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 to multiply a (MxK) by a (KxN) matrix:
# do in parallel for m in range(0, M, BLOCK_SIZE_M): # do in parallel for n in range(0, N, BLOCK_SIZE_N): acc = zeros((BLOCK_SIZE_M, BLOCK_SIZE_N), dtype=float32) for k in range(0, K, BLOCK_SIZE_K): a = A[m : m+BLOCK_SIZE_M, k : k+BLOCK_SIZE_K] b = B[k : k+BLOCK_SIZE_K, n : n+BLOCK_SIZE_N] acc += dot(a, b) C[m : m+BLOCK_SIZE_M, n : n+BLOCK_SIZE_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 b
y &X[i, j] = X + i*stride_x_0 + j*stride_x_1.
Therefore, blocks of pointers for A[m : m+BLOCK_SIZE_M, k:k+BLOCK_SIZE_K] and
B[k : k+BLOCK_SIZE_K, n : n+BLOCK_SIZE_N] can be defined in pseudo-code as:
&A[m : m+BLOCK_SIZE_M, k:k+BLOCK_SIZE_K] = A + (m : m+BLOCK_SIZE_M)[:, None]*A.stride(0) + (k : k+BLOCK_SIZE_K)[None, :]*A.stride(1); &B[k : k+BLOCK_SIZE_K, n:n+BLOCK_SIZE_N] = B + (k : k+BLOCK_SIZE_K)[:, None]*B.stride(0) + (n : n+BLOCK_SIZE_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_SIZE_M + triton.arange(0, BLOCK_SIZE_M) rn = pid_n * BLOCK_SIZE_N + triton.arange(0, BLOCK_SIZE_N) rk = triton.arange(0, BLOCK_SIZE_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_SIZE_K * stride_a_1; pb += BLOCK_SIZE_K * stride_b_0;
L2 Cache Optimizations¶
As mentioned above, each program instance computes a [BLOCK_SIZE_M, BLOCK_SIZE_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
a simple row-major ordering
pid = triton.program_id(0); grid_m = (M + BLOCK_SIZE_M - 1) // BLOCK_SIZE_M; grid_n = (N + BLOCK_SIZE_N - 1) // BLOCK_SIZE_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_SIZE_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);
For example, in the following matmul where each matrix is 9 blocks by 9 blocks, we can see that if we compute the output in row-major ordering, we need to load 90 blocks into SRAM to compute the first 9 output blocks, but if we do it in grouped ordering, we only need to load 54 blocks.
In practice, this can improve the performance of our matrix multiplication kernel by more than 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_SIZE_M) and compilation options (e.g., num_warps) to try
# - An autotuning *key* whose change in values will trigger evaluation of all the
# provided configs
@triton.autotune(
configs=[
triton.Config({'BLOCK_SIZE_M': 128, 'BLOCK_SIZE_N': 256, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=3, num_warps=8),
triton.Config({'BLOCK_SIZE_M': 256, 'BLOCK_SIZE_N': 128, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=3, num_warps=8),
triton.Config({'BLOCK_SIZE_M': 256, 'BLOCK_SIZE_N': 64, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4, num_warps=4),
triton.Config({'BLOCK_SIZE_M': 64 , 'BLOCK_SIZE_N': 256, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4, num_warps=4),
triton.Config({'BLOCK_SIZE_M': 128, 'BLOCK_SIZE_N': 128, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4, num_warps=4),
triton.Config({'BLOCK_SIZE_M': 128, 'BLOCK_SIZE_N': 64 , 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4, num_warps=4),
triton.Config({'BLOCK_SIZE_M': 64 , 'BLOCK_SIZE_N': 128, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4, num_warps=4),
