Autotuning sparse matrix kernels
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Transcript of Autotuning sparse matrix kernels
Autotuning sparse matrix kernels
Richard Vuduc Center for Applied Scientific Computing (CASC) Lawrence Livermore National Laboratory
February 28, 2007
Predictions (2003)
Need for “autotuning” will increase over time Improve performance for given app & machine using
automated experiments
Example: Sparse matrix-vector multiply (SpMV), 1987 to present Untuned: 10% of peak or less, decreasing Tuned: 2x speedup, increasing over time Tuning is getting harder (qualitative)
More complex machines & workloads Parallelism
Trends in uniprocessor SpMV performance (Mflop/s), pre-2004
Trends in uniprocessor SpMV performance (Mflop/s), pre-2004
Trends in uniprocessor SpMV performance (fraction of peak)
Is tuning getting easier?
// y <-- y + A*x
for all A(i,j):
y(i) += A(i,j) * x(j)
// Compressed sparse row (CSR)
for each row i:
t = 0
for k=ptr[i] to ptr[i+1]-1:
t += A[k] * x[J[k]]
y[i] = t
• Exploit 8x8 dense blocks
• As r x c Mflop/s
Speedups on Itanium 2: The need for search
ReferenceMflop/s (7.6%)
Mflop/s (31.1%)
Speedups on Itanium 2: The need for search
ReferenceMflop/s (7.6%)
Mflop/s (31.1%)
Best: 4x2
SpMV Performance—raefsky3
SpMV Performance—raefsky3
Better, worse, or about the same?Itanium 2, 900 MHz 1.3 GHz
Better, worse, or about the same?Itanium 2, 900 MHz 1.3 GHz
* Reference improves *
Better, worse, or about the same?Itanium 2, 900 MHz 1.3 GHz
* Best possible worsens slightly *
Better, worse, or about the same?Pentium M Core 2 Duo (1-core)
Better, worse, or about the same?Pentium M Core 2 Duo (1-core)
* Reference & best improve; relative speedup improves (~1.4 to 1.6) *
Better, worse, or about the same?Pentium M Core 2 Duo (1-core)
* Note: Best fraction of peak decreased from 11% 9.6% *
Better, worse, or about the same?Power4 Power5
Better, worse, or about the same?Power4 Power5
* Reference worsens! *
Better, worse, or about the same?Power4 Power5
* Relative importance of tuning increases *
A framework for performance tuningSource: SciDAC Performance Engineering Research Institute (PERI)
Outline
Motivation OSKI: An autotuned sparse kernel library Application-specific optimization “in the wild” Toward end-to-end application autotuning Summary and future work
Outline
Motivation OSKI: An autotuned sparse kernel library Application-specific optimization “in the wild” Toward end-to-end application autotuning Summary and future work
OSKI: Optimized Sparse Kernel Interface
Autotuned kernels for user’s matrix & machine BLAS-style interface: mat-vec (SpMV), tri. solve (TrSV), … Hides complexity of run-time tuning Includes fast locality-aware kernels: ATA*x, …
Faster than standard implementations Standard SpMV < 10% peak, vs. up to 31% with OSKI Up to 4x faster SpMV, 1.8x TrSV, 4x ATA*x, …
For “advanced” users & solver library writers PETSc extension available (OSKI-PETSc) Kokkos (for Trilinos) by Heroux Adopted by ClearShape, Inc. for shipping product (2x
speedup)
Tunable matrix-specific optimization techniques 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: 3x 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 AAT*x, ATA*x: 4x over CSR, 1.8x over RB A*x: 2x over CSR, 1.5x over RB
Tuning for workloads
Bi-conjugate gradients - equal mix of A*x and AT*y 3x1: Ax, ATy = 1053, 343 Mflop/s 517 Mflop/s 3x3: Ax, ATy = 806, 826 Mflop/s 816 Mflop/s
Higher-level operation - (Ax, ATy) kernel 3x1: 757 Mflop/s 3x3: 1400 Mflop/s
Matrix powers (Ak*x) with data structure transformations A2*x: up to 2x faster New latency-tolerant solvers? (Hoemmen’s thesis, on-going at
UCB)
How OSKI tunes (Overview)
Library Install-Time (offline) Application Run-Time
How OSKI tunes (Overview)
