20180728 halide-study
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Halide勉強会 @フィックスターズの資料です
![TC in PyTorch
$ conda create –y –name pytorch python=3.6
$ conda activate pytorch
$ conda install -y -c pytorch -c tensorcomp tensor_comprehensions
$ python ./matmul.py
Variable containing:
-2.4028 2.8492 7.6141 3.3159 3.7171
1.3839 0.6650 -1.7253 0.7447 1.3988
0.1396 -0.0661 -1.0574 0.2163 0.1711
[torch.cuda.FloatTensor of size 3x5 (GPU 0)]
import tensor_comprehensions as tc
import torch
mm = """
def matmul(float(M,K) A, float(K,N) B) -> (output) {
output(m, n) +=! A(m, r_k) * B(r_k, n)
}
"""
matmul = tc.define(mm, name="matmul")
A, B = torch.randn(3, 4).cuda(), torch.randn(4, 5).cuda()
C = matmul(A, B, options=tc.Options("naive"))
print(C)
1. TCをセットアップ
2. TC言語でカスタム
レイヤを書く
3. 実行する](https://image.slidesharecdn.com/20180728-halide-study-180731010243/85/20180728-halide-study-9-320.jpg)
![num ::= <number literal with C syntax>
id ::= [_a-zA-Z0-9]*[_a-zA-Z][_a-zA-Z0-9]*
exp ::= num
| ( '-' | '!' | ... ) exp
| exp ( [+-*/%] | '==' | '!=' | '<=' | ... ) exp
| exp '?' exp ':' exp
| id '.' num # range of num-th dimension of id
| id '(' exp_list ')' # builtin call or tensor access
reduction ::= <associative reduction operator>
| '+=' | '*=' | 'min=' | 'max='
| '+=!' | '*=!' | 'min=!' | 'max=!'
range_constraint ::= id 'in' exp ':' exp
stmt ::= id '(' id_list ')' [ '=' | reduction ] exp
[ 'where' range_constraint_list ]
| id_list = id '('id_list ')' # TC function call
arg ::= type id
return ::= id # inferred return type and range
scalar_type ::= 'double' | 'float' | 'half'
| 'int32' | 'byte' | 'uint32' | ...
type ::= scalar_type [ '(' id_list ')' ]
func ::= # TC function definition
'def' id '(' arg_list ')' '->' '(' return_list ')' '{'
stmt_list
'}'
id_list ::= <comma separated id list>
exp_list ::= <comma separated exp list>
arg_list ::= <comma separated arg list>
stmt_list ::= <whitespace separated stmt list>
return_list ::= <comma separated return list>
range_constraint_list ::= <non-empty comma separated
range_constraint list>](https://image.slidesharecdn.com/20180728-halide-study-180731010243/85/20180728-halide-study-11-320.jpg)
![domain(
[K, M, N] -> { S_0[output_s0_m, output_s0_n] : 0 <= output_s0_m < M and 0 <= output_s0_n < N }
[K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : 0 <= output_s1_m < M and 0 <=
output_s1_n < N and 0 <= output_s1_r_k < K })
sequence()
filter(
[K, M, N] -> { S_0[output_s0_m, output_s0_n] })
band(n(1) permutable(0) coincident(0) unroll(0)
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_m)] }
-----------------------------------------------------------------------
band(n(1) permutable(0) coincident(0) unroll(0)
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_n)] }
-----------------------------------------------------------------------
filter(
[K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] })
band(n(1) permutable(0) coincident(0) unroll(0)
-----------------------------------------------------------------------
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_m)] }
-----------------------------------------------------------------------
band(n(1) permutable(0) coincident(0) unroll(0)
-----------------------------------------------------------------------
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_n)] }
-----------------------------------------------------------------------
band(n(1) permutable(0) coincident(0) unroll(0)
-----------------------------------------------------------------------
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_r_k)] }
-----------------------------------------------------------------------
