix Along Diagonal
51
Post-RCM Reordering
52
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 1.0.1b,
  3/06)
• Available as PETSc extension (OSKI-PETSc .1d,
  3/06)
• Bebop.cs.berkeley.edu/oski

56
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.
57
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

58
Optimizations in the Initial OSKI Release

• 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

59
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 )
60
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 )
61
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 /
62
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 /
64
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
65
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

66
Accelerator Cavity Matrix
67
OSKI-PETSc Performance Accel. Cavity
68
Linear Programming Matrix

69
OSKI-PETSc Performance LP Matrix
70
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,
  

71
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

72
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
73
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)
74
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)
75
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)
76
Example Computing Ax,A2x,A3x,,Akx for
Laplacian
A 5pt Laplacian in 2D, Communicated point for
k3 shown
77
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
78
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

79
Numerical example (1)
Diagonal matrix with n1000, Aii from 1 down to
10-5 Instability as k grows, after many iterations
80
Numerical Example (2)
Partial remedy restarting periodically (every
120 iterations) Other remedies high precision,
different basis than v , A v , , Ak v
81
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)
82
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

83
Extra Slides
84
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.)
85
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

86
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.

87
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

88
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

89
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

90
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)

91
Review of Tuning by Illustration

• (Extra Slides)

92
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

93
Example Variable Block Row (Matrix 12)
2.1x over CSR 1.8x over RB
94
Example Row-Segmented Diagonals
2x over CSR
95
Mixed Diagonal and Block Structure
96
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)

97
Extra Slides
98
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.)
99
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

100
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

101
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

102
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
103
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

104
Historical Trends in SpMV Performance

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

105
SpMV Historical Trends Mflop/s
106
Example The Difficulty of Tuning

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

107
Still More Surprises

• More complicated non-zero structure in general

108
Still More Surprises

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

109
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?

110
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

111
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

112
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)

113
Accuracy of the Tuning Heuristics (1/4)
DGEMV
NOTE Fair flops used (ops on explicit zeros
not counted as work)
114
Accuracy of the Tuning Heuristics (2/4)
DGEMV
115
Accuracy of the Tuning Heuristics (3/4)
DGEMV
116
Accuracy of the Tuning Heuristics (4/4)
DGEMV
117
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

118
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?

119
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

120
Example Bounds on Itanium 2
121
Example Bounds on Itanium 2
122
Example Bounds on Itanium 2
123
Fraction of Upper Bound Across Platforms
124
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

125
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126
Where Does the Time Go?

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

127
Execution Time Breakdown Matrix 40
128
Speedups with Increasing Line Size
129
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

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
• Tuning other sparse kernels
• Statistical models of performance
• FDO 00 IJHPCA 04a

131
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

132
Example Select a Matmul Implementation
133
Example Support Vector Classification
134
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

135
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

136
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

137
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

138
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.

139
TSP-based Reordering Before
(Pinar 97 Moon, et al 04)
140
TSP-based Reordering After
(Pinar 97 Moon, et al 04) Up to
2x speedups over CSR
141
Example L2 Misses on Itanium 2
Misses measured using PAPI Browne 00
142
Example Distribution of Blocked Non-Zeros
143
Register Profile Itanium 2
1190 Mflop/s
190 Mflop/s
144
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
145
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
146
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

147
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

148
SpTS Performance Power3
149
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150
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

151
Register Blocking Speedup
152
Register Blocking Performance
153
Register Blocking Fraction of Peak
154
Example Confidence Interval Estimation
155
Costs of Tuning
156
Splitting UBCSR Pentium III
157
Splitting UBCSR Power4
158
SplittingUBCSR Storage Power4
159
(No Transcript)
160
Example Variable Block Row (Matrix 13)
161
Dense Tuning is Hard, Too

• Even dense matrix multiply can be notoriously
  difficult to tune

162
Dense matrix multiply surprising performance as
register tile size varies.
163
(No Transcript)
164
Preliminary Results (Matrix Set 2) Itanium 2
Dense
FEM
FEM (var)
Bio
LP
Econ
Stat
165
Multiple Vector Performance
166
(No Transcript)
167
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 3 billion
• 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

