215 lines
8.4 KiB
Python
215 lines
8.4 KiB
Python
"""
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Fused Softmax
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=================
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In this tutorial, you will write a fused softmax operation (that outperforms PyTorch) and learn about:
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- The benefits of kernel fusion for bandwidth-bound operations.
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- The syntax and usage of reduction operators in Triton.
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- The automatic vectorization capabilities of the Triton compiler.
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"""
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# %%
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# Motivations
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# ------------
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# Custom GPU kernels for elementwise additions are educationally valuable but won't get you very far in practice.
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# Let us consider instead the case of a simple (numerically stabilized) softmax operation:
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import torch
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# Compute the row-wise softmax of x
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def naive_softmax(x):
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# read MN elements ; write M elements
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x_max = torch.max(x, axis=1)[0]
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# read 2MN elements ; write MN elements
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z = x - x_max[:, None]
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# read MN elements ; write MN elements
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numerator = torch.exp(x)
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# read MN elements ; write M elements
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denominator = torch.sum(numerator, axis=1)
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# read 2MN elements ; write MN elements
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ret = numerator / denominator[:, None]
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# in total: read 7MN elements ; wrote 3MN + 2M elements
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return ret
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# %%
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# 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.
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# 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.
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# In this case, we would be reading and writing back only :math:`MN` bytes, so we could expect a theoretical speed-up of ~5x (i.e., :math:`(10MN + 2M) / 2MN`).
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# In practice, though, we would be getting a bit less as our kernel computes exponentials and internally moves data around in shared memory.
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# %%
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# Compute Kernel
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# ----------------
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# 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.
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# Note that one important limitation of Triton is that each block must have a power-of-two number of elements,
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# so we need to internally "pad" tiles and guard the memory operations properly if we want to handle any possible input shapes:
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#
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# .. code-block:: C
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#
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# __global__ void softmax(float* Y, float* X, int stride_xm, int stride_ym, int M, int N){
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# // row index
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# int m = get_program_id(0);
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# // column indices
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# int n [BLOCK] = 0 ... BLOCK;
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# // the memory address of all the elements
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# // that we want to load can be computed as follows
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# float* px [BLOCK] = X + m*stride_xm + n;
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# // because BLOCK has to be a power of two
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# // (per Triton-C specs), it is important
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# // to guard each memory operation with predicates
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# // or we will read out of bounds
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# bool check[BLOCK] = n < N;
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# float x [BLOCK] = check ? *px : -F32_INFINITY;
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# // syntax for reduction in Triton is:
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# // x[:, :, OPERATOR, :, :]
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# // ^
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# // index
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# // where operator is in {min, max, +}
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# // for 1D vectors, this is just x[OPERATOR].
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# float z [BLOCK] = x - x[max];
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# // Note that exponentials in Triton are fast
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# // but approximate (i.e., think __expf in CUDA)
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# float num [BLOCK] = exp(z);
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# float denom = num[+];
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# // The result of the reduction is now stored in y
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# float y [BLOCK] = num / denom;
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# // We write it back
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# float* py [BLOCK] = Y + m*stride_ym + n;
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# *?(check)py = y;
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# }
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# %%
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# Torch Bindings
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# ---------------
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# Here our torch bindings is quite similar to that of the vector addition mentioned in the previous tutorial.
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# 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.