初期状態のスケジューリングツリー
雑な解説: band=ループ、 filter=ステートメント](https://image.slidesharecdn.com/20180728-halide-study-180731010243/85/20180728-halide-study-37-320.jpg)
![domain(
[K, M, N] -> { S_0[output_s0_m, output_s0_n] : 0 <= output_s0_m < M and 0 <= output_s0_n < N }
[K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : 0 <= output_s1_m < M and 0 <=
output_s1_n < N and 0 <= output_s1_r_k < K })
band(n(3) permutable(1) coincident(1, 1, 0) unroll(0, 0, 0)
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_m)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_m)] }
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_n)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_n)] }
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_r_k)] }
-----------------------------------------------------------------------
sequence()
filter(
[K, M, N] -> { S_0[output_s0_m, output_s0_n] })
filter(
[K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] })
ループ融合後](https://image.slidesharecdn.com/20180728-halide-study-180731010243/85/20180728-halide-study-38-320.jpg)
![domain(
[K, M, N] -> { S_0[output_s0_m, output_s0_n] : 0 <= output_s0_m < M and 0 <= output_s0_n < N }
[K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : 0 <= output_s1_m < M and 0 <=
output_s1_n < N and 0 <= output_s1_r_k < K })
band(n(3) permutable(1) coincident(1, 1, 0) unroll(0, 0, 0)
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_m)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_m)] }
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(0)] }
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(0)] }
-----------------------------------------------------------------------
band(n(2) permutable(1) coincident(1, 0) unroll(0, 0)
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_n)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_n)] }
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_r_k)] }
-----------------------------------------------------------------------
sequence()
filter(
[K, M, N] -> { S_0[output_s0_m, output_s0_n] })
filter(
[K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] })
タイリング後](https://image.slidesharecdn.com/20180728-halide-study-180731010243/85/20180728-halide-study-39-320.jpg)
![domain(
[K, M, N] -> { S_0[output_s0_m, output_s0_n] : K = 4 and M = 3 and N = 5 and 0 <= output_s0_m <= 2 and 0 <= output_s0_n <= 4 }
[K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : K = 4 and M = 3 and N = 5 and 0 <= output_s1_m <= 2 and
0 <= output_s1_n <= 4 and 0 <= output_s1_r_k <= 3 })
context([K, M, N, t1, t0, t2, b2, b1, b0] -> { [] : t1 = 0 and t2 = 0 and b2 = 0 and b1 = 0 and 0 <= t0 <= 127 and 0 <= b0 <= 127 })
band(n(3) permutable(1) coincident(1, 1, 0) unroll(0, 0, 0)
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_m)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_m)] }
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(0)] }
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(0)] }
-----------------------------------------------------------------------
mapping_filter(ids(t0, )
[K, M, N, t0] -> { S_0[output_s0_m, output_s0_n] : (-t0 + output_s0_n) mod 128 = 0 and 0 <= t0 <= 127 }
[K, M, N, t0] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : (-t0 + output_s1_n) mod 128 = 0 and 0 <= t0 <= 127 })
band(n(1) permutable(1) coincident(1) unroll(0)
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_n)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_n)] }
-----------------------------------------------------------------------
thread_specific()
band(n(1) permutable(1) coincident(0) unroll(0)
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_r_k)] }