Benchmarkdata
1. Build forTargetArch.
2. Benchmark
Generatedcode
variants
Library Install-Time (offline) Application Run-Time
How OSKI tunes (Overview)
Benchmarkdata
1. Build forTargetArch.
2. Benchmark
Heuristicmodels
1. EvaluateModels
Generatedcode
variants
Library Install-Time (offline) Application Run-Time
Workloadfrom program
monitoring HistoryMatrix
How OSKI tunes (Overview)
Benchmarkdata
1. Build forTargetArch.
2. Benchmark
Heuristicmodels
1. EvaluateModels
Generatedcode
variants
2. SelectData Struct.
& Code
Library Install-Time (offline) Application Run-Time
To user:Matrix handlefor kernelcalls
Workloadfrom program
monitoring
Extensibility: Advanced users may write & dynamically add “Code variants” and “Heuristic models” to system.
HistoryMatrix
OSKI’s place in the tuning framework
Examples of OSKI’s early impact
Early adopter: ClearShape, Inc. Core product: lithography simulator 2x speedup on full simulation after using OSKI
Proof-of-concept: SLAC T3P accelerator cavity design simulator SpMV dominates execution time Symmetry, 2x2 block structure 2x speedups
OSKI-PETSc Performance: Accel. Cavity
Strengths and limitations of the library approach
Strengths Isolates optimization in the library for portable
performance Exploits domain-specific information aggressively Handles run-time tuning naturally
Limitations “Generation Me”: What about my application and its
abstractions? Run-time tuning: run-time overheads Limited context for optimization (without delayed
evaluation) Limited extensibility (fixed interfaces)
Outline
Motivation OSKI: An autotuned sparse kernel library Application-specific optimization “in the
wild” Toward end-to-end application autotuning Summary and future work
Tour of application-specific optimizations
Five case studies Common characteristics
Complex code Heavy use of abstraction Use generated code (e.g., SWIG C++/Python bindings)
Benefit from extensive code and data restructuring Multiple bottlenecks
[1] Loop transformations for SMG2000
SMG2000, implements semi-coarsening multigrid on structured grids (ASC Purple benchmark) Residual computation has an SpMV bottleneck Loop below looks simple but non-trivial to extract
for (si = 0; si < NS; ++si) for (k = 0; k < NZ; ++k) for (j = 0; j < NY; ++j) for (i = 0; i < NX; ++i) r[i + j*JR + k*KR] -= A[i + j*JA + k*KA + SA[si]] * x[i + j*JX + k*KX + Sx[si]]
[1] SMG2000 demo
[1] Before transformation
for (si = 0; si < NS; si++) /* Loop1 */ for (kk = 0; kk < NZ; kk++) { /* Loop2 */ for (jj = 0; jj < NY; jj++) { /* Loop3 */
for (ii = 0; ii < NX; ii++) { /* Loop4 */
r[ii + jj*Jr + kk*Kr] -= A[ii + jj*JA + kk*KA + SA[si]] * x[ii + jj*JA + kk*KA + SA[si]];
} /* Loop4 */
} /* Loop3 */ } /* Loop2 */ } /* Loop1 */
[1] After transformation, including interchange, unrolling, and prefetching
for (kk = 0; kk < NZ; kk++) { /* Loop2 */ for (jj = 0; jj < NY; jj++) { /* Loop3 */ for (si = 0; si < NS; si++) { /* Loop1 */ double* rp = r + kk*Kr + jj*Jr; const double* Ap = A + kk*KA + jj*JA + SA[si]; const double* xp = x + kk*Kx + jj*Jx + Sx[si]; for (ii = 0; ii <= NX-3; ii += 3) { /* core Loop4 */ _mm_prefetch (Ap + PFD_A, _MM_HINT_NTA); _mm_prefetch (xp + PFD_X, _MM_HINT_NTA); rp[0] -= Ap[0] * xp[0]; rp[1] -= Ap[1] * xp[1]; rp[2] -= Ap[2] * xp[2]; rp += 3; Ap += 3; xp += 3; } /* core Loop4 */ for ( ; ii < NX; ii++) { /* fringe Loop4 */ rp[0] -= Ap[0] * xp[0]; rp++; Ap++; xp++; } /* fringe Loop4 */ } /* Loop1 */ } /* Loop3 */ } /* Loop2 */
[1] Loop transformations for SMG2000
2x speedup on kernel from specialization, loop interchange, unrolling, prefetching But only 1.25x overall---multiple bottlenecks
Lesson: Need complex sequences of transformations Use profiling to guide Inspect run-time data for specialization Transformations are automatable
Research topic: Automated specialization of hypre?