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# This means that different values of BLOCK will result in different kernels
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import torch
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import triton
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# Source code for the Triton kernel
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_src = """
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__global__ void softmax(float* Y, float* X, int stride_ym, int stride_xm, int M, int N){
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int m = get_program_id(0);
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int n [BLOCK] = 0 ... BLOCK;
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float* px [BLOCK] = X + m*stride_xm + n;
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bool check[BLOCK] = n < N;
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float x [BLOCK] = check ? *px : -F32_INFINITY;
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float z [BLOCK] = x - x[max];
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float num [BLOCK] = exp(z);
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float denom = num[+];
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float y [BLOCK] = num / denom;
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float* py [BLOCK] = Y + m*stride_ym + n;
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*?(check)py = y;
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}
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"""
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# helper function to get the smaller power-of-two larger than a given number
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def next_power_of_2(n):
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n -= 1
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n |= n >> 1
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n |= n >> 2
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n |= n >> 4
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n |= n >> 8
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n |= n >> 16
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n += 1
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return n
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# kernel caching mechanism
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def make_kernel(N, device):
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cache = make_kernel.cache
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# Now are kernels are indexed not only by the provided device but also
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# by the rounded number of columns in the input matrix
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BLOCK = next_power_of_2(N)
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key = (BLOCK, device)
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if key not in cache:
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defines = {'BLOCK': BLOCK}
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cache[key] = triton.kernel(_src, device=device, defines=defines)
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return cache[key]
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make_kernel.cache = dict()
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class _softmax(torch.autograd.Function):
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@staticmethod
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def forward(ctx, x):
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# constraints of the op
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assert x.dtype == torch.float32
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y = torch.empty_like(x)
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# The launch grid is simple: we have one kernel instance per row of the input matrix
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M, N = y.shape
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grid = lambda opt: (M, )
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# Launch kernel
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kernel = make_kernel(N, y.device)
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kernel(y.data_ptr(), x.data_ptr(), y.stride(0), x.stride(0), M, N, grid=grid)
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return y
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softmax = _softmax.apply
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# %%
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# We can use the above softmax function to compute the row-wise softmax of a given matrix.
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# %%
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# Unit Test
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# ----------
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# %%
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# We make sure that we test our kernel on a matrix with an irregular number of rows and columns.
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# This will allow us to verify that our padding mechanism works.
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torch.manual_seed(0)
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x = torch.randn(1823, 781, device='cuda')
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y_tri = softmax(x)
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y_ref = torch.softmax(x, axis=1)
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print(torch.allclose(y_tri, y_ref))
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#%%
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# As expected, the results are identical.
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# %%
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# Benchmarking
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# -------------
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# Here we will benchmark our operation as a function of the number of columns in the input matrix -- assuming 4096 rows.
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# We will then compare its performance against (1) :code:`torch.softmax` and (2) the :code:`naive_softmax` defined above.
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@triton.testing.perf_report(
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triton.testing.Benchmark(
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x_names=['N'], # argument names to use as an x-axis for the plot
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x_vals=[256 * i for i in range(2, 50)], # different possible values for `x_name`
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y_name='provider', # argument name whose value corresponds to a different line in the plot
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y_vals=['torch', 'triton', 'naive'], # possible keys for `y_name`
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y_lines=["Torch", "Triton", 'Naive'], # label name for the lines
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ylabel="GB/s", # label name for the y-axis
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plot_name="softmax-performance", # name for the plot. Used also as a file name for saving the plot.
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args={'M': 4096} # values for function arguments not in `x_names` and `y_name`
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)
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)
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def benchmark(M, N, provider):
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x = torch.randn(M, N, device='cuda', dtype=torch.float32)
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if provider == 'torch':
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ms, min_ms, max_ms = triton.testing.do_bench(lambda: torch.softmax(x, axis=-1))
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if provider == 'triton':
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ms, min_ms, max_ms = triton.testing.do_bench(lambda: softmax(x))
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if provider == 'naive':
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ms, min_ms, max_ms = triton.testing.do_bench(lambda: naive_softmax(x))
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gbps = lambda ms: 2 * x.nelement() * x.element_size() * 1e-9 / (ms * 1e-3)
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return gbps(ms), gbps(max_ms), gbps(min_ms)
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benchmark.run(show_plots=True)
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# %%
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# In the above plot, we can see that:
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#
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# - Triton is 4-5x faster than the naive implementation, which is consistent with our theoretical predictions.
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# - 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.
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# 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.
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# Note that our Triton kernel is not only faster than PyTorch's CUDA kernel, it is also **easier to read, understand and maintain**. |