1. Automatic Performance Tuning of Sparse-Matrix-Vector-Multiplication (SpMV) and Iterative Sparse Solvers James Demmel www.cs.berkeley.edu/~demmel/cs267_Spr09

2. Outline Motivation for Automatic Performance Tuning Results for sparse matrix kernels –Sparse Matrix Vector Multiplication (SpMV) –Sequential and Multicore results OSKI = Optimized Sparse Kernel Interface Tuning Entire Sparse Solvers

3. Berkeley Benchmarking and OPtimization (BeBOP) Prof. Katherine Yelick Current members –Kaushik Datta, Mark Hoemmen, Marghoob Mohiyuddin, 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

4. Automatic Performance Tuning Goal: Let machine do hard work of writing fast code What does tuning of dense BLAS, FFTs, signal processing, 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 dimensions, size of FFT, etc.) Can’t always do tuning off-line –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

5. Source: Accelerator Cavity Design Problem (Ko via Husbands)

6. Linear Programming Matrix …

7. A Sparse Matrix You Encounter Every Day

8. Matrix-vector multiply kernel: y (i)  y (i) + A (i,j) *x (j) for each row i for k=ptr[i] to ptr[i+1] do y[i] = y[i] + val[k]*x[ind[k]] 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 k=ptr[i] to ptr[i+1] do y[i] = y[i] + val[k]*x[ind[k]] Only 2 flops per 2 mem_refs: Limited by getting data from memory

9. Example: The Difficulty of Tuning n = 21200 nnz = 1.5 M kernel: SpMV Source: NASA structural analysis problem

10. Example: The Difficulty of Tuning n = 21200 nnz = 1.5 M kernel: SpMV Source: NASA structural analysis problem 8x8 dense substructure: exploit this to limit #mem_refs

11. Speedups on Itanium 2: The Need for Search Reference Best: 4x2 Mflop/s

12. Register Profile: Itanium 2 190 Mflop/s 1190 Mflop/s

13. Register Profiles: IBM and Intel IA-64 Power3 - 17%Power4 - 16% Itanium 2 - 33%Itanium 1 - 8% 252 Mflop/s 122 Mflop/s 820 Mflop/s 459 Mflop/s 247 Mflop/s 107 Mflop/s 1.2 Gflop/s 190 Mflop/s

14. Register Profiles: Sun and Intel x86 Ultra 2i - 11%Ultra 3 - 5% Pentium III-M - 15%Pentium III - 21% 72 Mflop/s 35 Mflop/s 90 Mflop/s 50 Mflop/s 108 Mflop/s 42 Mflop/s 122 Mflop/s 58 Mflop/s

15. Another example of tuning challenges More complicated non-zero structure in general N = 16614 NNZ = 1.1M

16. Zoom in to top corner More complicated non-zero structure in general N = 16614 NNZ = 1.1M

17. 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”

18. 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 1.5 2 = 2.25 higher

19. 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)

20. Accurate and Efficient Adaptive Fill Estimation Idea: Sample matrix –Fraction of matrix to sample: s  [0,1] –Control cost = O(s * nnz ) by controlling s Search at run-time: the constant matters! Control s automatically by computing statistical confidence intervals, by monitoring variance Cost of tuning –Lower bound: convert matrix in 5 to 40 unblocked SpMVs –Heuristic: 1 to 11 SpMVs Tuning only useful when we do many SpMVs –Common case, eg in sparse solvers

21. Accuracy of the Tuning Heuristics (1/4) NOTE: “Fair” flops used (ops on explicit zeros not counted as “work”) See p. 375 of Vuduc’s thesis for matrices

22. Accuracy of the Tuning Heuristics (2/4)

23. DGEMV

24. Upper Bounds on Performance for blocked SpMV P = (flops) / (time) –Flops = 2 * nnz(A) Upper bound on speed: 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”  i at L i cache &  mem –e.g., Saavedra-Barrera and PMaC MAPS benchmarks Upper bound on time: assume all references to x( ) miss

25. Example: Bounds on Itanium 2

28. 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 –A·A T ·x, A T ·A·x: 4x over CSR, 1.8x over RB –A  ·x: 2x over CSR, 1.5x over RB –[A·x, A  ·x, A  ·x,.., A k ·x] …. more to say later

29. 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 IBM Power 4

30. Source: Accelerator Cavity Design Problem (Ko via Husbands)

31. Post-RCM Reordering

32. 100x100 Submatrix Along Diagonal

33. Before: Green + Red After: Green + Blue “Microscopic” Effect of RCM Reordering

34. “Microscopic” Effect of Combined RCM+TSP Reordering Before: Green + Red After: Green + Blue

35. (Omega3P)

36. Optimized Sparse Kernel Interface - OSKI Provides sparse kernels automatically tuned for user’s matrix & machine –BLAS-style functionality: SpMV, Ax & A T y, TrSV –Hides complexity of run-time tuning –Includes new, faster locality-aware kernels: A T Ax, A k x Faster than standard implementations –Up to 4x faster matvec, 1.8x trisolve, 4x A T A*x 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

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

38. 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 +  ·A·x, 500 times */ for( i = 0; i < 500; i++ ) my_matmult( ptr, ind, val, , x, , y );

39. 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 +  ·A·x, 500 times */ for( i = 0; i < 500; i++ ) my_matmult( ptr, ind, val, , x, , y );