-----------------------------------------------------------------------
sequence()
filter(
[K, M, N] -> { S_0[output_s0_m, output_s0_n] })
filter(
[K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] })
スレッドマッピング後](https://image.slidesharecdn.com/20180728-halide-study-180731010243/85/20180728-halide-study-40-320.jpg)
![domain(
[K, M, N] -> { S_0[output_s0_m, output_s0_n] : K = 4 and M = 3 and N = 5 and 0 <= output_s0_m <= 2 and 0 <= output_s0_n <= 4 }
[K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : K = 4 and M = 3 and N = 5 and 0 <= output_s1_m <= 2 and 0 <=
output_s1_n <= 4 and 0 <= output_s1_r_k <= 3 })
context([K, M, N, t1, t0, t2, b2, b1, b0] -> { [] : t1 = 0 and t2 = 0 and b2 = 0 and b1 = 0 and 0 <= t0 <= 127 and 0 <= b0 <= 127 })
mapping_filter(ids(b0, )
[K, M, N, b0] -> { S_0[output_s0_m, output_s0_n] : (-b0 + output_s0_m) mod 128 = 0 and 0 <= b0 <= 127 }
[K, M, N, b0] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : (-b0 + output_s1_m) mod 128 = 0 and 0 <= b0 <= 127 })
band(n(3) permutable(1) coincident(1, 1, 0) unroll(0, 0, 0)
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_m)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_m)] }
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(0)] }
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(0)] }
-----------------------------------------------------------------------
mapping_filter(ids(t0, )
[K, M, N, t0] -> { S_0[output_s0_m, output_s0_n] : (-t0 + output_s0_n) mod 128 = 0 and 0 <= t0 <= 127 }
[K, M, N, t0] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : (-t0 + output_s1_n) mod 128 = 0 and 0 <= t0 <= 127 })
band(n(1) permutable(1) coincident(1) unroll(0)
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_n)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_n)] }
-----------------------------------------------------------------------
thread_specific()
band(n(1) permutable(1) coincident(0) unroll(0)
-----------------------------------------------------------------------
| [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] }
| [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_r_k)] }
-----------------------------------------------------------------------
sequence()
filter(
[K, M, N] -> { S_0[output_s0_m, output_s0_n] })
filter(
[K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] })
ブロックマッピング後](https://image.slidesharecdn.com/20180728-halide-study-180731010243/85/20180728-halide-study-41-320.jpg)
![template<typename T> inline __device__ T floord(T n, T d) {
return n < 0 ? - (-n + d - 1)/d : n / d;
}
#define if_then_else(cond,a,b) ((cond) ? (a) : (b))
// Can't include system dependencies with NVRTC
// Can't include cuda_fp16.h with NVRTC due to transitive system dependencies
// #include <cuda_fp16.h>
// Halide type handling
typedef char int8;
typedef short int16;
typedef int int32;
typedef long int64;
typedef unsigned char uint8;
typedef unsigned short uint16;
typedef unsigned int uint32;
typedef unsigned long uint64;
// typedef half float16;
typedef float float32;
typedef double float64;
#define inff __int_as_float(0x7f800000)
#define inf __longlong_as_double(0x7ff0000000000000LL)
// Before CUDA 9, syncwarp is a noop since warps are always synchronized.
#if __CUDACC_VER_MAJOR__ < 9
__device__ void __syncwarp(unsigned mask = 0xFFFFFFFF) {}
#endif
extern "C" {
__global__ void matmul_4_3_5(int32 K, int32 M, int32 N, float32* poutput, const float32* pA, const float32* pB) {
int b0 = blockIdx.x; int b1 = blockIdx.y; int b2 = blockIdx.z;
int t0 = threadIdx.x; int t1 = threadIdx.y; int t2 = threadIdx.z;
float32 (*output)[5] = reinterpret_cast<float32 (*)[5]>(poutput);