[2] Slicing and dicing 3P
Accelerator design code from SLAC calcBasis() very expensive Scaling problems as |
Eigensystem| grows In principle, loop interchange or
precomputation via slicing possible
/* Post-processing phase */foreach mode in Eigensystem foreach elem in Mesh b = calcBasis (elem) f = calcField (b, mode)
[2] Slicing and dicing 3P
Accelerator design code calcBasis() very expensive Scaling problems as |
Eigensystem| grows In principle, loop interchange or
precomputation via slicing possible
Challenges in practice “Loop nest” ~ 500+ LOC 150+ LOC to calcBasis() calcBasis() in 6-deep call chain,
4-deep loop nest, 2 conditionals File I/O Changes must be unobtrusive
/* Post-processing phase */foreach mode in Eigensystem foreach elem in Mesh // { … b = calcBasis (elem) // } f = calcField (b, mode) writeDataToFiles (…);
[2] 3P: Impact and lessons
4-5x speedup for post-processing step; 1.5x overall
Changes “checked-in” Lesson: Need clean source-level transformations
To automate, need robust program analysis and developer guidance
Research: Annotation framework for developers [w/ Quinlan, Schordan, Yi: POHLL’06]
[3] Structure splitting
Convert (array of structs) into (struct of arrays) Improve spatial locality through increased stride-1 accesses Make code hardware-prefetch and vector/SIMD unit “friendly”c
struct Type { double p; double x, y, z; double E; int k;} X[N], Y[N];
for (i = 0; i < N; i++) Y[i].E += Y[X[i].k].p;
double Xp[N];double Xx[N], Xy[N], Xz[N];double XE[N];int Xk[N];// … same for Y …
for (i = 0; i < N; i++) YE[i] += sqrt (Yp[Xk[i]]);
[3] Structure splitting: Impact and challenges
2x speedup on a KULL benchmark (suggested by Brian Miller)
Implementation challenges Potentially affects entire code Can apply only locally, at a cost
Extra storage Overhead of copying
Tedious to do by hand
Lesson: Extensive data restructuring may be necessary
Research: When and how best to split?
[4] Finding a loop-fusion needle in a haystack
Interprocedural loop fusion finder [w/ B. White : Cornell U.] Known example had 2x speedup on benchmark (Miller) Built “abstraction-aware” analyzer using ROSE
First pass: Associate “loop signatures” with each function Second pass: Propagate signatures through call chains
for (Zone::iterator z = zones.begin (); z != zones.end (); ++z) for (Corner::iterator c = (*z).corners().begin (); …) for (int s = 0; s < c->sides().size(); s++) …
[4] Finding a loop-fusion needle in a haystack
Found 6 examples of 3- and 4-deep nested loops “Analysis-only” tool Finds, though does not verify/transform
Lesson: “Classical” optimizations relevant to abstraction use
Research Recognizing and optimizing abstractions [White’s thesis,
on-going] Extending traditional optimizations to abstraction use
[5] Aggregating messages (on-going)
Idea: Merge sends (suggested by Miller)
Implementing a fully automated translator to find and transform
Research: When and how best to aggregate?