40. 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 +  ·A·x, 500 times */ for( i = 0; i < 500; i++ ) oski_MatMult(A_tunable, OP_NORMAL, , x_view, , y_view); /* Step 2 */

41. 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_tunable, 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_tunable); /* Ask OSKI to tune */ for( i = 0; i < 500; i++ ) oski_MatMult(A_tunable, OP_NORMAL, , x_view, , y_view);

42. 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 < 500; i++ ) { oski_MatMult(A_tunable, OP_NORMAL, , x_view, , y_view); oski_TuneMat(A_tunable); /* Ask OSKI to tune */ }

43. 43 Multicore SMPs Used for Tuning SpMV AMD Opteron 2356 (Barcelona)Intel Xeon E5345 (Clovertown) IBM QS20 Cell BladeSun T2+ T5140 (Victoria Falls)

44. 44 Multicore SMPs with Conventional cache-based memory hierarchy AMD Opteron 2356 (Barcelona)Intel Xeon E5345 (Clovertown) IBM QS20 Cell Blade Sun T2+ T5140 (Victoria Falls)

45. 45 Multicore SMPs with local store-based memory hierarchy AMD Opteron 2356 (Barcelona)Intel Xeon E5345 (Clovertown) IBM QS20 Cell Blade Sun T2+ T5140 (Victoria Falls)

46. 46 Multicore SMPs with CMT = Chip-MultiThreading AMD Opteron 2356 (Barcelona)Intel Xeon E5345 (Clovertown) IBM QS20 Cell Blade Sun T2+ T5140 (Victoria Falls)

47. 47 Multicore SMPs: Number of threads AMD Opteron 2356 (Barcelona)Intel Xeon E5345 (Clovertown) IBM QS20 Cell BladeSun T2+ T5140 (Victoria Falls) 8 threads 16 * threads 128 threads * SPEs only

48. 48 Multicore SMPs: peak double precision flops AMD Opteron 2356 (Barcelona)Intel Xeon E5345 (Clovertown) IBM QS20 Cell BladeSun T2+ T5140 (Victoria Falls) 75 GFlop/s74 Gflop/s 29* GFlop/s19 GFlop/s * SPEs only

49. 49 Multicore SMPs: total DRAM bandwidth AMD Opteron 2356 (Barcelona)Intel Xeon E5345 (Clovertown) IBM QS20 Cell BladeSun T2+ T5140 (Victoria Falls) 21 GB/s (read) 10 GB/s (write) 21 GB/s 51 GB/s 42 GB/s (read) 21 GB/s (write) * SPEs only

50. 50 Multicore SMPs with Non-Uniform Memory Access - NUMA AMD Opteron 2356 (Barcelona)Intel Xeon E5345 (Clovertown) IBM QS20 Cell BladeSun T2+ T5140 (Victoria Falls) * SPEs only

51. 51 Set of 14 test matrices All bigger than the caches of our SMPs Dense Protein FEM / Spheres FEM / Cantilever Wind Tunnel FEM / Harbor QCD FEM / Ship EconomicsEpidemiology FEM / Accelerator Circuitwebbase LP 2K x 2K Dense matrix stored in sparse format Well Structured (sorted by nonzeros/row) Poorly Structured hodgepodge Extreme Aspect Ratio (linear programming)

52. 52 SpMV Performance: Naive parallelization Out-of-the box SpMV performance on a suite of 14 matrices Scalability isn’t great: Compare to # threads 8 8 128 16 Naïve Pthreads Naïve

53. SpMV Performance: NUMA and Software Prefetching 53  NUMA-aware allocation is essential on NUMA SMPs.  Explicit software prefetching can boost bandwidth and change cache replacement policies  used exhaustive search

54. SpMV Performance: “Matrix Compression” 54  Compression includes  register blocking  other formats  smaller indices  Use heuristic rather than search

55. 55 SpMV Performance: cache and TLB blocking +Cache/LS/TLB Blocking +Matrix Compression +SW Prefetching +NUMA/Affinity Naïve Pthreads Naïve

56. 56 SpMV Performance: Architecture specific optimizations +Cache/LS/TLB Blocking +Matrix Compression +SW Prefetching +NUMA/Affinity Naïve Pthreads Naïve

57. 57 SpMV Performance: max speedup Fully auto-tuned SpMV performance across the suite of matrices Included SPE/local store optimized version Why do some optimizations work better on some architectures? +Cache/LS/TLB Blocking +Matrix Compression +SW Prefetching +NUMA/Affinity Naïve Pthreads Naïve 2.7x 4.0x 2.9x 35x

58. Avoiding Communication in Sparse Linear Algebra k-steps of typical iterative solver for Ax=b or Ax=λx –Does k SpMVs with starting vector (eg with b, if solving Ax=b) –Finds “best” solution among all linear combinations of these k+1 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

59. x Ax A2xA2x A3xA3x A4xA4x A5xA5x A6xA6x A7xA7x A8xA8x Locally Dependent Entries for [x,Ax], A tridiagonal, 2 processors Can be computed without communication Proc 1 Proc 2

60. x Ax A2xA2x A3xA3x A4xA4x A5xA5x A6xA6x A7xA7x A8xA8x Can be computed without communication Proc 1 Proc 2 Locally Dependent Entries for [x,Ax,A 2 x], A tridiagonal, 2 processors