const float32 (*A)[4] = reinterpret_cast<const float32 (*)[4]>(pA);
const float32 (*B)[5] = reinterpret_cast<const float32 (*)[5]>(pB);
output[b0][t0] = 0.000000f;
for (int c4 = 0; c4 <= 3; c4 += 1) {
output[b0][t0] = (output[b0][t0] + (A[b0][c4]*B[c4][t0]));
}
}
}
コード生成](https://image.slidesharecdn.com/20180728-halide-study-180731010243/85/20180728-halide-study-42-320.jpg)
20180728 halide-study
- 1. Tensor Comprehensions にみる Halide IRの汎用性 Fixstars Solutions, Inc. Takuro Iizuka
- 2. Takuro Iizuka / @iitaku 北米子会社のFixstars Solutions, Inc. にて HalideのFPGAバックエンドおよびツールチェイン”GENESIS”の開発やってます
- 3. もくじ TC: Tensor Comprehensions 概要 TC言語 Inside TC まとめ
- 4. TC: Tensor Comprehensionsとは？ テンソル計算の記述言語および 最適化コンパイラフレームワーク 2018.2.14にFacebook AI Researchからリリース TC言語でアルゴリズムを書くと ライブラリがいい感じに最適化してくれる PyTorchとシームレスに統合できる
- 5. TC: Tensor Comprehensionsとは？ テンソル計算の記述言語および 最適化コンパイラフレームワーク 2018.2にFacebook AI Researchからリリース TC言語でアルゴリズムを書くと ライブラリがいい感じに最適化してくれる PyTorchとシームレスに統合できる コンパイラの中間表現としてHalide IRを採用
- 7. TCベンチマーク結果 MLP: Multi-Layer Perceptron TMM: Transposed Matrix Multiplication TBMM: Transposed Batched Matrix Multiplication GCOV: Grouped Convolutions https://research.fb.com/announcing-tensor-comprehensions/
- 9. TC in PyTorch $ conda create –y –name pytorch python=3.6 $ conda activate pytorch $ conda install -y -c pytorch -c tensorcomp tensor_comprehensions $ python ./matmul.py Variable containing: -2.4028 2.8492 7.6141 3.3159 3.7171 1.3839 0.6650 -1.7253 0.7447 1.3988 0.1396 -0.0661 -1.0574 0.2163 0.1711 [torch.cuda.FloatTensor of size 3x5 (GPU 0)] import tensor_comprehensions as tc import torch mm = """ def matmul(float(M,K) A, float(K,N) B) -> (output) { output(m, n) +=! A(m, r_k) * B(r_k, n) } """ matmul = tc.define(mm, name="matmul") A, B = torch.randn(3, 4).cuda(), torch.randn(4, 5).cuda() C = matmul(A, B, options=tc.Options("naive")) print(C) 1. TCをセットアップ 2. TC言語でカスタム レイヤを書く 3. 実行する
- 10. TC言語
- 11. num ::= <number literal with C syntax> id ::= [_a-zA-Z0-9]*[_a-zA-Z][_a-zA-Z0-9]* exp ::= num | ( '-' | '!' | ... ) exp | exp ( [+-*/%] | '==' | '!=' | '<=' | ... ) exp | exp '?' exp ':' exp | id '.' num # range of num-th dimension of id | id '(' exp_list ')' # builtin call or tensor access reduction ::= <associative reduction operator> | '+=' | '*=' | 'min=' | 'max=' | '+=!' | '*=!' | 'min=!' | 'max=!' range_constraint ::= id 'in' exp ':' exp stmt ::= id '(' id_list ')' [ '=' | reduction ] exp [ 'where' range_constraint_list ] | id_list = id '('id_list ')' # TC function call arg ::= type id return ::= id # inferred return type and range scalar_type ::= 'double' | 'float' | 'half' | 'int32' | 'byte' | 'uint32' | ... type ::= scalar_type [ '(' id_list ')' ] func ::= # TC function definition 'def' id '(' arg_list ')' '->' '(' return_list ')' '{' stmt_list '}' id_list ::= <comma separated id list> exp_list ::= <comma separated exp list> arg_list ::= <comma separated arg list> stmt_list ::= <whitespace separated stmt list> return_list ::= <comma separated return list> range_constraint_list ::= <non-empty comma separated range_constraint list>
- 12. TC言語の特徴 テンソル計算記述特化言語 Halide言語よりさらに簡素な言語体型 超ミニマルなプリミティブ型 畳み込み演算用の特殊なオペレータ群 where節でレンジ制約を記述
- 13. def matmul(float(M,K) A, float(K,N) B) -> (output) { output(m, n) +=! A(m, r_k) * B(r_k, n) } def 関数名 {} で関数定義
- 14. def matmul(float(M,K) A, float(K,N) B) -> (output) { output(m, n) +=! A(m, r_k) * B(r_k, n) } type(X,Y,…)で入力引数に制約 （要素型制約およびレンジ制約）を付与
- 15. def matmul(float(M,K) A, float(K,N) B) -> (output) { output(m, n) +=! A(m, r_k) * B(r_k, n) } レンジ制約に同一シンボルを使用することで、 異なる引数間の制約を表現できる
- 16. def matmul(float(M,K) A, float(K,N) B) -> (output) { output(m, n) +=! A(m, r_k) * B(r_k, n) } 出力の制約はコンパイラによって自動推論される
