# 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 (M, K) by a (K, N) 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 is performed by a dedicated 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_xi + j*stride_xj`. 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_ptr + (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_ptr + (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:

```offs_am = pid_m * BLOCK_SIZE_M + tl.arange(0, BLOCK_SIZE_M)
offs_bn = pid_n * BLOCK_SIZE_N + tl.arange(0, BLOCK_SIZE_N)
offs_k = tl.arange(0, BLOCK_SIZE_K)
a_ptrs = a_ptr + (offs_am[:, None]*stride_am + offs_k [None, :]*stride_ak)
b_ptrs = b_ptr + (offs_k [:, None]*stride_bk + offs_bn[None, :]*stride_bn)
```

And then updated in the inner loop as follows:

```pa += BLOCK_SIZE_K * stride_ak;
pb += BLOCK_SIZE_K * stride_bk;
```

### 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:

```# program ID
pid = tl.program_id(axis=0)
# number of program ids along the M axis
num_pid_m = tl.cdiv(M, BLOCK_SIZE_M)
# number of programs ids along the N axis
num_pid_n = tl.cdiv(N, BLOCK_SIZE_N)
# number of programs in group
num_pid_in_group = GROUP_SIZE_M * num_pid_n
# id of the group this program is in
group_id = pid // num_pid_in_group
# row-id of the first program in the group
first_pid_m = group_id * GROUP_SIZE_M
# if `num_pid_m` isn't divisible by `GROUP_SIZE_M`, the last group is smaller
group_size_m = min(num_pid_m - first_pid_m, GROUP_SIZE_M)
# *within groups*, programs are ordered in a column-major order
# row-id of the program in the *launch grid*
pid_m = first_pid_m + (pid % group_size_m)
# col-id of the program in the *launch grid*
pid_n = (pid % num_pid_in_group) // group_size_m
```

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-parameters
**meta,
):
"""Kernel for computing the matmul C = A x B.
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

# -----------------------------------------------------------
# Map program ids `pid` to the block of C it should compute.
# This is done in a grouped ordering to promote L2 data reuse
# See above `L2 Cache Optimizations` section for details
pid = tl.program_id(axis=0)
num_pid_m = tl.cdiv(M, BLOCK_SIZE_M)
num_pid_n = tl.cdiv(N, BLOCK_SIZE_N)
num_pid_in_group = GROUP_SIZE_M * num_pid_n
group_id = pid // num_pid_in_group
first_pid_m = group_id * GROUP_SIZE_M
group_size_m = min(num_pid_m - first_pid_m, GROUP_SIZE_M)
pid_m = first_pid_m + (pid % group_size_m)
pid_n = (pid % num_pid_in_group) // group_size_m

# ----------------------------------------------------------
# Create pointers for the first blocks of A and B.
# We will advance this pointer as we move in the K direction
# and accumulate
# a_ptrs is a block of [BLOCK_SIZE_M, BLOCK_SIZE_K] pointers
# b_ptrs is a block of [BLOCK_SIZE_K, BLOCK_SIZE_n] pointers
# see above `Pointer Arithmetics` section for details
offs_am = pid_m * BLOCK_SIZE_M + tl.arange(0, BLOCK_SIZE_M)
offs_bn = pid_n * BLOCK_SIZE_N + tl.arange(0, BLOCK_SIZE_N)
offs_k = tl.arange(0, BLOCK_SIZE_K)
a_ptrs = a_ptr + (offs_am[:, None]*stride_am + offs_k [None, :]*stride_ak)
b_ptrs = b_ptr + (offs_k [:, None]*stride_bk + offs_bn[None, :]*stride_bn)

# -----------------------------------------------------------
# Iterate to compute a block of the C matrix
# We accumulate into a `[BLOCK_SIZE_M, BLOCK_SIZE_N]` block
# of fp32 values for higher accuracy.
# `accumulator` will be converted back to fp16 after the loop
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 produce an
# error or (worse!) incorrect results.
# 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
# you can fuse arbitrary activation functions here
# while the accumulator is still in FP32 !
if meta['ACTIVATION']:
accumulator = meta['ACTIVATION'](accumulator)
c = accumulator.to(tl.float16)

# -----------------------------------------------------------
# Write back the block of the output matrix C
offs_cm = pid_m * BLOCK_SIZE_M + tl.arange(0, BLOCK_SIZE_M)
offs_cn = pid_n * BLOCK_SIZE_N + tl.arange(0, BLOCK_SIZE_N)
c_ptrs = c_ptr + stride_cm * offs_cm[:, None] + stride_cn * offs_cn[None, :]
c_mask = (offs_cm[:, None] < M) & (offs_cn[None, :] < N)

# 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(2, 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    256.0   2.730667  ...   2.978909              2.978909
1    384.0   7.372800  ...   8.507077              8.507077
2    512.0  14.563555  ...  16.384000             16.384000
3    640.0  22.260869  ...  24.380953             24.380953
4    768.0  32.768000  ...  34.028308             34.028308
5    896.0  37.971025  ...  40.140799             39.025776
6   1024.0  49.932191  ...  53.773130             52.428801
7   1152.0  44.566925  ...  46.656000             46.656000
8   1280.0  51.200001  ...  56.888887             56.109587
9   1408.0  64.138541  ...  67.305878             67.305878
10  1536.0  80.430545  ...  79.526831             79.526831
11  1664.0  63.372618  ...  62.492442             62.492442
12  1792.0  72.983276  ...  72.047592             72.047592
13  1920.0  69.120002  ...  70.172588             70.172588
14  2048.0  73.908442  ...  76.959706             76.608294
15  2176.0  83.500614  ...  86.367588             85.632545
16  2304.0  68.446623  ...  76.809875             76.809875
17  2432.0  71.305746  ...  74.918570             85.393507
18  2560.0  78.019048  ...  80.908642             80.709358
19  2688.0  83.552988  ...  89.149366             89.464755
20  2816.0  82.602666  ...  83.074685             83.233226
21  2944.0  82.646820  ...  82.373605             82.784108
22  3072.0  81.943708  ...  88.612060             87.516392
23  3200.0  78.914919  ...  91.822093             93.567248
24  3328.0  81.530349  ...  84.003845             84.496824
25  3456.0  82.519518  ...  91.200871             90.943675
26  3584.0  85.633710  ...  94.847460             96.579370
27  3712.0  85.528545  ...  85.019017             87.706180
28  3840.0  81.980725  ...  86.130841             91.247522
29  3968.0  85.993854  ...  91.747320             86.053553
30  4096.0  93.727466  ...  88.417474             84.307617

[31 rows x 5 columns]
```

Total running time of the script: ( 5 minutes 41.070 seconds)

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