DataType A;// … operations on A …A.allToAll();
// …
DataType B;// … operations on B …B.allToAll();
DataType A;// … operations on A …// …DataType B;// … operations on B …
bulkAllToAll(A, B);
Summary of application-specific optimizations
Like library-based approach, exploit knowledge for big gains Guidance from developer Use run-time information
Would benefit from automated transformation tools Real code is hard to process Changes may become part of software re-engineering Need robust analysis and transformation infrastructure Range of tools possible: analysis and/or transformation
No silver bullets or magic compilers
Outline
Motivation OSKI: An autotuned sparse kernel library “Real world” optimization Toward end-to-end application autotuning Summary and future work
A framework for performance tuningSource: SciDAC Performance Engineering Research Institute (PERI)
OSKI’s place in the tuning framework
An empirical tuning framework using ROSE
gprof,HPCtoolkitOpen SpeedShop
POET
Search engine
Empirical TuningFramework using ROSE
An end-to-end autotuning framework using ROSE
Guiding philosophy Leverage external stand-alone components Provide open components and tools for community
User or “system” profiles to collect data and/or analyses In ROSE
Mark-up AST with data/analysis, to identify optimizable target(s) Outline target into stand-alone dynamically loadable library
routine Make “benchmark” by inserting checkpoint library calls into app Generate parameterized representation of target
Independent search engine performs search
Interfaces to performance tools
Mark-up AST with data, analysis, to identify optimizable target(s) gprof HPCToolkit [Mellor-Crummey : Rice] VizzAnalyzer / Vizz3D [Panas : LLNL] In progress: Open SpeedShop [Schulz : LLNL]
Needed: Analysis to identify targets
Outlining
Outline target into dynamically loadable library routine Extends initial implementations by Liao [U. Houston], Jula
[TAMU] Handles many details of C & C++
Wraps up variables, inserts declarations, generates call Produces suitable interfaces for dynamic loading Handles non-local control flow
void OUT_38725__ (double* r, int JR, int KR, const double* A, …) { int si, j, k, i; for (si = 0; si < NS; si++) … r[i + j*JR + k*KR] -= A[i + …
Making a benchmark
Make “benchmark” by inserting checkpoint library calls Measure application behavior “in context” Use ckpt (user-level) [Zander : U. Wisc.] Insert timing code (cycle counter) May insert arbitrary code to distinguish calling contexts
Reasonably fast in practice Checkpoint read/write bandwidth: 500 MB/s on my Pentium-M For SMG2000: Problem consuming ~500 MB footprint takes ~30s to
run
Needed Best procedure to get accurate and fair comparisons?
Do restarts resume in comparable states? More portable checkpoint library
Example of “benchmark” (pseudo)code
static int num_calls = 0; // no. of invocations of outlined codeif (!num_calls) { ckpt (); // Checkpoint/resume OUT_38725__ = dlsym (…); // Load an implementation startTimer (); }
OUT_38725__ (…); // outlined call-site
if (++num_calls == CALL_LIMIT) { // Measured CALL_LIMIT calls stopTimer (); outputTime (); exit (0); }
Generating parameterized representations
Generate parameterized representation of target POET: Embedded scripting language for expressing
parameterized code variations [see POHLL’07] Loop optimizer will generate POET for each target
Hand-coded POET for SMG2000 Interchange Machine-specific: Unrolling, prefetching Source-specific: register & restrict keywords, C pointer
