164 lines
6.6 KiB
Python
164 lines
6.6 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 reduction operators in Triton.
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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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# 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`).
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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 matrix 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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import triton
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import triton.language as tl
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@triton.jit
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def _softmax(Y, X, stride_xm, stride_ym, M, N, **meta):
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# row index
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m = tl.program_id(0)
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# col indices
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n = tl.arange(0, meta['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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X = X + m * stride_xm + n
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x = tl.load(X, mask=n < N, other=-float('inf'))
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# Substract maximum for numerical stability
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z = x - tl.max(x, axis=0)
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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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num = tl.exp(z)
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denom = tl.sum(num, axis=0)
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y = num / denom
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# Write back to Y
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Y = Y + m * stride_ym + n
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tl.store(Y, y, mask=n < N)
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# %%
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# We can create a helper function that enqueues the kernel and its (meta-)arguments for any given input tensor.
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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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def softmax(x):
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M, N = x.shape
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# The block size is the smallest power of two greater than the number of columns in `x`
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BLOCK = next_power_of_2(N)
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# Another trick we can use is to ask the compiler to parallelize each
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# row-normalization more aggressively -- i.e., with more warps -- vectors
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# that are longer
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# You will see in the next tutorial how to auto-tune this value in a more natural
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# way so you don't have to come up with manual heuristics yourself
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num_warps = 4
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if BLOCK >= 2048: num_warps = 8
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if BLOCK >= 4096: num_warps = 16
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# Allocate output
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y = torch.empty_like(x)
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# Enqueue kernel. The launch grid is simple: we have one kernel instance per row of the input matrix
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_softmax[(M, )](y, x, x.stride(0), y.stride(0), M, N, num_warps=num_warps, BLOCK=BLOCK)
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return y
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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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# Benchmark
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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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line_arg='provider', # argument name whose value corresponds to a different line in the plot
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line_vals=['torch', 'triton', 'naive'], # possible values for `line_arg``
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line_names=["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**. |