Note
Click here to download the full example code
Matrix Multiplication¶
In this tutorial, you will write a 25-lines high-performance FP16 matrix multiplication kernel that achieves performance on par with cuBLAS. You will specifically learn about:
Block-level matrix multiplications
Multi-dimensional pointer arithmetic
Program re-ordering for improved L2 cache hit rate
Automatic performance tuning
Motivations¶
Matrix multiplications are a key building block of most modern high-performance computing systems. They are notoriously hard to optimize, hence their implementation is generally done by hardware vendors themselves as part of so-called “kernel libraries” (e.g., cuBLAS). Unfortunately, these libraries are often proprietary and cannot be easily customized to accomodate the needs of modern deep learning workloads (e.g., fused activation functions). In this tutorial, you will learn how to implement efficient matrix multiplications by yourself with Triton, in a way that is easy to customize and extend.
Roughly speaking, the kernel that we will write will implement the following blocked algorithm:
# do in parallel for m in range(0, M, BLOCK_M): # do in parallel for n in range(0, N, BLOCK_N): acc = zeros((BLOCK_M, BLOCK_N), dtype=float32) for k in range(0, K, BLOCK_K): a = A[m : m+BLOCK_M, k : k+BLOCK_K] b = B[k : k+BLOCK_K, n : n+BLOCK_N] acc += dot(a, b) C[m : m+BLOCK_M, n : n+BLOCK_N] = acc;
where each iteration of the doubly-nested for-loop corresponds to a Triton program instance.
Compute Kernel¶
The above algorithm is, actually, fairly straightforward to implement in Triton.
The main difficulty comes from the computation of the memory locations at which blocks of A and B must be read in the inner loop. For that, we need multi-dimensional pointer arithmetics.
Pointer Arithmetics¶
For a row-major 2D tensor X, the memory location of X[i, j] is given by &X[i, j] = X + i*stride_x_0 + j*stride_x_1.
Therefore, blocks of pointers for A[m : m+BLOCK_M, k:k+BLOCK_K] and B[k : k+BLOCK_K, n : n+BLOCK_N] can be defined in pseudo-code as:
&A[m : m+BLOCK_M, k:k+BLOCK_K] = A + (m : m+BLOCK_M)[:, None]*A.stride(0) + (k : k+BLOCK_K)[None, :]*A.stride(1); &B[k : k+BLOCK_K, n:n+BLOCK_N] = B + (k : k+BLOCK_K)[:, None]*B.stride(0) + (n : n+BLOCK_N)[None, :]*B.stride(1);
Which means that pointers for blocks of A and B can be initialized (i.e., k=0) in Triton as:
pid_m = triton.program_id(0) pid_n = triton.program_id(1) rm = pid_m * BLOCK_M + triton.arange(0, BLOCK_M) rn = pid_n * BLOCK_N + triton.arange(0, BLOCK_N) rk = triton.arange(0, BLOCK_K) // pointer for A operand pa = A + (rm[:, None] * stride_a_0 + rk[None, :] * stride_a_1); // pointer for B operand pb = B + (rk[:, None] * stride_b_0 + rn[None, :] * stride_b_1);
And then updated in the inner loop as follows:
pa += BLOCK_K * stride_a_1; pb += BLOCK_K * stride_b_0;
L2 Cache Optimizations¶
As mentioned above, each program instance computes an [BLOCK_M, BLOCK_N] block of C.
It is important to remember that the order in which these blocks are computed does matter, since it affects the L2 cache hit rate of our program.
And unfortunately, a simple row-major ordering
pid = triton.program_id(0); grid_m = (M + BLOCK_M - 1) // BLOCK_M; grid_n = (N + BLOCK_N - 1) // BLOCK_N; pid_m = pid / grid_n; pid_n = pid % grid_n;
is just not going to cut it.
One possible solution is to launch blocks in an order that promotes data reuse.