- 17. def matmul(float(M,K) A, float(K,N) B) -> (output) { output(m, n) +=! A(m, r_k) * B(r_k, n) } 初期化付き畳み込み用記述のオペレータ
- 18. def matmul(float(M,K) A, float(K,N) B) -> (output) { output(m, n) +=! A(m, r_k) * B(r_k, n) } 左辺で定義した誘導変数を右辺で使用すると ループが形成される for m := 0, M for n := 0, N
- 19. def matmul(float(M,K) A, float(K,N) B) -> (output) { output(m, n) +=! A(m, r_k) * B(r_k, n) } ループのレンジは入力制約から決定される e.g. m := [0. M)
- 20. def matmul(float(M,K) A, float(K,N) B) -> (output) { output(m, n) +=! A(m, r_k) * B(r_k, n) } 左辺で定義していない誘導変数を右辺で使用した場合、 既存ループの最内に新たにループが形成される for m := 0, M for n := 0, N for r_k := 0, K
- 21. def matmul(float(M,K) A, float(K,N) B) -> (output) { output(m, n) +=! A(m, r_k) * B(r_k, n) } その場合も入力の制約をもとに制約チェックが行われる
- 22. Halide言語と比較してみる def matmul(float(M,K) A, float(K,N) B) -> (output) { output(m, n) +=! A(m, r_k) * B(r_k, n) } TC言語 Func matmul; ImageParam A{float, 2}; ImageParam B{float, 2}; Var m, n; Rdom r_k{0, K}; matmul(m, n) = sum(A(m, r_k) * B(r_k, n)); matmul.realize({M, N}); Halide言語
- 23. Halide言語と比較してみる def matmul(float(M,K) A, float(K,N) B) -> (output) { output(m, n) +=! A(m, r_k) * B(r_k, n) } Func matmul; ImageParam A{float, 2}; ImageParam B{float, 2}; Var m, n; Rdom r_k{0, K}; matmul(m, n) = sum(A(m, r_k) * B(r_k, n)); matmul.realize({M, N}); レンジ制約は前向きに推論 レンジ制約は後ろ向きに推論
- 24. Inside TC
- 25. TC GPU Backendコンパイルフロー TC言語 AST Halide IR パース/AST構築: tc/core/compiler.cc, parse(…) Halide IR変換: tc/core/tc2halide.cc, translate(…) Halide IR レンジ推論/ループ形成/簡約化: tc/core/tc2halide.cc, translate(…) Polyhedral IR CUDA C Polyhedral IR変換: tc/core/polyhedral/scop.cc, makeScop(…) Polyhedral IR ループ変形/スレッディング: tc/core/polyhedral/mapped_scop.cc, makeWithOuterBlockInnerThreadStrategy(…) コード生成: tc/core/polyhedral/mapped_scop.cc, codegen(…)
- 26. Halide IR オリジナルのHalide実装では、 中間表現の変換過程においてHalide IRを２つに大別できる Halide IR Halide IR Halide IR Halide IR Halide IR Call/Provide系 Load/Store系 storage_flattening以前/以後
- 27. Call/Provide系 抽象度高い Load/Store系 抽象度低い for (y, 0, out.extent.1) { for (x, 0, out.extent.0) { Provide(out, {x, y}) = Call(in, {x, y}) } } for (y, 0, out.extent.1) { for (x, 0, out.extent.0) { Store(out, y*out.stride.1+x, Load(in, y*in.stride.1.x)) } }
- 28. TCが取り扱うHalide IRは 抽象度の高いCall/Provide系のみ for (y, 0, out.extent.1) { for (x, 0, out.extent.0) { Provide(out, {x, y}) = Call(in, {x, y}) } } for (y, 0, out.extent.1) { for (x, 0, out.extent.0) { Store(out, y*out.stride.1+x, Load(in, y*in.stride.1.x)) } } ターゲットコードに近い中間表現は Halide IRではなくPolyhedral IRを使う
- 29. Halide IR上でのLowering コンパイラインフラストラクチャとしてのHalide IR – Halide中にはHalide IR (Halide::Stmt/Expr)を操作する 関数が多数実装済み • 簡約器 • ソルバー • 範囲演算 • CSE • などなど – IRMutatorやIRVisitorクラスを使用すれば Halide IRに対する変換や解析を独自に実装できる TCのHalide IR Loweringは以下を行っている – レンジ推論 – ループ形成 – 簡約化
- 30. レンジ推論で使用されるHalide API solve_for_inner_interval(c, v) – 条件式cを必ず満たす変数vの最大範囲を計算する and_condition_over_domain(c, varying) – 変数範囲varyingの仮定のもとで条件式cを簡約化する simplify(e) – 式eを簡約化する これらを組み合わせてレンジ推論を行い、後段で行われる Polyhedral最適化に必要な条件を満たすかをテストしておく
- 31. ループ形成で使用されるHalide API realization_order(…) – Provide/Call間の依存グラフを トポロジカルソートによって順序付けをする schedule_functions(…) – 出力が依存するすべての計算ループを形成し、 Halide IR (Halide::Stmt) を返す 後の解析や変換のベースとなるループ構築を行う