triton.Config({'BLOCK_SIZE_M': 128, 'BLOCK_SIZE_N': 32 , 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4, num_warps=4),
triton.Config({'BLOCK_SIZE_M': 64 , 'BLOCK_SIZE_N': 32 , 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=5, num_warps=2),
triton.Config({'BLOCK_SIZE_M': 32 , 'BLOCK_SIZE_N': 64 , 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=5, num_warps=2),
],
key=['M', 'N', 'K'],
)
# %
# We can now define our kernel as normal, using all the techniques presented above
@triton.jit
def matmul_kernel(
# Pointers to matrices
a_ptr,
b_ptr,
c_ptr,
# Matrix dimensions
M,
N,
K,
# The stride variables represent how much to increase the ptr by when moving by 1
# element in a particular dimension. E.g. stride_am is how much to increase a_ptr
# by to get the element one row down (A has M rows)
stride_am,
stride_ak,
stride_bk,
stride_bn,
stride_cm,
stride_cn,
**meta,
):
"""Kernel for computing the matmul AB = C
A has shape (M, K), B has shape (K, N) and C has shape (M, N)
"""
# extract meta-parameters
BLOCK_SIZE_M = meta['BLOCK_SIZE_M']
BLOCK_SIZE_N = meta['BLOCK_SIZE_N']
BLOCK_SIZE_K = meta['BLOCK_SIZE_K']
GROUP_SIZE_M = 8
pid = tl.program_id(axis=0)
# the number of blocks is the ceil(M / BLOCK_SIZE_M) since we need an extra block
# Note that this will lead to some quantization in performance where time-taken jumps
# when you need to add a new block
n_blocks_m = (M + BLOCK_SIZE_M - 1) // BLOCK_SIZE_M
n_blocks_n = (N + BLOCK_SIZE_N - 1) // BLOCK_SIZE_N
# Map PIDs to the block they should compute. This is done in a grouped ordering
# to promote L2 cache reuse.
n_output_blocks_in_group = GROUP_SIZE_M * n_blocks_n
group_id = pid // n_output_blocks_in_group
first_m_block_in_group = group_id * GROUP_SIZE_M
# If the number of blocks is not divisible by the group size, the last group is smaller
group_size_m = min(n_blocks_m - first_m_block_in_group, GROUP_SIZE_M)
# Within a group, we compute in col-major ordering, block_m and block_n are the
# output row and col that this program is computing in terms of blocks
block_m = first_m_block_in_group + (pid % group_size_m)
block_n = (pid % n_output_blocks_in_group) // group_size_m
# Convert from block indices back to element indices
m_start = block_m * BLOCK_SIZE_M
n_start = block_n * BLOCK_SIZE_N
# Expand out to all the offsets for each of the elements in this block.
m_offsets_a = (m_start + tl.arange(0, BLOCK_SIZE_M))[:, None]
n_offsets_b = (n_start + tl.arange(0, BLOCK_SIZE_N))[None, :]
k_offsets = tl.arange(0, BLOCK_SIZE_K)
# Get the pointers for the first block of each. We will advance this pointer
# as we move in the K direction and accumulate.
# a_ptrs should contain BLOCK_SIZE_M * BLOCK_SIZE_K pointers
a_ptrs = a_ptr + (stride_am * m_offsets_a + stride_ak * k_offsets[None, :])
# b_ptrs should contain BLOCK_SIZE_K * BLOCK_SIZE_N pointers
b_ptrs = b_ptr + (stride_bk * k_offsets[:, None] + stride_bn * n_offsets_b)
# We accumulate internally in fp32, but the output is written out in the dtype
# of the tensor when it is stored
accumulator = tl.zeros((BLOCK_SIZE_M, BLOCK_SIZE_N), dtype=tl.float32)
for k in range(0, K, BLOCK_SIZE_K):