This can be done by ‘super-grouping’ blocks in groups of GROUP_M rows before switching to the next column:
pid = triton.program_id(0); width = GROUP_M * grid_n; group_id = pid // width; # we need to handle the case where M % (GROUP_M*BLOCK_M) != 0 group_size = min(grid_m - group_id * GROUP_M, GROUP_M); pid_m = group_id * GROUP_M + (pid % group_size); pid_n = (pid % width) // (group_size);
In practice, this can improve the performance of our matrix multiplication kernel by >10% on some hardware architecture (e.g., 220 to 245 TFLOPS on A100).
Final Result¶
import torch
import triton
import triton.language as tl
# %
# :code:`triton.jit`'ed functions can be auto-tuned by using the `triton.autotune` decorator, which consumes:
# - A list of :code:`triton.Config` objects that define different configurations of meta-parameters (e.g., BLOCK_M) and compilation options (e.g., num_warps) to try
# - A autotuning *key* whose change in values will trigger evaluation of all the provided configs
@triton.autotune(
configs=[
triton.Config({'BLOCK_M': 128, 'BLOCK_N': 256, 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=3, num_warps=8),
triton.Config({'BLOCK_M': 256, 'BLOCK_N': 128, 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=3, num_warps=8),
triton.Config({'BLOCK_M': 256, 'BLOCK_N': 64, 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=4, num_warps=4),
triton.Config({'BLOCK_M': 64 , 'BLOCK_N': 256, 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=4, num_warps=4),\
triton.Config({'BLOCK_M': 128, 'BLOCK_N': 128, 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=4, num_warps=4),\
triton.Config({'BLOCK_M': 128, 'BLOCK_N': 64 , 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=4, num_warps=4),\
triton.Config({'BLOCK_M': 64 , 'BLOCK_N': 128, 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=4, num_warps=4),
triton.Config({'BLOCK_M': 128, 'BLOCK_N': 32 , 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=4, num_warps=4),\
triton.Config({'BLOCK_M': 64 , 'BLOCK_N': 32 , 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=5, num_warps=2),\
triton.Config({'BLOCK_M': 32 , 'BLOCK_N': 64 , 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=5, num_warps=2),
#triton.Config({'BLOCK_M': 64, 'BLOCK_N': 128, 'BLOCK_K': 32, 'GROUP_M': 8}, num_warps=4),
],
key=['M', 'N', 'K'],
)
# %
# We can now define our kernel as normal, using all the techniques presented above
@triton.jit
def _matmul(A, B, C, M, N, K, stride_am, stride_ak, stride_bk, stride_bn, stride_cm, stride_cn, **META):
# extract meta-parameters
BLOCK_M = META['BLOCK_M']
BLOCK_N = META['BLOCK_N']
BLOCK_K = META['BLOCK_K']
GROUP_M = 8
# matrix multiplication
pid = tl.program_id(0)
grid_m = (M + BLOCK_M - 1) // BLOCK_M
grid_n = (N + BLOCK_N - 1) // BLOCK_N
# re-order program ID for better L2 performance
width = GROUP_M * grid_n
group_id = pid // width
group_size = min(grid_m - group_id * GROUP_M, GROUP_M)
pid_m = group_id * GROUP_M + (pid % group_size)
pid_n = (pid % width) // (group_size)
# do matrix multiplication
rm = pid_m * BLOCK_M + tl.arange(0, BLOCK_M)
rn = pid_n * BLOCK_N + tl.arange(0, BLOCK_N)
rk = tl.arange(0, BLOCK_K)
A = A + (rm[:, None] * stride_am + rk[None, :] * stride_ak)
B = B + (rk[:, None] * stride_bk + rn[None, :] * stride_bn)
acc = tl.zeros((BLOCK_M, BLOCK_N), dtype=tl.float32)
for k in range(K, 0, -BLOCK_K):
a = tl.load(A)
b = tl.load(B)
acc += tl.dot(a, b)