61. x Ax A2xA2x A3xA3x A4xA4x A5xA5x A6xA6x A7xA7x A8xA8x Can be computed without communication Proc 1 Proc 2 Locally Dependent Entries for [x,Ax,…,A 3 x], A tridiagonal, 2 processors

62. x Ax A2xA2x A3xA3x A4xA4x A5xA5x A6xA6x A7xA7x A8xA8x Can be computed without communication Proc 1 Proc 2 Locally Dependent Entries for [x,Ax,…,A 4 x], A tridiagonal, 2 processors

63. x Ax A2xA2x A3xA3x A4xA4x A5xA5x A6xA6x A7xA7x A8xA8x Locally Dependent Entries for [x,Ax,…,A 8 x], A tridiagonal, 2 processors Can be computed without communication k=8 fold reuse of A Proc 1 Proc 2

64. Remotely Dependent Entries for [x,Ax,…,A 8 x], A tridiagonal, 2 processors x Ax A2xA2x A3xA3x A4xA4x A5xA5x A6xA6x A7xA7x A8xA8x One message to get data needed to compute remotely dependent entries, not k=8 Minimizes number of messages = latency cost Price: redundant work  “surface/volume ratio” Proc 1 Proc 2

65. Remotely Dependent Entries for [x,Ax,A 2 x,A 3 x], A irregular, multiple processors

66. Sequential [x,Ax,…,A 4 x], with memory hierarchy v One read of matrix from slow memory, not k=4 Minimizes words moved = bandwidth cost No redundant work

67. Performance results on 8-Core Clovertown

68. Optimizing Communication Complexity of Sparse Solvers Example: GMRES for Ax=b on “2D Mesh” –x lives on n-by-n mesh –Partitioned on p ½ -by- p ½ grid of p processors –A has “5 point stencil” (Laplacian) (Ax)(i,j) = linear_combination(x(i,j), x(i,j±1), x(i±1,j)) –Ex: 18-by-18 mesh on 3-by-3 grid of 9 processors

69. Minimizing Communication of GMRES What is the cost = (#flops, #words, #mess) of k steps of standard GMRES? GMRES, ver.1: for i=1 to k w = A * v(i-1) MGS(w, v(0),…,v(i-1)) update v(i), H endfor solve LSQ problem with H n/p ½ Cost(A * v) = k * (9n 2 /p, 4n / p ½, 4 ) Cost(MGS) = k 2 /2 * ( 4n 2 /p, log p, log p ) Total cost ~ Cost( A * v ) + Cost (MGS) Can we reduce the latency?

70. Minimizing Communication of GMRES Cost(GMRES, ver.1) = Cost(A*v) + Cost(MGS) Cost(W) = ( ~ same, ~ same, 8 ) Latency cost independent of k – optimal Cost (MGS) unchanged Can we reduce the latency more? = ( 9kn 2 /p, 4kn / p ½, 4k ) + ( 2k 2 n 2 /p, k 2 log p / 2, k 2 log p / 2 ) How much latency cost from A*v can you avoid? Almost all GMRES, ver. 2: W = [ v, Av, A 2 v, …, A k v ] [Q,R] = MGS(W) Build H from R, solve LSQ problem k = 3

71. Minimizing Communication of GMRES Cost(GMRES, ver. 2) = Cost(W) + Cost(MGS) = ( 9kn 2 /p, 4kn / p ½, 8 ) + ( 2k 2 n 2 /p, k 2 log p / 2, k 2 log p / 2 ) How much latency cost from MGS can you avoid? Almost all Cost(TSQR) = ( ~ same, ~ same, log p ) Latency cost independent of s - optimal GMRES, ver. 3: W = [ v, Av, A 2 v, …, A k v ] [Q,R] = TSQR(W) … “Tall Skinny QR” Build H from R, solve LSQ problem W = W1W2W3W4W1W2W3W4 R1R2R3R4R1R2R3R4 R 12 R 34 R 1234

72. Minimizing Communication of GMRES Cost(GMRES, ver. 2) = Cost(W) + Cost(MGS) = ( 9kn 2 /p, 4kn / p ½, 8 ) + ( 2k 2 n 2 /p, k 2 log p / 2, k 2 log p / 2 ) How much latency cost from MGS can you avoid? Almost all GMRES, ver. 3: W = [ v, Av, A 2 v, …, A k v ] [Q,R] = TSQR(W) … “Tall Skinny QR” Build H from R, solve LSQ problem W = W1W2W3W4W1W2W3W4 R1R2R3R4R1R2R3R4 R 12 R 34 R 1234 Cost(TSQR) = ( ~ same, ~ same, log p ) Oops

73. Minimizing Communication of GMRES Cost(GMRES, ver. 2) = Cost(W) + Cost(MGS) = ( 9kn 2 /p, 4kn / p ½, 8 ) + ( 2k 2 n 2 /p, k 2 log p / 2, k 2 log p / 2 ) How much latency cost from MGS can you avoid? Almost all GMRES, ver. 3: W = [ v, Av, A 2 v, …, A k v ] [Q,R] = TSQR(W) … “Tall Skinny QR” Build H from R, solve LSQ problem W = W1W2W3W4W1W2W3W4 R1R2R3R4R1R2R3R4 R 12 R 34 R 1234 Cost(TSQR) = ( ~ same, ~ same, log p ) Oops – W from power method, precision lost!