idiom New parameterization for loop fusion [Zhao,
Kennedy : Rice, Yi : UTSA]
SMG2000 kernel POET instantiation
for (kk = 0; kk < NZ; kk++) { /* L4 */ for (jj = 0; jj < NY; jj++) { /* L3 */ for (si = 0; si < NS; si++) { /* L1 */ double* rp = r + kk*Kr + jj*Jr; const double* Ap = A + kk*KA + jj*JA + SA[si]; const double* xp = x + kk*Kx + jj*Jx + Sx[si]; for (ii = 0; ii <= NX-3; ii += 3) { /* core L2 */ _mm_prefetch (Ap + PFD_A, _MM_HINT_NTA); _mm_prefetch (xp + PFD_X, _MM_HINT_NTA); rp[0] -= Ap[0] * xp[0]; rp[1] -= Ap[1] * xp[1]; rp[2] -= Ap[2] * xp[2]; rp += 3; Ap += 3; xp += 3; } /* core L2 */ for ( ; ii < NX; ii++) { /* fringe L2 */ rp[0] -= Ap[0] * xp[0]; rp++; Ap++; xp++; } /* fringe L2 */ } /* L1 */ } /* L3 */ } /* L4 */
Search
We are search-engine agnostics Many possible hybrid modeling/search techniques
Summary of autotuning compiler approach
End-to-end framework leverages existing work ROSE provides a heavy-duty (robust) source-level
infrastructure Assemble stand-alone components
Current and future work Assembling a more complete end-to-end example Interfaces between components? Extending basic ROSE infrastructure, particularly
program analysis
Current and future research directions
Autotuning End-to-end autotuning compiler framework Tuning for novel architectures (e.g., multicore) Tools for generating domain-specific libraries
Performance modeling Kernel- and machine-specific analytical and statistical
models Hybrid symbolic/empirical modeling Implications for applications and architectures?
Tools for debugging massively parallel applications JitterBug [w/ Schulz, Quinlan, de Supinski, Saebjoernsen] Static/dynamic analyses for debugging MPI
End
What is ROSE?
Research: Develop techniques to optimize applications that rely heavily on high-level abstractions Target scientific computing apps relevant to DOE/LLNL Domain-specific analysis and optimization Optimize use of object-oriented abstractions Performance portability via empirical tuning
Infrastructure: Tool for building source-to-source optimizers Full compiler: basic program analysis, loop optimizer, OpenMP
[UH] Support for C, C++; Fortran90 in progress Target “non-compiler audience” Open-source
What is ROSE?
Research: Develop techniques to optimize applications that rely heavily on high-level abstractions Target scientific computing apps relevant to DOE/LLNL Domain-specific analysis and optimization Optimize use of object-oriented abstractions Performance portability via empirical tuning
Infrastructure: Tool for building source-to-source optimizers Full compiler: basic program analysis, loop optimizer, OpenMP
[UH] Support for C, C++; Fortran90 in progress Target “non-compiler audience” Open-source
Bug hunting in MPI programs
Motivation: MPI is a large, complex API Bug pattern detectors
Check basic API usage Adapt existing tools: MPI-CHECK; FindBugs; Farchi, et al.
VC’05 Tasks requiring deeper program analysis
Properly matched sends/receives, barriers, collectives Buffer errors, e.g., overruns, read before non-blocking op
completes Temporal usage properties See error survey by DeSouza, Kuhn, & de Supinski ‘05 Extend existing analyses by Shires, et al., PDPTA’99;
Strout, et al. ICPP‘06
Compiler-based testing tools
Instrumentation and dynamic analysis to measure coverage [IBM]
Measurement-unit validation via Osprey [Jiang and Su, UC Davis]