A += BLOCK_K * stride_ak
B += BLOCK_K * stride_bk
# triton can accept arbitrary activation function
# via metaparameters!
if META['ACTIVATION']:
acc = META['ACTIVATION'](acc)
# rematerialize rm and rn to save registers
rm = pid_m * BLOCK_M + tl.arange(0, BLOCK_M)
rn = pid_n * BLOCK_N + tl.arange(0, BLOCK_N)
C = C + (rm[:, None] * stride_cm + rn[None, :] * stride_cn)
mask = (rm[:, None] < M) & (rn[None, :] < N)
tl.store(C, acc, mask=mask)
# we can fuse `leaky_relu` by providing it as an `ACTIVATION` meta-parameter in `_matmul`
@triton.jit
def leaky_relu(x):
return tl.where(x >= 0, x, 0.01*x)
We can now create a convenience wrapper function that only takes two input tensors and (1) checks any shape constraint; (2) allocates the output; (3) launches the above kernel
def matmul(a, b, activation=None):
# checks constraints
assert a.shape[1] == b.shape[0], "incompatible dimensions"
assert a.is_contiguous(), "matrix A must be contiguous"
assert b.is_contiguous(), "matrix B must be contiguous"
M, K = a.shape
_, N = b.shape
# allocates output
c = torch.empty((M, N), device=a.device, dtype=a.dtype)
# launch kernel
grid = lambda META: (triton.cdiv(M, META['BLOCK_M']) * triton.cdiv(N, META['BLOCK_N']), )
pgm = _matmul[grid](
a, b, c, M, N, K, \
a.stride(0), a.stride(1), b.stride(0), b.stride(1), c.stride(0), c.stride(1),\
ACTIVATION = activation
)
# done; return the output tensor
return c
Unit Test¶
We can test our custom matrix multiplication operation against a native torch implementation (i.e., cuBLAS)
torch.manual_seed(0)
a = torch.randn((512, 512), device='cuda', dtype=torch.float16)
b = torch.randn((512, 512), device='cuda', dtype=torch.float16)
c_0 = matmul(a, b, activation=None)
c_1 = torch.matmul(a, b)
print(c_0)
print(c_1)
print(triton.testing.allclose(c_0, c_1))
Out:
tensor([[ 1.1045, -36.9688, 31.4688, ..., -11.3984, 24.4531, -32.3438],
[ 6.3555, -19.6094, 34.0938, ..., -5.8945, 5.2891, 6.8867],
[-32.0625, 5.9492, 15.3984, ..., -21.3906, -23.9844, -10.1328],
...,
[ -5.7031, 7.4492, 8.2656, ..., -10.6953, -40.0000, 17.7500],
[ 25.5000, 24.3281, -8.4688, ..., -18.9375, 32.5312, -29.9219],
[ -5.3477, 4.9844, 11.8906, ..., 5.5898, 6.4023, -17.3125]],
device='cuda:0', dtype=torch.float16)
tensor([[ 1.1045, -36.9688, 31.4688, ..., -11.3906, 24.4531, -32.3438],
[ 6.3516, -19.6094, 34.0938, ..., -5.8906, 5.2812, 6.8828],
[-32.0625, 5.9531, 15.3984, ..., -21.4062, -23.9844, -10.1328],
...,
[ -5.7070, 7.4492, 8.2656, ..., -10.6953, -40.0000, 17.7500],
[ 25.5000, 24.3438, -8.4609, ..., -18.9375, 32.5312, -29.9219],
[ -5.3477, 4.9805, 11.8828, ..., 5.5859, 6.4023, -17.3125]],
device='cuda:0', dtype=torch.float16)
tensor(True, device='cuda:0')
Benchmark¶
Square Matrix Performance¶
We can now compare the performance of our kernel against that of cuBLAS. Here we focus on square matrices, but feel free to arrange this script as you wish to benchmark any other matrix shape.