75. Minimizing Communication of GMRES Cost(GMRES, ver. 3) = Cost(W) + Cost(TSQR) = ( 9kn 2 /p, 4kn / p ½, 8 ) + ( 2k 2 n 2 /p, k 2 log p / 2, log p ) Latency cost independent of k, 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, A 2 v, …], instead … Newton Basis W N = [v, (A – θ 1 I)v, (A – θ 2 I)(A – θ 1 I)v, …] or Chebyshev Basis W C = [v, T 1 (v), T 2 (v), …]

77. Speed ups on 8-core Clovertown

78. Conclusions Fast code must minimize communication –Especially for sparse matrix computations because communication dominates Generating fast code for a single SpMV –Design space of possible algorithms must be searched at run-time, when sparse matrix available –Design space should be searched automatically Biggest speedups from minimizing communication in an entire sparse solver –Many more opportunities to minimize communication in multiple SpMVs than in one –Requires transforming entire algorithm –Lots of open problems

79. EXTRA SLIDES

80. 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

81. 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

82. Accelerator Cavity Matrix

83. OSKI-PETSc Performance: Accel. Cavity

84. Linear Programming Matrix …

85. OSKI-PETSc Performance: LP Matrix

86. 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

87. Speedups on Intel Clovertown (8 core)

88. Extensions Other Krylov methods –Arnoldi, CG, Lanczos, … Preconditioning –Solve MAx=Mb 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

89. Design Space for [x,Ax,…,A k x] (1/3) Mathematical Operation –Keep all vectors Krylov Subspace Methods –Improving conditioning of basis W = [x, p 1 (A)x, p 2 (A)x,…,p k (A)x] p i (A) = degree i polynomial chosen to reduce cond(W) –Preconditioning (Ay=b  MAy=Mb) [x,Ax,MAx,AMAx,MAMAx,…,(MA) k x] –Keep last vector A k x only Jacobi, Gauss Seidel

90. Design Space for [x,Ax,…,A k x] (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 patternImplicit pattern Explicit nonzerosGeneral sparse matrixImage segmentation Implicit nonzerosLaplacian(graph), for graph partitioning “Stencil matrix” Ex: tridiag(-1,2,-1) Representation of dense preconditioners M – Low rank off-diagonal blocks (semiseparable)

91. Design Space for [x,Ax,…,A k x] (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!

92. Summary Communication-Avoiding Linear Algebra (CALA) Lots of related work –Some going back to 1960’s –Reports discuss this comprehensively, not here Our contributions –Several new algorithms, improvements on old ones –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?

93. Possible Class Projects Come to BEBOP meetings (T 9 – 10:30, 606 Soda) Incorporate multicore optimizations into OSKI 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,…,A k x] kernel –GPU, multicore, distributed memory –On matrices of interest –Experiment with new solvers using this kernel

94. Extra Slides

95. Tuning Higher Level Algorithms So far we have tuned a single sparse matrix kernel y = A T *A*x motivated by higher level algorithm (SVD) What can we do by extending tuning to a higher level? Consider Krylov subspace methods for Ax=b, Ax = x Conjugate Gradients (CG), GMRES, Lanczos, … Inner loop does y=A*x, 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, …

96. Krylov Subspace Methods for Solving Ax=b Compute a basis for a subspace V by doing y = A*x k times Find “best” solution in that Krylov subspace V Given starting vector x 1, V spanned by x 2 = A*x 1, x 3 = A*x 2, …, x k = A*x k-1 GMRES finds an orthogonal basis of V by Gram-Schmidt, so it actually does a different set of SpMVs than in last bullet

97. Example: Standard GMRES r = b - A*x 1,  = length(r), v 1 = r /  … length(r) = sqrt(  r i 2 ) for m= 1 to k do w = A*v m … at least O(1) messages for i = 1 to m do … Gram-Schmidt h im = dotproduct(v i, w ) … at least O(1) messages, or log(p) w = w – h im * v i end for h m+1,m = length(w) … at least O(1) messages, or log(p) v m+1 = w / h m+1,m end for find y minimizing length( H k * y –  e 1 ) … small, local problem new x = x 1 + V k * y … V k = [v 1, v 2, …, v k ] O(k 2 ), or O(k 2 log p ), messages altogether

98. Latency-Avoiding GMRES (1) r = b - A*x 1,  = length(r), v 1 = r /  …  O(log p) messages W k+1 = [v 1, A * v 1, A 2 * v 1, …, A k * v 1 ] … O(1) messages [Q, R] = qr(W k+1 ) … QR decomposition, O(log p) messages H k = R(:, 2:k+1) * (R(1:k,1:k)) -1 … small, local problem find y minimizing length( H k * y –  e 1 ) … small, local problem new x = x 1 + Q k * y … local problem O(log p) messages altogether Independent of k

99. Latency-Avoiding GMRES (2) [Q, R] = qr(W k+1 ) … QR decomposition, O(log p) messages Easy, but not so stable way to do it: X(myproc) = W k+1 T (myproc) * W k+1 (myproc) … local computation Y = sum_reduction(X(myproc)) … O(log p) messages … Y = W k+1 T * W k+1 R = (cholesky(Y)) T … small, local computation Q(myproc) = W k+1 (myproc) * R -1 … local computation