# Note that for simplicity, we don't apply a mask here. This means that if K is
# not a multiple of BLOCK_SIZE_K, this will access out-of-bounds memory and
# accumulate it incorrectly.
a = tl.load(a_ptrs)
b = tl.load(b_ptrs)
# We accumulate along the K dimension
accumulator += tl.dot(a, b)
# Advance the ptrs to the next K block
a_ptrs += BLOCK_SIZE_K * stride_ak
b_ptrs += BLOCK_SIZE_K * stride_bk
# triton can accept arbitrary activation function via metaparameters!
if meta['ACTIVATION']:
accumulator = meta['ACTIVATION'](accumulator)
m_offsets_c = (m_start + tl.arange(0, BLOCK_SIZE_M))[:, None]
n_offsets_c = (n_start + tl.arange(0, BLOCK_SIZE_N))[None, :]
c_ptrs = c_ptr + stride_cm * m_offsets_c + stride_cn * n_offsets_c
mask = (m_offsets_c < M) & (n_offsets_c < N)
tl.store(c_ptrs, accumulator, 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
K, N = b.shape
assert (
K % 32 == 0
), "We don't check memory-out-of-bounds with K so K must be divisible by BLOCK_SIZE_K"
# allocates output
c = torch.empty((M, N), device=a.device, dtype=a.dtype)
# 1D launch kernel where each block gets its own program.
grid = lambda META: (
triton.cdiv(M, META['BLOCK_SIZE_M']) * triton.cdiv(N, META['BLOCK_SIZE_N']),
)
matmul_kernel[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,
)
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)
triton_output = matmul(a, b, activation=None)
torch_output = torch.matmul(a, b)
print(f"triton_output={triton_output}")
print(f"torch_output={torch_output}")
if triton.testing.allclose(triton_output, torch_output):
print("✅ Triton and Torch match")
else:
print("❌ Triton and Torch differ")
Out:
triton_output=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)
torch_output=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)
✅ Triton and Torch match
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
# possible values for `line_arg``
line_vals=['cublas', 'cublas + relu', 'triton', 'triton + relu'],
# label name for the lines
line_names=["cuBLAS", "cuBLAS (+ torch.nn.LeakyReLU)", "Triton", "Triton (+ LeakyReLU)"],
# line styles
styles=[('green', '-'), ('green', '--'), ('blue', '-'), ('blue', '--')],
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 ... 3.276800 2.978909
2 384.0 7.372800 ... 8.507077 8.507077
3 512.0 14.563555 ... 16.384000 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 ... 40.140799 39.025776
7 1024.0 49.932191 ... 53.773130 52.428801
8 1152.0 45.242181 ... 46.656000 46.656000
9 1280.0 51.200001 ... 56.888887 56.109587
10 1408.0 64.138541 ... 64.902096 64.902096
11 1536.0 80.430545 ... 76.106321 76.106321
12 1664.0 63.372618 ... 62.061463 62.061463
13 1792.0 72.983276 ... 69.810085 69.810085
14 1920.0 69.120002 ... 70.172588 69.120002
15 2048.0 73.584279 ... 74.898285 73.584279
16 2176.0 82.813365 ... 78.916269 79.540109
17 2304.0 68.056616 ... 73.275679 73.275679
18 2432.0 71.125224 ... 81.197876 81.908060
19 2560.0 77.649287 ... 76.560748 75.676673
20 2688.0 84.108772 ... 81.053536 84.108772
21 2816.0 80.469019 ... 79.298560 78.726003
22 2944.0 81.832567 ... 79.737653 79.104810
23 3072.0 82.540970 ... 80.890151 83.146995
24 3200.0 84.210524 ... 89.260810 87.791493
25 3328.0 83.130825 ... 86.736504 82.275764
26 3456.0 78.578525 ... 83.459178 85.767626
27 3584.0 87.808000 ... 92.410473 95.148565
28 3712.0 85.601834 ... 85.601834 88.876645
29 3840.0 84.615146 ... 86.875096 87.148936
30 3968.0 92.512459 ... 83.807647 83.520835
31 4096.0 93.596744 ... 90.748973 90.321484
[32 rows x 5 columns]
Total running time of the script: ( 2 minutes 16.405 seconds)