168
(No Transcript)
169
MAPS Benchmark Example Power4
170
MAPS Benchmark Example Itanium 2
171
Saavedra-Barrera Example Ultra 2i
172
(No Transcript)
173
Summary of Results Pentium III
174
Summary of Results Pentium III (3/3)
175
Execution Time Breakdown (PAPI) Matrix 40
176
Preliminary Results (Matrix Set 1) Itanium 2
LP
FEM
FEM (var)
Assorted
Dense
177
Tuning Sparse Triangular Solve (SpTS)

• Compute xL-1b where L sparse lower triangular,
  x b dense
• L from sparse LU has rich dense substructure
• Dense trailing triangle can account for 2090 of
  matrix non-zeros
• SpTS optimizations
• Split into sparse trapezoid and dense trailing
  triangle
• Use tuned dense BLAS (DTRSV) on dense triangle
• Use Sparsity register blocking on sparse part
• Tuning parameters
• Size of dense trailing triangle
• Register block size

178
Sparse Kernels and Optimizations

• Kernels
• Sparse matrix-vector multiply (SpMV) yAx
• Sparse triangular solve (SpTS) xT-1b
• yAATx, yATAx
• Powers (yAkx), sparse triple-product (RART),
  
• Optimization techniques (implementation space)
• Register blocking
• Cache blocking
• Multiple dense vectors (x)
• A has special structure (e.g., symmetric, banded,
  )
• Hybrid data structures (e.g., splitting,
  switch-to-dense, )
• Matrix reordering
• How and when do we search?
• Off-line Benchmark implementations
• Run-time Estimate matrix properties, evaluate
  performance models based on benchmark data

179
Cache Blocked SpMV on LSI Matrix Ultra 2i
A 10k x 255k 3.7M non-zeros Baseline 16
Mflop/s Best block size performance 16k x
64k 28 Mflop/s
180
Cache Blocking on LSI Matrix Pentium 4
A 10k x 255k 3.7M non-zeros Baseline 44
Mflop/s Best block size performance 16k x
16k 210 Mflop/s
181
Cache Blocked SpMV on LSI Matrix Itanium
A 10k x 255k 3.7M non-zeros Baseline 25
Mflop/s Best block size performance 16k x
32k 72 Mflop/s
182
Cache Blocked SpMV on LSI Matrix Itanium 2
A 10k x 255k 3.7M non-zeros Baseline 170
Mflop/s Best block size performance 16k x
65k 275 Mflop/s
183
Inter-Iteration Sparse Tiling (1/3)

• Strout, et al., 01
• Let A be 6x6 tridiagonal
• Consider yA2x
• tAx, yAt
• Nodes vector elements
• Edges matrix elements aij

184
Inter-Iteration Sparse Tiling (2/3)

• Strout, et al., 01
• Let A be 6x6 tridiagonal
• Consider yA2x
• tAx, yAt
• Nodes vector elements
• Edges matrix elements aij
• Orange everything needed to compute y1
• Reuse a11, a12

185
Inter-Iteration Sparse Tiling (3/3)

• Strout, et al., 01
• Let A be 6x6 tridiagonal
• Consider yA2x
• tAx, yAt
• Nodes vector elements
• Edges matrix elements aij
• Orange everything needed to compute y1
• Reuse a11, a12
• Grey y2, y3
• Reuse a23, a33, a43

186
Inter-Iteration Sparse Tiling Issues

• Tile sizes (colored regions) grow with no. of
  iterations and increasing out-degree
• G likely to have a few nodes with high out-degree
  (e.g., Yahoo)
• Mathematical tricks to limit tile size?
• Judicious dropping of edges Ng01

187
Summary and Questions

• Need to understand matrix structure and machine
• BeBOP suite of techniques to deal with different
  sparse structures and architectures
• Google matrix problem
• Established techniques within an iteration
• Ideas for inter-iteration optimizations
• Mathematical structure of problem may help
• Questions
• Structure of G?
• What are the computational bottlenecks?
• Enabling future computations?
• E.g., topic-sensitive PageRank ? multiple vector
  version Haveliwala 02
• See www.cs.berkeley.edu/richie/bebop/intel/google
  for more info, including more complete Itanium 2
  results.