100. Numerical example (1) Diagonal matrix with n=1000, A ii from 1 down to 10 -5 Instability as k grows, after many iterations

101. Numerical Example (2) Partial remedy: restarting periodically (every 120 iterations) Other remedies: high precision, different basis than [v, A * v, …, A k * v ]

102. Operation Counts for [Ax,A 2 x,A 3 x,…,A k x] on p procs ProblemPer-proc costStandardOptimized 1D mesh#messages2k2 (tridiagonal)#words sent2k #flops5kn/p 5kn/p + 5k 2 memory(k+4)n/p(k+4)n/p + 8k 3D mesh#messages26k26 27 pt stencil#words sent 6kn 2 p -2/3 + 12knp -1/3 + O(k) 6kn 2 p -2/3 + 12k 2 np -1/3 + O(k 3 ) #flops 53kn 3 /p53kn 3 /p + O(k 2 n 2 p -2/3 ) memory (k+28)n 3 /p+ 6n 2 p -2/3 + O(np -1/3 ) (k+28)n 3 /p + 168kn 2 p -2/3 + O(k 2 np -1/3 )

103. 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

104. Extra Slides

105. 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.)

106. 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 =  G + (1-  )(1/n)uu T –Markov model: Surfer follows link with probability , jumps to a random page with probability 1-  –G is n x n connectivity matrix [n  billions] g ij is non-zero if page i links to page j Normalized so each column sums to 1 Very sparse: about 7—8 non-zeros per row (power law dist.) –u is a vector of all 1 values –Steady-state probability x i of landing on page i is solution to x = Ax Approximate x by power method: x = A k x 0 –In practice, k  25

107. 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 ( A T A*X ; LBL) Systematic heuristics for data structure selection? Evaluation of emerging architectures –Revisiting vector micros Other sparse kernels –Matrix triple products, A k *x Parallelism Sparse benchmarks (with UTK) [Gahvari & Hoemmen] Automatic tuning of MPI collective ops [Nishtala, et al.]

108. 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

109. 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

110. 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

111. 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)

112. Review of Tuning by Illustration (Extra Slides)

113. Splitting for Variable Blocks and Diagonals Decompose A = A 1 + A 2 + … A t –Detect “canonical” structures (sampling) –Split –Tune each A i –Improve performance and save storage New data structures –Unaligned block CSR Relax alignment in rows & columns –Row-segmented diagonals

114. Example: Variable Block Row (Matrix #12) 2.1x over CSR 1.8x over RB

115. Example: Row-Segmented Diagonals 2x over CSR

116. Mixed Diagonal and Block Structure

117. Summary Automated block size selection –Empirical modeling and search –Register blocking for SpMV, triangular solve, A T A*x 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)

118. Extra Slides

119. 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.)

120. 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

121. 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 A T A*x; 2x for A 2 *x 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

122. 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

123. Matrix-vector multiply kernel: y (i)  y (i) + A (i,j) *x (j) for each row i for k=ptr[i] to ptr[i+1] do y[i] = y[i] + val[k]*x[ind[k]] Compressed Sparse Row (CSR) Storage Matrix-vector multiply kernel: y (i)  y (i) + A (i,j) *x (j) for each row i for k=ptr[i] to ptr[i+1] do y[i] = y[i] + val[k]*x[ind[k]]

124. 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

125. Historical Trends in SpMV Performance The Data –Uniprocessor SpMV performance since 1987 –“Untuned” and “Tuned” implementations –Cache-based superscalar micros; some vectors –LINPACK

126. SpMV Historical Trends: Mflop/s

127. Example: The Difficulty of Tuning n = 21216 nnz = 1.5 M kernel: SpMV Source: NASA structural analysis problem

128. Still More Surprises More complicated non-zero structure in general

129. Still More Surprises More complicated non-zero structure in general Example: 3x3 blocking –Logical grid of 3x3 cells

130. Historical Trends: Mixed News Observations –Good news: Moore’s 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?

131. Road Map Sparse matrix-vector multiply (SpMV) in a nutshell Historical trends and the need for search Automatic tuning techniques –SpMV [SC’02; IJHPCA ’04b] –Sparse triangular solve (SpTS) [ICS/POHLL ’02] –A T A*x [ICCS/WoPLA ’03] Upper bounds on performance Statistical models of performance

132. 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 What’s 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, A T A*x, A  *x

133. 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)

134. Accuracy of the Tuning Heuristics (1/4) NOTE: “Fair” flops used (ops on explicit zeros not counted as “work”) DGEMV

135. Accuracy of the Tuning Heuristics (2/4) DGEMV

136. Accuracy of the Tuning Heuristics (3/4) DGEMV

137. Accuracy of the Tuning Heuristics (4/4) DGEMV

138. Road Map Sparse matrix-vector multiply (SpMV) in a nutshell Historical trends and the need for search Automatic tuning techniques Upper bounds on performance –SC’02 Statistical models of performance

139. 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?