- 32. ループ形成結果 後の解析や変換のベースとなるループ構築を行う produce output { let output.s0.n.loop_max = output.s0.n.max let output.s0.n.loop_min = output.s0.n.min let output.s0.n.loop_extent = ((output.s0.n.max + 1) - output.s0.n.min) let output.s0.m.loop_max = output.s0.m.max let output.s0.m.loop_min = output.s0.m.min let output.s0.m.loop_extent = ((output.s0.m.max + 1) - output.s0.m.min) for (output.s0.m, output.s0.m.loop_min, output.s0.m.loop_extent) { for (output.s0.n, output.s0.n.loop_min, output.s0.n.loop_extent) { output(output.s0.m, output.s0.n) = 0.000000f } } let output.s1.r_k.loop_extent = ((output.s1.r_k.max - output.s1.r_k.min) + 1) let output.s1.r_k.loop_max = output.s1.r_k.max let output.s1.r_k.loop_min = output.s1.r_k.min let output.s1.n.loop_max = output.s1.n.max let output.s1.n.loop_min = output.s1.n.min let output.s1.n.loop_extent = ((output.s1.n.max + 1) - output.s1.n.min) let output.s1.m.loop_max = output.s1.m.max let output.s1.m.loop_min = output.s1.m.min let output.s1.m.loop_extent = ((output.s1.m.max + 1) - output.s1.m.min) for (output.s1.m, output.s1.m.loop_min, output.s1.m.loop_extent) { for (output.s1.n, output.s1.n.loop_min, output.s1.n.loop_extent) { for (output.s1.r_k, output.s1.r_k.loop_min, output.s1.r_k.loop_extent) { output(output.s1.m, output.s1.n) = ReductionUpdate((output(output.s1.m, output.s1.n) + (A(output.s1.m, output.s1.r_k)* B(output.s1.r_k, output.s1.n)))) } } } } 初期化 計算
- 33. 簡約化で使用されるHalide API LetStmt::make(n, e, s) – 文s中で式eをシンボルnに束縛する simplify(s) – 文sを簡約化する レンジ制約を適用しループ範囲の簡約化を行う
- 34. 簡約化結果 後段のPolyhedral Transformationで 解析・変形可能なループ構造に簡約できた for (output.s0.m, 0, M) { for (output.s0.n, 0, K) { output(output.s0.m, output.s0.n) = 0.000000f } } for (output.s1.m, 0, M) { for (output.s1.n, 0, K) { for (output.s1.r_k, 0, K) { output(output.s1.m, output.s1.n) = ReductionUpdate((output(output.s1.m, output.s1.n) + (A(output.s1.m, output.s1.r_k)* B(output.s1.r_k, output.s1.n)))) } } } ループ範囲がTC言語のレンジ制約と対応している
- 35. tc::polyhedral::Scop – ISL (Integer Set Library) /Polyhedral Compilation Libraryを用いて計算されたスケジューリング – RAW, WAR, WAW 依存関係 – メモリ配置 – Halide IR関連 • パラメータ • 入出力 • Stmt Polyhedral IR Polyhedral IR Halide IR パラメータ/ 入出力 Polyhedral IR変換 Stmt of Provide A Stmt of Provide B パラメータ/ 入出力 スケジューリング 依存関係 メモリ配置
- 36. Polyhedral IR 上でのLowering 1. ループ融合 2. タイリング – パラメータ: タイリング戦略 3. スレッドマッピング – パラメータ: スレッドサイズ 4. ブロックマッピング – パラメータ: ブロックサイズ 5. メモリマッピング – パラメータ: 有効/無効、マッピング先、マッピング量 Polyhedral Transformationとは、 ループ構造をPolytope=多面体と見立ててアフィン変換を施すことで Legalなループ変形を行う最適化手法
- 37. domain( [K, M, N] -> { S_0[output_s0_m, output_s0_n] : 0 <= output_s0_m < M and 0 <= output_s0_n < N } [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : 0 <= output_s1_m < M and 0 <= output_s1_n < N and 0 <= output_s1_r_k < K }) sequence() filter( [K, M, N] -> { S_0[output_s0_m, output_s0_n] }) band(n(1) permutable(0) coincident(0) unroll(0) ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_m)] } ----------------------------------------------------------------------- band(n(1) permutable(0) coincident(0) unroll(0) ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_n)] } ----------------------------------------------------------------------- filter( [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] }) band(n(1) permutable(0) coincident(0) unroll(0) ----------------------------------------------------------------------- | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_m)] } ----------------------------------------------------------------------- band(n(1) permutable(0) coincident(0) unroll(0) ----------------------------------------------------------------------- | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_n)] } ----------------------------------------------------------------------- band(n(1) permutable(0) coincident(0) unroll(0) ----------------------------------------------------------------------- | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_r_k)] } ----------------------------------------------------------------------- 初期状態のスケジューリングツリー 雑な解説: band=ループ、 filter=ステートメント