Numerical interval/bounds analysis [Sun] Interface to MOPS model-checker [Collingbourne,
Imperial College] Interactive program visualization via VizzAnalyzer
[Panas, LLNL]
Trends in uniprocessor SpMV performance (absolute Mflop/s)
Trends in uniprocessor SpMV performance (fraction of peak)
Motivation: The Difficulty of Tuning SpMV
// y <-- y + A*x
for all A(i,j):
y(i) += A(i,j) * x(j)
Motivation: The Difficulty of Tuning SpMV
// y <-- y + A*x
for all A(i,j):
y(i) += A(i,j) * x(j)
// Compressed sparse row (CSR)
for each row i:
t = 0
for k=ptr[i] to ptr[i+1]-1:
t += A[k] * x[J[k]]
y[i] = t
Motivation: The Difficulty of Tuning SpMV
// y <-- y + A*x
for all A(i,j):
y(i) += A(i,j) * x(j)
// Compressed sparse row (CSR)
for each row i:
t = 0
for k=ptr[i] to ptr[i+1]-1:
t += A[k] * x[J[k]]
y[i] = t
• Exploit 8x8 dense blocks
Speedups on Itanium 2: The Need for Search
ReferenceMflop/s (7.6%)
Mflop/s (31.1%)
Speedups on Itanium 2: The Need for Search
ReferenceMflop/s (7.6%)
Mflop/s (31.1%)
Best: 4x2
SpMV Performance—raefsky3
SpMV Performance—raefsky3
Better, worse, or about the same?Pentium 4, 1.5 GHz Xeon, 3.2 GHz
Better, worse, or about the same?Pentium 4, 1.5 GHz Xeon, 3.2 GHz
* Faster, but relative improvement increases (20% ~50%) *
Problem-Specific Performance Tuning
Problem-Specific Optimization Techniques 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: 3x 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 AAT*x, ATA*x: 4x over CSR, 1.8x over RB A*x: 2x over CSR, 1.5x over RB
Problem-Specific Optimization Techniques 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: 3x 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 AAT*x, ATA*x: 4x over CSR, 1.8x over RB A*x: 2x over CSR, 1.5x over RB
BCSR Captures Regularly Aligned Blocks
n = 21216 nnz = 1.5 M Source: NASA
structural analysis problem
8x8 dense substructure
Reduces storage
Problem: Forced Alignment
BCSR(2x2) Stored / true nz = 1.24
Problem: Forced Alignment
BCSR(2x2) Stored / true nz = 1.24
BCSR(3x3) Stored / true nz = 1.46
Problem: Forced Alignment Implies UBCSR
BCSR(2x2) Stored / true nz = 1.24
BCSR(3x3) Stored / true nz = 1.46
Forces i mod 3 = j mod 3 = 0
Unaligned BCSR format: Store row indices
The Speedup GapThe Speedup Gap: BCSR vs. CSR
Speedup:BCSR/CSR
Machine
1.1—1.5x gap
Approach: Splitting + Relaxed Block Alignment
Goal: Close the gap between FEM classes
Our approach: Capture actual structure more precisely Split: A = A1 + A2 + … + As
Store each Ai in unaligned BCSR (UBCSR) format Relax both row and column alignment Buttari, et al. (2005) show improvements from relaxed
column alignment 2.1x over no blocking, 1.8x over blocking When not faster than BCSR, may still reduce storage
Variable Block Row (VBR) Analysis
Partition by grouping consecutive rows/columns having same pattern
From VBR, Identify Multiple Natural Block Sizes
VBR with Fill
Can also pad by matching rows/columns with nearly similar patterns
Define VBR() = VBR where consecutive
rows grouped when “similarity”
01
VBR with Fill
Fill of 1%
A Complex Tuning Problem
Many parameters need “tuning” Fill threshold, .5 1 Number of splittings, 2 s 4 Ordering of block sizes, rici; rscs = 11
See paper in HPCC 2005 for proof-of-concept experiments based on a semi-exhaustive search Heuristic in progress (uses Buttari, et al. (2005) work)
FEM 2 MatricesMatrix Dimensio
n# non-zeros
Dominant blocks
10-ct20stifEngine block
52k 2.7M 6x6 (39%), 3x3 (15%)
12-raefsky4Buckling
20k 1.3M 3x3 (96%)
13-ex11Fluid flow