140. 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”  i at L i cache &  mem –e.g., Saavedra-Barrera and PMaC MAPS benchmarks

141. Example: Bounds on Itanium 2

144. Fraction of Upper Bound Across Platforms

145. 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

147. Where Does the Time Go? Most time assigned to memory Caches “disappear” when line sizes are equal –Strictly increasing line sizes

148. Execution Time Breakdown: Matrix 40

149. Speedups with Increasing Line Size

150. 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

151. 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

152. 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  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

153. Example: Select a Matmul Implementation

154. Example: Support Vector Classification

155. 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

156. 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

157. 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

158. 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

159. Acknowledgements Super-advisors: Jim and Kathy Undergraduate R.A.s: Attila, Ben, Jen, Jin, Michael, Rajesh, Shoaib, Sriram, Tuyet-Linh See pages xvi—xvii of dissertation.

160. TSP-based Reordering: Before (Pinar ’97; Moon, et al ‘04)

161. TSP-based Reordering: After (Pinar ’97; Moon, et al ‘04) Up to 2x speedups over CSR

162. Example: L2 Misses on Itanium 2 Misses measured using PAPI [Browne ’00]

163. Example: Distribution of Blocked Non-Zeros

164. Sparse/Dense Partitioning for SpTS Partition L into sparse (L 1,L 2 ) and dense L D : Perform SpTS in three steps: Sparsity optimizations for (1)—(2); DTRSV for (3) Tuning parameters: block size, size of dense triangle

165. SpTS Performance: Power3

167. Summary of SpTS and AA T *x Results SpTS — Similar to SpMV –1.8x speedups; limited benefit from low-level tuning AA T x, A T Ax –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: 60—80% Opportunity for better low-level tuning a la PHiPAC/ATLAS Matrix triple products? A k *x ? –Preliminary work

168. Register Blocking: Speedup

169. Register Blocking: Performance

170. Register Blocking: Fraction of Peak

171. Example: Confidence Interval Estimation

172. Costs of Tuning

173. Splitting + UBCSR: Pentium III

174. Splitting + UBCSR: Power4

175. Splitting+UBCSR Storage: Power4

177. Example: Variable Block Row (Matrix #13)

179. Preliminary Results (Matrix Set 2): Itanium 2 Web/IR DenseFEMFEM (var)BioLPEconStat

180. Multiple Vector Performance

182. 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 =  G + (1-  )(1/n)uu T –Markov model: Surfer follows link with probability , jumps to a random page with probability 1-  –G is n x n connectivity matrix [n  3 billion] g ij is non-zero if page i links to page j Normalized so each column sums to 1 Very sparse: about 7—8 non-zeros per row (power law dist.) –u is a vector of all 1 values –Steady-state probability x i of landing on page i is solution to x = Ax Approximate x by power method: x = A k x 0 –In practice, k  25

184. MAPS Benchmark Example: Power4

185. MAPS Benchmark Example: Itanium 2

186. Saavedra-Barrera Example: Ultra 2i

187. Execution Time Breakdown (PAPI): Matrix 40

188. Preliminary Results (Matrix Set 1): Itanium 2 LPFEMFEM (var)AssortedDense

189. Tuning Sparse Triangular Solve (SpTS) Compute x=L -1 *b where L sparse lower triangular, x & b dense L from sparse LU has rich dense substructure –Dense trailing triangle can account for 20—90% 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

190. Sparse Kernels and Optimizations Kernels –Sparse matrix-vector multiply (SpMV): y=A*x –Sparse triangular solve (SpTS): x=T -1 *b –y=AA T *x, y=A T A*x –Powers (y=A k *x), sparse triple-product (R*A*R T ), … 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

191. 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

192. 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

193. 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

194. 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

195. Inter-Iteration Sparse Tiling (1/3) [Strout, et al., ‘01] Let A be 6x6 tridiagonal Consider y=A 2 x –t=Ax, y=At Nodes: vector elements Edges: matrix elements a ij y1y1 y2y2 y3y3 y4y4 y5y5 t1t1 t2t2 t3t3 t4t4 t5t5 x1x1 x2x2 x3x3 x4x4 x5x5

196. Inter-Iteration Sparse Tiling (2/3) [Strout, et al., ‘01] Let A be 6x6 tridiagonal Consider y=A 2 x –t=Ax, y=At Nodes: vector elements Edges: matrix elements a ij Orange = everything needed to compute y 1 –Reuse a 11, a 12 y1y1 y2y2 y3y3 y4y4 y5y5 t1t1 t2t2 t3t3 t4t4 t5t5 x1x1 x2x2 x3x3 x4x4 x5x5

197. Inter-Iteration Sparse Tiling (3/3) [Strout, et al., ‘01] Let A be 6x6 tridiagonal Consider y=A 2 x –t=Ax, y=At Nodes: vector elements Edges: matrix elements a ij Orange = everything needed to compute y 1 –Reuse a 11, a 12 Grey = y 2, y 3 –Reuse a 23, a 33, a 43 y1y1 y2y2 y3y3 y4y4 y5y5 t1t1 t2t2 t3t3 t4t4 t5t5 x1x1 x2x2 x3x3 x4x4 x5x5

198. 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 [Ng’01] y1y1 y2y2 y3y3 y4y4 y5y5 t1t1 t2t2 t3t3 t4t4 t5t5 x1x1 x2x2 x3x3 x4x4 x5x5

199. 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.