- 38. domain( [K, M, N] -> { S_0[output_s0_m, output_s0_n] : 0 <= output_s0_m < M and 0 <= output_s0_n < N } [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : 0 <= output_s1_m < M and 0 <= output_s1_n < N and 0 <= output_s1_r_k < K }) band(n(3) permutable(1) coincident(1, 1, 0) unroll(0, 0, 0) ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_m)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_m)] } ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_n)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_n)] } ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_r_k)] } ----------------------------------------------------------------------- sequence() filter( [K, M, N] -> { S_0[output_s0_m, output_s0_n] }) filter( [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] }) ループ融合後
- 39. domain( [K, M, N] -> { S_0[output_s0_m, output_s0_n] : 0 <= output_s0_m < M and 0 <= output_s0_n < N } [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : 0 <= output_s1_m < M and 0 <= output_s1_n < N and 0 <= output_s1_r_k < K }) band(n(3) permutable(1) coincident(1, 1, 0) unroll(0, 0, 0) ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_m)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_m)] } ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(0)] } ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(0)] } ----------------------------------------------------------------------- band(n(2) permutable(1) coincident(1, 0) unroll(0, 0) ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_n)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_n)] } ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_r_k)] } ----------------------------------------------------------------------- sequence() filter( [K, M, N] -> { S_0[output_s0_m, output_s0_n] }) filter( [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] }) タイリング後
- 40. domain( [K, M, N] -> { S_0[output_s0_m, output_s0_n] : K = 4 and M = 3 and N = 5 and 0 <= output_s0_m <= 2 and 0 <= output_s0_n <= 4 } [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : K = 4 and M = 3 and N = 5 and 0 <= output_s1_m <= 2 and 0 <= output_s1_n <= 4 and 0 <= output_s1_r_k <= 3 }) context([K, M, N, t1, t0, t2, b2, b1, b0] -> { [] : t1 = 0 and t2 = 0 and b2 = 0 and b1 = 0 and 0 <= t0 <= 127 and 0 <= b0 <= 127 }) band(n(3) permutable(1) coincident(1, 1, 0) unroll(0, 0, 0) ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_m)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_m)] } ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(0)] } ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(0)] } ----------------------------------------------------------------------- mapping_filter(ids(t0, ) [K, M, N, t0] -> { S_0[output_s0_m, output_s0_n] : (-t0 + output_s0_n) mod 128 = 0 and 0 <= t0 <= 127 } [K, M, N, t0] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : (-t0 + output_s1_n) mod 128 = 0 and 0 <= t0 <= 127 }) band(n(1) permutable(1) coincident(1) unroll(0) ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_n)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_n)] } ----------------------------------------------------------------------- thread_specific() band(n(1) permutable(1) coincident(0) unroll(0) ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_r_k)] } ----------------------------------------------------------------------- sequence() filter( [K, M, N] -> { S_0[output_s0_m, output_s0_n] }) filter( [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] }) スレッドマッピング後