16k 1.1M 1x1 (38%), 3x3 (23%)
15-Vavasis32D PDE
41k 1.7M 2x1 (81%), 2x2 (19%)
17-rimFluid flow
23k 1.0M 1x1 (75%), 3x1 (12%)
A-bmw7st_1Car chassis
141k 7.3M 6x6 (82%)
B-cop20k_mAccel. Cavity
121k 4.8M 2x1 (26%), 1x2 (26%),1x1 (26%), 2x2 (22%)
C-pwtkWind tunnel
218k 11.6M 6x6 (94%)
D-rma10Charleston Harbor
47k 2.4M 2x2 (17%), 3x2 (15%),2x3 (15%), 4x2 (9%), 2x4 (9%)
E-s3dkqm4Cylindrical shell
90k 4.8M 6x6 (99%)
Power 4 Performance
Storage Savings
Traveling Salesman Problem-based Reordering
Application: Stanford accelerator design problem (Omega3P)
Reorder by approximately solving TSP [Pinar & Heath ‘97] 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] Also: symmetric storage, register blocking
Manually selected optimizations Problem: High-cost of computing approximate
solution to TSP
100x100 Submatrix Along Diagonal
“Microscopic” Effect of Combined RCM+TSP Reordering
Before: Green + RedAfter: Green + Blue
Inter-Iteration Sparse Tiling (1/3)
y1
y2
y3
y4
y5
t1
t2
t3
t4
t5
x1
x2
x3
x4
x5
Idea: Strout, et al., ICCS 2001
Let A be 5x5 tridiagonal
Consider y=A2x t=Ax, y=At
Nodes: vector elements
Edges: matrix elements aij
Inter-Iteration Sparse Tiling (2/3)
y1
y2
y3
y4
y5
t1
t2
t3
t4
t5
x1
x2
x3
x4
x5
Idea: Strout, et al., ICCS 2001
Let A be 5x5 tridiagonal
Consider y=A2x t=Ax, y=At
Nodes: vector elements Edges: matrix elements
aij
Orange = everything needed to compute y1
Reuse a11, a12
Inter-Iteration Sparse Tiling (3/3)
Idea: Strout, et al., ICCS 2001
Let A be 5x5 tridiagonal Consider y=A2x
t=Ax, y=At Nodes: vector elements Edges: matrix elements aij
Orange = everything needed to compute y1
Reuse a11, a12
Grey = y2, y3 Reuse a23, a33, a43
y1
y2
y3
y4
y5
t1
t2
t3
t4
t5
x1
x2
x3
x4
x5
Serial Sparse Tiling Performance (Itanium 2)
OSKI Software Architecture and API
Empirical Model Evaluation
Tuning loop Compute a “tuning time budget” based on workload While (time remains and no tuning chosen)
Try a heuristic
Heuristic for blocked SpMV: Choose r x c to minimize
predicted time(A,r,c)estimated flops(A,r,c)
benchmark Mflop /s(r,c)
Tuning for workloads Weighted sums of empirical models Dynamic programming for alternatives
Example: Combined y=ATAx vs. separate (w=Ax, y=ATw)
Cost of Tuning
Non-trivial run-time cost: up to ~40 mat-vecs Dominated by conversion time (~ 80%)
Design point: user calls “tune” routine explicitly Exposes cost Tuning time limited using estimated workload
Provided by user or inferred by library
User may save tuning results To apply on future runs with similar matrix Stored in “human-readable” format
Interface supports legacy app migrationint* ptr = …, *ind = …; double* val = …; /* Matrix A, in CSR format */
double* x = …, *y = …; /* Vectors */
/* Compute y = ·y + ·A·x, 500 times */for( i = 0; i < 500; i++ )
my_matmult( ptr, ind, val, , x, , y );r = ddot (x, y); /* Some dense BLAS op on vectors */
Interface supports legacy app migrationint* ptr = …, *ind = …; double* val = …; /* Matrix A, in CSR format */
double* x = …, *y = …; /* Vectors */
/* Step 1: Create OSKI wrappers */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 + ·A·x, 500 times */for( i = 0; i < 500; i++ )
my_matmult( ptr, ind, val, , x, , y );r = ddot (x, y);
Interface supports legacy app migrationint* ptr = …, *ind = …; double* val = …; /* Matrix A, in CSR format */
double* x = …, *y = …; /* Vectors */
/* Step 1: Create OSKI wrappers */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);