200. 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 2 i x 2 j sparse submatrices Effective when x vector is large Currently, search to find fastest size

201. Symmetric SpMV Performance: Pentium 4

202. SpMV with Split Matrices: Ultra 2i

203. Cache Blocking on Random Matrices: Itanium Speedup on four banded random matrices.

204. Sparse Kernels and Optimizations Kernels –Sparse matrix-vector multiply (SpMV): y=A*x –Sparse triangular solve (SpTS): x=T -1 *b –y=AA T *x, y=A T A*x –Powers (y=A k *x), sparse triple-product (R*A*R T ), … 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

205. Register Blocked SpMV: Pentium III

206. Register Blocked SpMV: Ultra 2i

207. Register Blocked SpMV: Power3

208. Register Blocked SpMV: Itanium

209. Possible Optimization Techniques Within an iteration, i.e., computing (G+uu T )*x once –Cache block G*x On linear programming matrices and matrices with random structure (e.g., LSI), 1.5—4x 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., (G+uu T ) 2 x –(G+uu T ) 2 x = G 2 x + (Gu)u T x + u(u T G)x + u(u T u)u T x Compute Gu, u T G, u T u once for all iterations G 2 x: Inter-iteration tiling to read G only once

210. Multiple Vector Performance: Itanium

211. Sparse Kernels and Optimizations Kernels –Sparse matrix-vector multiply (SpMV): y=A*x –Sparse triangular solve (SpTS): x=T -1 *b –y=AA T *x, y=A T A*x –Powers (y=A k *x), sparse triple-product (R*A*R T ), … 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

212. SpTS Performance: Itanium (See POHLL ’02 workshop paper, at ICS ’02.)

213. Sparse Kernels and Optimizations Kernels –Sparse matrix-vector multiply (SpMV): y=A*x –Sparse triangular solve (SpTS): x=T -1 *b –y=AA T *x, y=A T A*x –Powers (y=A k *x), sparse triple-product (R*A*R T ), … 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

214. Optimizing AA T *x Kernel: y=AA T *x, where A is sparse, x & y dense –Arises in linear programming, computation of SVD –Conventional implementation: compute z=A T *x, y=A*z Elements of A can be reused: When a k represent blocks of columns, can apply register blocking.

215. Optimized AA T *x Performance: Pentium III

216. Current Directions Applying new optimizations –Other split data structures (variable block, diagonal, …) –Matrix reordering to create block structure –Structural symmetry New kernels (triple product RAR T, powers A k, …) 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

217. 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 Johnsson ’00] –MPI collective operations [Vadhiyar & Dongarra ’01] Code generation –FLAME [Gunnels & van de Geijn, ’01] –Sparse compilers: [Bik ’99], Bernoulli [Pingali, et al., ’97] –Generic programming: Blitz++ [Veldhuizen ’98], MTL [Siek & Lumsdaine ’98], GMCL [Czarnecki, et al. ’98], … Sparse performance modeling –[Temam & Jalby ’92], [White & Saddayappan ’97], [Navarro, et al., ’96], [Heras, et al., ’99], [Fraguela, et al., ’99], …

218. More Related Work Compiler analysis, models –CROPS [Carter, Ferrante, et al.]; Serial sparse tiling [Strout ’01] –TUNE [Chatterjee, et al.] –Iterative compilation [O’Boyle, et al., ’98] –Broadway compiler [Guyer & Lin, ’99] –[Brewer ’95], ADAPT [Voss ’00] Sparse BLAS interfaces –BLAST Forum (Chapter 3) –NIST Sparse BLAS [Remington & Pozo ’94]; SparseLib++ –SPARSKIT [Saad ’94] –Parallel Sparse BLAS [Fillipone, et al. ’96]

219. Context: Creating High-Performance Libraries Application performance dominated by a few computational kernels Today: Kernels hand-tuned by vendor or user Performance tuning challenges –Performance is a complicated function of kernel, architecture, compiler, and workload –Tedious and time-consuming Successful automated approaches –Dense linear algebra: ATLAS/PHiPAC –Signal processing: FFTW/SPIRAL/UHFFT

220. Cache Blocked SpMV on LSI Matrix: Itanium

221. Sustainable Memory Bandwidth

222. Multiple Vector Performance: Pentium 4

223. Multiple Vector Performance: Itanium

224. Multiple Vector Performance: Pentium 4

225. Optimized AA T *x Performance: Ultra 2i

226. Optimized AA T *x Performance: Pentium 4

227. High Precision GEMV (XBLAS)

228. High Precision Algorithms (XBLAS) Double-double ( High precision word represented as pair of doubles) –Many variations on these algorithms; we currently use Bailey’s Exploiting Extra-wide Registers –Suppose s(1), …, s(n) have f-bit fractions, SUM has F>f bit fraction –Consider following algorithm for S =  i=1,n s(i) Sort so that |s(1)|  |s(2)|  …  |s(n)| SUM = 0, for i = 1 to n SUM = SUM + s(i), end for, sum = SUM –Theorem (D., Hida) Suppose F<2f (less than double precision) If n  2 F-f + 1, then error  1.5 ulps If n = 2 F-f + 2, then error  2 2f-F ulps (can be  1) If n  2 F-f + 3, then error can be arbitrary (S  0 but sum = 0 ) –Examples s(i) double (f=53), SUM double extended (F=64) – accurate if n  2 11 + 1 = 2049 Dot product of single precision x(i) and y(i) –s(i) = x(i)*y(i) (f=2*24=48), SUM double extended (F=64)  –accurate if n  2 16 + 1 = 65537

