Note

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# Fused Softmax¶

In this tutorial, you will write a fused softmax operation that is significantly faster than PyTorch’s native op for a particular class of matrices: those whose rows can fit in the GPU’s SRAM.

In doing so, you will learn about:

The benefits of kernel fusion for bandwidth-bound operations.

Reduction operators in Triton.

## Motivations¶

Custom GPU kernels for elementwise additions are educationally valuable but won’t get you very far in practice. Let us consider instead the case of a simple (numerically stabilized) softmax operation:

```
import torch
import triton
import triton.language as tl
@torch.jit.script
def naive_softmax(x):
"""Compute row-wise softmax of X using native pytorch
We subtract the maximum element in order to avoid overflows. Softmax is invariant to
this shift.
"""
# read MN elements ; write M elements
x_max = x.max(dim=1)[0]
# read MN + M elements ; write MN elements
z = x - x_max[:, None]
# read MN elements ; write MN elements
numerator = torch.exp(z)
# read MN elements ; write M elements
denominator = numerator.sum(dim=1)
# read MN + M elements ; write MN elements
ret = numerator / denominator[:, None]
# in total: read 5MN + 2M elements ; wrote 3MN + 2M elements
return ret
```

When implemented naively in PyTorch, computing `y = naive_softmax(x)`

for \(x \in R^{M \times N}\)
requires reading \(5MN + 2M\) elements from DRAM and writing back \(3MN + 2M\) elements.
This is obviously wasteful; we’d prefer to have a custom “fused” kernel that only reads
X once and does all the necessary computations on-chip.
Doing so would require reading and writing back only \(MN\) bytes, so we could
expect a theoretical speed-up of ~4x (i.e., \((8MN + 4M) / 2MN\)).
The torch.jit.script flags aims to perform this kind of “kernel fusion” automatically
but, as we will see later, it is still far from ideal.

## Compute Kernel¶

Our softmax kernel works as follows: each program loads a row of the input matrix X, normalizes it and writes back the result to the output Y.

Note that one important limitation of Triton is that each block must have a power-of-two number of elements, so we need to internally “pad” each row and guard the memory operations properly if we want to handle any possible input shapes:

```
@triton.jit
def softmax_kernel(output_ptr, input_ptr, input_row_stride, output_row_stride, n_cols, BLOCK_SIZE: tl.constexpr):
# The rows of the softmax are independent, so we parallelize across those
row_idx = tl.program_id(0)
# The stride represents how much we need to increase the pointer to advance 1 row
row_start_ptr = input_ptr + row_idx * input_row_stride
# The block size is the next power of two greater than n_cols, so we can fit each
# row in a single block
col_offsets = tl.arange(0, BLOCK_SIZE)
input_ptrs = row_start_ptr + col_offsets
# Load the row into SRAM, using a mask since BLOCK_SIZE may be > than n_cols
row = tl.load(input_ptrs, mask=col_offsets < n_cols, other=-float('inf'))
# Subtract maximum for numerical stability
row_minus_max = row - tl.max(row, axis=0)
# Note that exponentiation in Triton is fast but approximate (i.e., think __expf in CUDA)
numerator = tl.exp(row_minus_max)
denominator = tl.sum(numerator, axis=0)
softmax_output = numerator / denominator
# Write back output to DRAM
output_row_start_ptr = output_ptr + row_idx * output_row_stride
output_ptrs = output_row_start_ptr + col_offsets
tl.store(output_ptrs, softmax_output, mask=col_offsets < n_cols)
```