188
Exploiting Matrix Structure

• Symmetry (numerical or structural)
• Reuse matrix entries
• Can combine with register blocking, multiple
  vectors,
• Matrix splitting
• Split the matrix, e.g., into r x c and 1 x 1
• No fill overhead
• Large matrices with random structure
• E.g., Latent Semantic Indexing (LSI) matrices
• Technique cache blocking
• Store matrix as 2i x 2j sparse submatrices
• Effective when x vector is large
• Currently, search to find fastest size

189
Symmetric SpMV Performance Pentium 4
190
SpMV with Split Matrices Ultra 2i
191
Cache Blocking on Random Matrices Itanium
Speedup on four banded random matrices.
192
Sparse Kernels and Optimizations

• Kernels
• Sparse matrix-vector multiply (SpMV) yAx
• Sparse triangular solve (SpTS) xT-1b
• yAATx, yATAx
• Powers (yAkx), sparse triple-product (RART),
  
• Optimization techniques (implementation space)
• Register blocking
• Cache blocking
• Multiple dense vectors (x)
• A has special structure (e.g., symmetric, banded,
  )
• Hybrid data structures (e.g., splitting,
  switch-to-dense, )
• Matrix reordering
• How and when do we search?
• Off-line Benchmark implementations
• Run-time Estimate matrix properties, evaluate
  performance models based on benchmark data

193
Register Blocked SpMV Pentium III
194
Register Blocked SpMV Ultra 2i
195
Register Blocked SpMV Power3
196
Register Blocked SpMV Itanium
197
Possible Optimization Techniques

• Within an iteration, i.e., computing (GuuT)x
  once
• Cache block Gx
• On linear programming matrices and matrices with
  random structure (e.g., LSI), 1.54x speedups
• Best block size is matrix and machine dependent
• Reordering and/or splitting of G to separate
  dense structure (rows, columns, blocks)
• Between iterations, e.g., (GuuT)2x
• (GuuT)2x G2x (Gu)uTx u(uTG)x u(uTu)uTx
• Compute Gu, uTG, uTu once for all iterations
• G2x Inter-iteration tiling to read G only once

198
Multiple Vector Performance Itanium
199
Sparse Kernels and Optimizations

• Kernels
• Sparse matrix-vector multiply (SpMV) yAx
• Sparse triangular solve (SpTS) xT-1b
• yAATx, yATAx
• Powers (yAkx), sparse triple-product (RART),
  
• Optimization techniques (implementation space)
• Register blocking
• Cache blocking
• Multiple dense vectors (x)
• A has special structure (e.g., symmetric, banded,
  )
• Hybrid data structures (e.g., splitting,
  switch-to-dense, )
• Matrix reordering
• How and when do we search?
• Off-line Benchmark implementations
• Run-time Estimate matrix properties, evaluate
  performance models based on benchmark data

200
SpTS Performance Itanium
(See POHLL 02 workshop paper, at ICS 02.)
201
Sparse Kernels and Optimizations

• Kernels
• Sparse matrix-vector multiply (SpMV) yAx
• Sparse triangular solve (SpTS) xT-1b
• yAATx, yATAx
• Powers (yAkx), sparse triple-product (RART),
  
• Optimization techniques (implementation space)
• Register blocking
• Cache blocking
• Multiple dense vectors (x)
• A has special structure (e.g., symmetric, banded,
  )
• Hybrid data structures (e.g., splitting,
  switch-to-dense, )
• Matrix reordering
• How and when do we search?
• Off-line Benchmark implementations
• Run-time Estimate matrix properties, evaluate
  performance models based on benchmark data

202
Optimizing AATx

• Kernel yAATx, where A is sparse, x y dense
• Arises in linear programming, computation of SVD
• Conventional implementation compute zATx,
  yAz
• Elements of A can be reused

• When ak represent blocks of columns, can apply
  register blocking.

203
Optimized AATx Performance Pentium III
204
Current Directions

• Applying new optimizations
• Other split data structures (variable block,
  diagonal, )
• Matrix reordering to create block structure
• Structural symmetry
• New kernels (triple product RART, powers Ak, )
• Tuning parameter selection
• Building an automatically tuned sparse matrix
  library
• Extending the Sparse BLAS
• Leverage existing sparse compilers as code
  generation infrastructure
• More thoughts on this topic tomorrow

205
Related Work

• Automatic performance tuning systems
• PHiPAC Bilmes, et al., 97, ATLAS Whaley
  Dongarra 98
• FFTW Frigo Johnson 98, SPIRAL Pueschel, et
  al., 00, UHFFT Mirkovic and Johnss