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 syntax and usage of reduction operators in Triton.

• The automatic vectorization capabilities of the Triton compiler.

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 y = naive_softmax(x) for \(x \in R^{M \times N}\) requires reading \(7MN\) 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. In this case, we would be reading and writing back only \(MN\) bytes, so we could expect a theoretical speed-up of ~5x (i.e., \((10MN + 2M) / 2MN\)). 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 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:

__global__ void softmax(float* Y, float* X, int stride_xm, int stride_ym, int M, int N){
  // row index
  int    m             = get_program_id(0);
  // column indices
  int    n    [BLOCK] = 0 ... BLOCK;
  // the memory address of all the elements
  // that we want to load can be computed as follows
  float* px   [BLOCK] = X + m*stride_xm + n;
  // because BLOCK has to be a power of two
  // (per Triton-C specs), it is important
  // to guard each memory operation with predicates
  // or we will read out of bounds
  bool   check[BLOCK] = n < N;
  float  x    [BLOCK] = check ? *px : -F32_INFINITY;
  // syntax for reduction in Triton is:
  // x[:, :, OPERATOR, :, :]
  //            ^
  //           index
  // where operator is in {min, max, +}
  // for 1D vectors, this is just x[OPERATOR].
  float  z    [BLOCK] = x - x[max];
  // Note that exponentials in Triton are fast
  // but approximate (i.e., think __expf in CUDA)
  float  num  [BLOCK] = exp(z);
  float  denom         = num[+];
  // The result of the reduction is now stored in y
  float  y    [BLOCK] = num / denom;
  // We write it back
  float* py   [BLOCK] = Y + m*stride_ym + n;
  *?(check)py = y;
}

Torch Bindings

Here our torch bindings is quite similar to that of the vector addition mentioned in the previous tutorial. We just need to make sure that BLOCK is the smallest power of two greater than the number of columns N of the input matrix. This means that different values of BLOCK will result in different kernels

import torch
import triton

# Source code for the Triton kernel
_src = """
__global__ void softmax(float* Y, float* X, int stride_ym, int stride_xm, int M, int N){
    int    m             = get_program_id(0);
    int    n    [BLOCK] = 0 ... BLOCK;
    float* px   [BLOCK] = X + m*stride_xm + n;
    bool   check[BLOCK] = n < N;
    float  x    [BLOCK] = check ? *px : -F32_INFINITY;
    float  z    [BLOCK] = x - x[max];
    float  num  [BLOCK] = exp(z);
    float  denom        = num[+];
    float  y    [BLOCK] = num / denom;
    float* py   [BLOCK] = Y + m*stride_ym + n;
    *?(check)py = y;
}
"""


# helper function to get the smaller power-of-two larger than a given number
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


# kernel caching mechanism
def make_kernel(N, device):
    cache = make_kernel.cache
    # Now are kernels are indexed not only by the provided device but also
    # by the rounded number of columns in the input matrix
    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
    # Each (BLOCK, num_warps, device) results in a different kernel
    key = (BLOCK, num_warps, device)
    if key not in cache:
        defines = {'BLOCK': BLOCK}
        cache[key] = triton.kernel(_src, device=device, defines=defines, num_warps=num_warps)
    return cache[key]


make_kernel.cache = dict()


class _softmax(torch.autograd.Function):
    @staticmethod
    def forward(ctx, x):
        # constraints of the op
        assert x.dtype == torch.float32
        y = torch.empty_like(x)
        # The launch grid is simple: we have one kernel instance per row of the input matrix
        M, N = y.shape
        grid = lambda opt: (M, )
        # Launch kernel
        kernel = make_kernel(N, y.device)
        kernel(y.data_ptr(), x.data_ptr(), y.stride(0), x.stride(0), M, N, grid=grid)
        return y


softmax = _softmax.apply

We can use the above softmax function to compute the row-wise softmax of a given matrix.

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))

Out:

True

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=[256 * i for i in range(2, 50)],  # different possible values for `x_name`
        y_name='provider',  # argument name whose value corresponds to a different line in the plot
        y_vals=['torch', 'triton', 'naive'],  # possible keys for `y_name`
        y_lines=["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 torch.softmax for very large input matrices. My guess from looking at the source-code of the PyTorch kernel 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.