@triton.testing.perf_report(
triton.testing.Benchmark(
x_names=['M', 'N', 'K'], # argument names to use as an x-axis for the plot
x_vals=[128 * i for i in range(1, 33)], # different possible values for `x_name`
line_arg='provider', # argument name whose value corresponds to a different line in the plot
line_vals=['cublas', 'cublas + relu', 'triton', 'triton + relu'], # possible values for `line_arg``
line_names=["cuBLAS", "cuBLAS (+ torch.nn.LeakyReLU)", "Triton", "Triton (+ LeakyReLU)"], # label name for the lines
styles=[('green', '-'), ('green', '--'), ('blue', '-'), ('blue', '--')], # line styles
ylabel="TFLOPS", # label name for the y-axis
plot_name="matmul-performance", # name for the plot. Used also as a file name for saving the plot.
args={}
)
)
def benchmark(M, N, K, provider):
a = torch.randn((M, K), device='cuda', dtype=torch.float16)
b = torch.randn((K, N), device='cuda', dtype=torch.float16)
if provider == 'cublas':
ms, min_ms, max_ms = triton.testing.do_bench(lambda: torch.matmul(a, b))
if provider == 'triton':
ms, min_ms, max_ms = triton.testing.do_bench(lambda: matmul(a, b))
if provider == 'cublas + relu':
torch_relu = torch.nn.ReLU(inplace=True)
ms, min_ms, max_ms = triton.testing.do_bench(lambda: torch_relu(torch.matmul(a, b)))
if provider == 'triton + relu':
ms, min_ms, max_ms = triton.testing.do_bench(lambda: matmul(a, b, activation=leaky_relu))
perf = lambda ms: 2 * M * N * K * 1e-12 / (ms * 1e-3)
return perf(ms), perf(max_ms), perf(min_ms)
benchmark.run(show_plots=True, print_data=True)
Out:
matmul-performance:
M cuBLAS ... Triton Triton (+ LeakyReLU)
0 128.0 0.455111 ... 0.512000 0.512000
1 256.0 2.978909 ... 2.978909 2.978909
2 384.0 7.372800 ... 8.507077 8.507077
3 512.0 14.563555 ... 16.384000 16.384000
4 640.0 22.260869 ... 24.380953 24.380953
5 768.0 32.768000 ... 34.028308 34.028308
6 896.0 39.025776 ... 39.025776 37.971025
7 1024.0 49.932191 ... 52.428801 52.428801
8 1152.0 44.566925 ... 46.656000 46.656000
9 1280.0 51.200001 ... 56.109587 56.109587
10 1408.0 64.138541 ... 66.485074 65.684049
11 1536.0 80.430545 ... 76.106321 76.106321
12 1664.0 63.372618 ... 62.061463 60.803457
13 1792.0 72.512412 ... 68.953520 68.533074
14 1920.0 69.120002 ... 69.467336 68.776119
15 2048.0 73.584279 ... 75.915006 75.915006
16 2176.0 83.500614 ... 80.817862 80.494588
17 2304.0 68.446623 ... 72.828879 72.607513
18 2432.0 71.125224 ... 79.139336 79.813818
19 2560.0 77.833728 ... 76.740048 75.328737
20 2688.0 84.108772 ... 79.524227 83.369354
21 2816.0 82.135981 ... 78.868366 76.516158
22 2944.0 81.967162 ... 81.034195 79.737653
23 3072.0 81.825298 ... 83.146995 83.761985
24 3200.0 84.210524 ... 89.887639 86.137280
25 3328.0 82.464255 ... 82.843841 86.424125
26 3456.0 82.604067 ... 87.442050 82.435141
27 3584.0 86.540320 ... 96.683219 90.458141
28 3712.0 84.946722 ... 82.084920 87.170458
29 3840.0 84.421376 ... 86.806905 86.063813
30 3968.0 92.652949 ... 87.472354 87.284643
31 4096.0 93.401342 ... 91.553703 91.678778
[32 rows x 5 columns]
Total running time of the script: ( 2 minutes 14.506 seconds)