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 2.730667 ... 3.276800 2.978909
2 384.0 7.372800 ... 7.899428 7.899428
3 512.0 14.563555 ... 15.420235 15.420235
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 39.025776
7 1024.0 49.932191 ... 52.428801 52.428801
8 1152.0 44.566925 ... 45.938215 45.938215
9 1280.0 51.200001 ... 56.109587 56.109587
10 1408.0 64.138541 ... 65.684049 58.621246
11 1536.0 79.526831 ... 76.106321 75.296679
12 1664.0 63.372618 ... 61.636381 62.061463
13 1792.0 72.983276 ... 69.379162 68.533074
14 1920.0 69.467336 ... 68.776119 69.120002
15 2048.0 73.908442 ... 75.573044 74.898285
16 2176.0 83.155572 ... 80.494588 79.855747
17 2304.0 68.446623 ... 73.051599 72.387489
18 2432.0 71.125224 ... 80.269900 79.139336
19 2560.0 77.833728 ... 76.740048 74.812787
20 2688.0 83.737433 ... 80.196737 82.463163
21 2816.0 83.552120 ... 78.442822 77.882512
22 2944.0 82.102191 ... 81.034195 78.979452
23 3072.0 80.202695 ... 84.010539 79.750851
24 3200.0 84.432717 ... 89.012517 86.720870
25 3328.0 79.114032 ... 78.851363 81.071278
26 3456.0 81.518272 ... 87.252780 82.773682
27 3584.0 84.905939 ... 95.654673 95.451583
28 3712.0 84.088676 ... 82.902362 84.159518
29 3840.0 83.655065 ... 84.036474 85.267542
30 3968.0 92.935215 ... 84.797731 83.807647
31 4096.0 93.336389 ... 91.616198 91.118618
[32 rows x 5 columns]
Total running time of the script: ( 2 minutes 15.188 seconds)