We can create a helper function that enqueues the kernel and its (meta-)arguments for any given input tensor.

```
def softmax(x):
n_rows, n_cols = x.shape
# The block size is the smallest power of two greater than the number of columns in `x`
BLOCK_SIZE = triton.next_power_of_2(n_cols)
# Another trick we can use is to ask the compiler to use more threads per row by
# increasing the number of warps (`num_warps`) over which each row is distributed.
# You will see in the next tutorial how to auto-tune this value in a more natural
# way so you don't have to come up with manual heuristics yourself.
num_warps = 4
if BLOCK_SIZE >= 2048:
num_warps = 8
if BLOCK_SIZE >= 4096:
num_warps = 16
# Allocate output
y = torch.empty_like(x)
# Enqueue kernel. The 1D launch grid is simple: we have one kernel instance per row o
# f the input matrix
softmax_kernel[(n_rows, )](
y,
x,
x.stride(0),
y.stride(0),
n_cols,
num_warps=num_warps,
BLOCK_SIZE=BLOCK_SIZE,
)
return y
```

## Unit Test¶

We make sure that we test our kernel on a matrix with an irregular number of rows and columns. This will allow us to verify that our padding mechanism works.

```
torch.manual_seed(0)
x = torch.randn(1823, 781, device='cuda')
y_triton = softmax(x)
y_torch = torch.softmax(x, axis=1)
assert torch.allclose(y_triton, y_torch), (y_triton, y_torch)
```

As expected, the results are identical.

## Benchmark¶

Here we will benchmark our operation as a function of the number of columns in the input matrix – assuming 4096 rows.
We will then compare its performance against (1) `torch.softmax`

and (2) the `naive_softmax`

defined above.

```
@triton.testing.perf_report(
triton.testing.Benchmark(
x_names=['N'], # argument names to use as an x-axis for the plot
x_vals=[128 * i for i in range(2, 100)], # different possible values for `x_name`
line_arg='provider', # argument name whose value corresponds to a different line in the plot
line_vals=[
'triton',
'torch-native',
'torch-jit',
], # possible values for `line_arg``
line_names=[
"Triton",
"Torch (native)",
"Torch (jit)",
], # label name for the lines
styles=[('blue', '-'), ('green', '-'), ('green', '--')], # line styles
ylabel="GB/s", # label name for the y-axis
plot_name="softmax-performance", # name for the plot. Used also as a file name for saving the plot.
args={'M': 4096}, # values for function arguments not in `x_names` and `y_name`
))
def benchmark(M, N, provider):
x = torch.randn(M, N, device='cuda', dtype=torch.float32)
quantiles = [0.5, 0.2, 0.8]
if provider == 'torch-native':
ms, min_ms, max_ms = triton.testing.do_bench(lambda: torch.softmax(x, axis=-1), quantiles=quantiles)
if provider == 'triton':
ms, min_ms, max_ms = triton.testing.do_bench(lambda: softmax(x), quantiles=quantiles)
if provider == 'torch-jit':
ms, min_ms, max_ms = triton.testing.do_bench(lambda: naive_softmax(x), quantiles=quantiles)
gbps = lambda ms: 2 * x.nelement() * x.element_size() * 1e-9 / (ms * 1e-3)
return gbps(ms), gbps(max_ms), gbps(min_ms)
benchmark.run(show_plots=True, print_data=True)
```

```
softmax-performance:
N Triton Torch (native) Torch (jit)
0 256.0 682.666643 744.727267 256.000001
1 384.0 877.714274 877.714274 311.088617
2 512.0 910.222190 910.222190 334.367358
3 640.0 975.238103 930.909084 372.363633
4 768.0 1068.521715 1023.999964 390.095241
.. ... ... ... ...
93 12160.0 1601.316858 1514.085562 464.343688
94 12288.0 1604.963246 1389.455824 465.344393
95 12416.0 1602.064538 1389.202822 463.067592
96 12544.0 1599.235121 1393.777728 463.519622
97 12672.0 1602.782573 1393.484517 463.963376
[98 rows x 4 columns]
```

- In the above plot, we can see that:
Triton is 4x faster than the Torch JIT. This confirms our suspicions that the Torch JIT does not do any fusion here.

Triton is noticeably faster than

`torch.softmax`

– in addition to being**easier to read, understand and maintain**. Note however that the PyTorch softmax operation is more general and will work on tensors of any shape.

**Total running time of the script:** (0 minutes 41.007 seconds)