229. More Extra Slides

230. Awards Best Paper, Intern. Conf. Parallel Processing, 2004 –“Performance models for evaluation and automatic performance tuning of symmetric sparse matrix-vector multiply” Best Student Paper, Intern. Conf. Supercomputing, Workshop on Performance Optimization via High-Level Languages and Libraries, 2003 –Best Student Presentation too, to Richard Vuduc –“Automatic performance tuning and analysis of sparse triangular solve” Finalist, Best Student Paper, Supercomputing 2002 –To Richard Vuduc –“Performance Optimization and Bounds for Sparse Matrix-vector Multiply” Best Presentation Prize, MICRO-33: 3 rd ACM Workshop on Feedback-Directed Dynamic Optimization, 2000 –To Richard Vuduc –“Statistical Modeling of Feedback Data in an Automatic Tuning System”

231. Accuracy of the Tuning Heuristics (4/4)

232. Can Match DGEMV Performance DGEMV

233. Fraction of Upper Bound Across Platforms

234. Achieved Performance and Machine Balance Machine balance [Callahan ’88; McCalpin ’95] –Balance = Peak Flop Rate / Bandwidth (flops / double) –Lower is better (I.e. can hope to get higher fraction of peak flop rate) Ideal balance for mat-vec:  2 flops / double –For SpMV, even less

236. Where Does the Time Go? Most time assigned to memory Caches “disappear” when line sizes are equal –Strictly increasing line sizes

237. Execution Time Breakdown: Matrix 40

238. Execution Time Breakdown (PAPI): Matrix 40

239. Speedups with Increasing Line Size

240. 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? –Help to have strictly increasing line sizes

241. Evaluating algorithms and machines for SpMV Metrics –Speedups –Mflop/s (“fair” flops) –Fraction of peak Questions –Speedups are good, but what is “the best?” Independent of instruction scheduling, selection Can SpMV be further improved or not? –What machines are “good” for SpMV?

243. 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  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

244. SpMV Historical Trends: Fraction of Peak

245. 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

246. Sparse CALA Summary Optimizing one SpMV too limiting Take k steps of Krylov subspace method –GMRES, CG, Lanczos, Arnoldi –Assume matrix “well-partitioned,” with modest surface-to-volume ratio –Parallel implementation Conventional: O(k log p) messages CALA: O(log p) messages - optimal –Serial implementation Conventional: O(k) moves of data from slow to fast memory CALA: O(1) moves of data – optimal –Can incorporate some preconditioners Need to be able to “compress” interactions between distant i, j Hierarchical, semiseparable matrices … –Lots of speed up possible (measured and modeled) Price: some redundant computation

247. 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?

248. Examples of Automatic Performance Tuning (1) Dense BLAS –Sequential –PHiPAC (UCB), then ATLAS (UTK) (used in Matlab) –math-atlas.sourceforge.net/ 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, …

249. 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

250. 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.

251. SpMV Performance (Matrix #2): Generation 2 Ultra 2i - 9%Ultra 3 - 5% Pentium III-M - 15%Pentium III - 19% 63 Mflop/s 35 Mflop/s 109 Mflop/s 53 Mflop/s 96 Mflop/s 42 Mflop/s 120 Mflop/s 58 Mflop/s

252. SpMV Performance (Matrix #2): Generation 1 Power3 - 13%Power4 - 14% Itanium 2 - 31%Itanium 1 - 7% 195 Mflop/s 100 Mflop/s 703 Mflop/s 469 Mflop/s 225 Mflop/s 103 Mflop/s 1.1 Gflop/s 276 Mflop/s

253. Taking advantage of block structure in SpMV Bottleneck is time to get matrix from memory Don’t store each nonzero with index, instead store each nonzero r-by-c block with index –Storage drops by up to 2x, if rc >> 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 r=c=8 best choice? –Minimizes storage, so looks like a good idea…

254. Example: L2 Misses on Itanium 2 Misses measured using PAPI [Browne ’00]

255. Example: Sparse Triangular Factor Raefsky4 (structural problem) + SuperLU + colmmd N=19779, nnz=12.6 M Dense trailing triangle: dim=2268, 20% of total nz Can be as high as 90+%! 1.8x over CSR

256. Cache Optimizations for AA T *x Cache-level: Interleave multiplication by A, A T –Only fetch A from memory once Register-level: a i T to be r  c block row, or diag row dot product “axpy” … …

257. Example: Combining Optimizations (1/2) Register blocking, symmetry, multiple (k) vectors –Three low-level tuning parameters: r, c, v v k X Y A c r += *

258. Example: Combining Optimizations (2/2) Register blocking, symmetry, and multiple vectors [Ben Lee @ 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

259. 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 user’s 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

260. 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 A T A*x with register-level blocking User may select other optimizations manually –Diagonal storage optimizations, reordering, splitting; tiled matrix powers kernel ( A k *x ) –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

261. Performance Results Measured –Sequential/OOC speedup up to 3x Modeled –Sequential/multicore speedup up to 2.5x –Parallel/Petascale speedup up to 6.9x –Parallel/Grid speedup up to 22x See bebop.cs.berkeley.edu/#pubs