triton/python/tutorials/02-fused-softmax.py

"""
Fused Softmax
=================
In this tutorial, you will write a fused softmax operation (that outperforms PyTorch) and learn about:

- The benefits of kernel fusion for bandwidth-bound operations.
- The 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


# Compute the row-wise softmax of x
def naive_softmax(x):
    # read  MN elements ; write M  elements
    x_max = torch.max(x, axis=1)[0]
    # read 2MN elements ; write MN elements
    z = x - x_max[:, None]
    # read  MN elements ; write MN elements
    numerator = torch.exp(x)
    # read  MN elements ; write M  elements
    denominator = torch.sum(numerator, axis=1)
    # read 2MN elements ; write MN elements
    ret = numerator / denominator[:, None]
    # in total: read 7MN elements ; wrote 3MN + 2M elements
    return ret


# %%
# When implemented naively in pytorch, computing :code:`y = naive_softmax(x)` for :math:`x \in R^{M \times N}` requires reading :math:`7MN` elements from DRAM and writing back :math:`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.
# This solution would require reading and writing back only :math:`MN` bytes, so we could expect a theoretical speed-up of ~5x (i.e., :math:`(10MN + 2M) / 2MN`).
# In practice, though, we would be getting a bit less as our kernel computes exponentials and internally moves data around in shared memory.

# %%
# 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" tiles and guard the memory operations properly if we want to handle any possible input shapes:

import triton
import triton.language as tl


@triton.jit
def _softmax(Y, X, stride_xm, stride_ym, M, N, **meta):
    # row index
    m = tl.program_id(0)
    # col indices
    n = tl.arange(0, meta['BLOCK'])
    # the memory address of all the elements
    # that we want to load can be computed as follows
    X = X + m * stride_xm + n
    x = tl.load(X, mask=n < N, other=-float('inf'))
    # Substract maximum for numerical stability
    z = x - tl.max(x, axis=0)
    # Note that exponentials in Triton are fast
    # but approximate (i.e., think __expf in CUDA)
    num = tl.exp(z)
    denom = tl.sum(num, axis=0)
    y = num / denom
    # Write back to Y
    Y = Y + m * stride_ym + n
    tl.store(Y, y, mask=n < N)


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


def next_power_of_2(n):
    n -= 1
    n |= n >> 1
    n |= n >> 2
    n |= n >> 4
    n |= n >> 8
    n |= n >> 16
    n += 1
    return n


def softmax(x):
    M, N = x.shape
    # The block size is the smallest power of two greater than the number of columns in `x`
    BLOCK = next_power_of_2(N)
    # Another trick we can use is to ask the compiler to parallelize each
    # row-normalization more aggressively -- i.e., with more warps -- vectors
    # that are longer
    # 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 >= 2048: num_warps = 8
    if BLOCK >= 4096: num_warps = 16
    # Allocate output
    y = torch.empty_like(x)
    # Enqueue kernel. The launch grid is simple: we have one kernel instance per row of the input matrix
    _softmax[(M, )](y, x, x.stride(0), y.stride(0), M, N, num_warps=num_warps, BLOCK=BLOCK)
    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_tri = softmax(x)
y_ref = torch.softmax(x, axis=1)
print(torch.allclose(y_tri, y_ref))

#%%
# 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) :code:`torch.softmax` and (2) the :code:`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=[256 * i for i in range(2, 50)],  # different possible values for `x_name`
        line_arg='provider',  # argument name whose value corresponds to a different line in the plot
        line_vals=['torch', 'triton', 'naive'],  # possible values for `line_arg``
        line_names=["Torch", "Triton", 'Naive'],  # label name for the lines
        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)
    if provider == 'torch':
        ms, min_ms, max_ms = triton.testing.do_bench(lambda: torch.softmax(x, axis=-1))
    if provider == 'triton':
        ms, min_ms, max_ms = triton.testing.do_bench(lambda: softmax(x))
    if provider == 'naive':
        ms, min_ms, max_ms = triton.testing.do_bench(lambda: naive_softmax(x))
    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)

# %%
# In the above plot, we can see that:
#
#  - Triton is 4-5x faster than the naive implementation, which is consistent with our theoretical predictions.
#  - Triton is significantly faster than :code:`torch.softmax` for very large input matrices. My guess from looking at the source-code of the `PyTorch kernel <https://github.com/pytorch/pytorch/blob/9409a3a39b7149bb2d833a89e0c944109bef7c27/caffe2/operators/softmax_ops.cu#L240>`_ is that PyTorch only partially fuses the computation of the softmax.
#    This means that -- when temporary data is too large to fit entirely in the GPU's cache -- it transfers almost twice the amount of data necessary.
#    Note that our Triton kernel is not only faster than PyTorch's CUDA kernel, it is also **easier to read, understand and maintain**.