- 41. domain( [K, M, N] -> { S_0[output_s0_m, output_s0_n] : K = 4 and M = 3 and N = 5 and 0 <= output_s0_m <= 2 and 0 <= output_s0_n <= 4 } [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : K = 4 and M = 3 and N = 5 and 0 <= output_s1_m <= 2 and 0 <= output_s1_n <= 4 and 0 <= output_s1_r_k <= 3 }) context([K, M, N, t1, t0, t2, b2, b1, b0] -> { [] : t1 = 0 and t2 = 0 and b2 = 0 and b1 = 0 and 0 <= t0 <= 127 and 0 <= b0 <= 127 }) mapping_filter(ids(b0, ) [K, M, N, b0] -> { S_0[output_s0_m, output_s0_n] : (-b0 + output_s0_m) mod 128 = 0 and 0 <= b0 <= 127 } [K, M, N, b0] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : (-b0 + output_s1_m) mod 128 = 0 and 0 <= b0 <= 127 }) band(n(3) permutable(1) coincident(1, 1, 0) unroll(0, 0, 0) ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_m)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_m)] } ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(0)] } ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(0)] } ----------------------------------------------------------------------- mapping_filter(ids(t0, ) [K, M, N, t0] -> { S_0[output_s0_m, output_s0_n] : (-t0 + output_s0_n) mod 128 = 0 and 0 <= t0 <= 127 } [K, M, N, t0] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] : (-t0 + output_s1_n) mod 128 = 0 and 0 <= t0 <= 127 }) band(n(1) permutable(1) coincident(1) unroll(0) ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(output_s0_n)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_n)] } ----------------------------------------------------------------------- thread_specific() band(n(1) permutable(1) coincident(0) unroll(0) ----------------------------------------------------------------------- | [K, M, N] -> { S_0[output_s0_m, output_s0_n] -> [(0)] } | [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] -> [(output_s1_r_k)] } ----------------------------------------------------------------------- sequence() filter( [K, M, N] -> { S_0[output_s0_m, output_s0_n] }) filter( [K, M, N] -> { S_1[output_s1_m, output_s1_n, output_s1_r_k] }) ブロックマッピング後
- 42. template<typename T> inline __device__ T floord(T n, T d) { return n < 0 ? - (-n + d - 1)/d : n / d; } #define if_then_else(cond,a,b) ((cond) ? (a) : (b)) // Can't include system dependencies with NVRTC // Can't include cuda_fp16.h with NVRTC due to transitive system dependencies // #include <cuda_fp16.h> // Halide type handling typedef char int8; typedef short int16; typedef int int32; typedef long int64; typedef unsigned char uint8; typedef unsigned short uint16; typedef unsigned int uint32; typedef unsigned long uint64; // typedef half float16; typedef float float32; typedef double float64; #define inff __int_as_float(0x7f800000) #define inf __longlong_as_double(0x7ff0000000000000LL) // Before CUDA 9, syncwarp is a noop since warps are always synchronized. #if __CUDACC_VER_MAJOR__ < 9 __device__ void __syncwarp(unsigned mask = 0xFFFFFFFF) {} #endif extern "C" { __global__ void matmul_4_3_5(int32 K, int32 M, int32 N, float32* poutput, const float32* pA, const float32* pB) { int b0 = blockIdx.x; int b1 = blockIdx.y; int b2 = blockIdx.z; int t0 = threadIdx.x; int t1 = threadIdx.y; int t2 = threadIdx.z; float32 (*output)[5] = reinterpret_cast<float32 (*)[5]>(poutput); const float32 (*A)[4] = reinterpret_cast<const float32 (*)[4]>(pA); const float32 (*B)[5] = reinterpret_cast<const float32 (*)[5]>(pB); output[b0][t0] = 0.000000f; for (int c4 = 0; c4 <= 3; c4 += 1) { output[b0][t0] = (output[b0][t0] + (A[b0][c4]*B[c4][t0])); } } } コード生成
- 44. まとめ
- 45. Halide IR良くできてる – 制約は正義 • Polyhedral Transformation等応用的な最適化手法を適用可能 • 解析時の計算量爆発がおきにくい – IRを操作する関数の実装が揃ってる • 自前の解析や変形を行う場合でも多くの機能を転用可能 – IRの変換が書きやすい • IRMutator/IRVisitorのクラスが割とシンプルで書きやすい – TC/TVM/Tiramisu等、 Halide IRを再利用する取り組みも出てきた High Level Compiler IRとしてのHalideに今後も要注目！