""" 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 @triton.jit def _softmax(Y, X, stride_xm, stride_ym, M, N, **meta): # row index m = triton.program_id(0) # col indices n = triton.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 = triton.load(X, mask=n < N, other=-float('inf')) # Substract maximum for numerical stability z = x - triton.max(x, axis=0) # Note that exponentials in Triton are fast # but approximate (i.e., think __expf in CUDA) num = triton.exp(z) denom = triton.sum(num, axis=0) y = num / denom # Write back to Y Y = Y + m * stride_ym + n triton.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` 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 :code:`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**.