/* Step 2: Call tune (with optional hints) */oski_SetHintMatMult (A_tunable, …, 500);oski_TuneMat (A_tunable);
/* Compute y = ·y + ·A·x, 500 times */for( i = 0; i < 500; i++ ) my_matmult( ptr, ind, val, , x, , y );r = ddot (x, y);
Interface supports legacy app migrationint* ptr = …, *ind = …; double* val = …; /* Matrix A, in CSR format */
double* x = …, *y = …; /* Vectors */
/* Step 1: Create OSKI wrappers */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);
/* Step 2: Call tune (with optional hints) */oski_setHintMatMult (A_tunable, …, 500);oski_TuneMat (A_tunable);
/* Compute y = ·y + ·A·x, 500 times */for( i = 0; i < 500; i++ ) oski_MatMult (A_tunable, OP_NORMAL, , x_view, , y_view);// Step 3r = ddot (x, y);
Quick-and-dirty Parallelism: OSKI-PETSc
Extend PETSc’s distributed memory SpMV (MATMPIAIJ)
p0
p1
p2
p3
PETSc Each process stores
diag (all-local) and off-diag submatrices
OSKI-PETSc: Add OSKI wrappers Each submatrix tuned
independently
OSKI-PETSc Proof-of-Concept Results
Matrix 1: Accelerator cavity design (R. Lee @ 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
Accelerator cavity matrix from SLAC’s T3P code
Embedded scripting language for selecting customized, complex transformations
Mechanism to save/restore transformations
# In file, “my_xform.txt”
# Compute Afast = P*A*PT using Pinar’s reordering algorithm
A_fast, P = reorder_TSP(InputMat);
# Split Afast = A1 + A2, where A1 in 2x2 block format, A2 in CSR
A1, A2 = A_fast.extract_blocks(2, 2);
return transpose(P)*(A1+A2)*P;
/* In “my_app.c” */fp = fopen(“my_xform.txt”, “rt”);fgets(buffer, BUFSIZE, fp);
oski_ApplyMatTransform(A_tunable, buffer);
oski_MatMult(A_tunable, …);
Additional Features: OSKI-Lua
Current Work and Future Directions
Current and Future Work on OSKI
OSKI 1.0.1 at bebop.cs.berkeley.edu/oski “Pre-alpha” version of OSKI-PETSc available; “Beta” for Kokkos
(Trilinos) Future work
Evaluation on full solves/apps Bay area lithography shop - 2x speedup in full solve Code generators Studying use of higher-level OSKI kernels
Port to additional architectures (e.g., vectors, SMPs) Additional heuristics [Buttari, et al. (2005)] Many BeBOP projects on-going
SpMV benchmark for HPC-Challenge [Gavari & Hoemmen] Evaluation of Cell [Williams] Higher-level kernels, solvers [Hoemmen, Nishtala] Tuning collective communications [Nishtala] Cache-oblivious stencils [Kamil]
ROSE: A Compiler-Based Approach to Tuning General Applications ROSE: Tool for building customized source-to-source tools (Quinlan,
et al.) Full support for C and C++; Fortran90 in development Targets users with little or no compiler background
Focus on performance optimization for scientific computing Domain-specific analysis and optimizations Object-oriented abstraction recognition Rich loop-transformation support Annotation language support Additional infrastructure support for s/w assurance, testing, and
debugging Toward an end-to-end empirical tuning compiler
Combines profiling, checkpointing, analysis, parameterized code generation, search
Joint work with Qing Yi (University of Texas at San Antonio) Sponsored by U.S. Department of Energy
ROSE Architecture
Front-end (EDG-based)
Back-end
Transformed application source
Application Library Interface
Mid-end
Source
fragmentAST fragment
AST fragmentSource
fragment
AST fragment
AST
AST
Annotations
Tools
Abtraction RecognitionAbstraction Aware Analysis
Abstraction EliminationExtended Traditional Optimizations
Source+AST Transformations