.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "getting-started/tutorials/03-matrix-multiplication.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here ` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_getting-started_tutorials_03-matrix-multiplication.py: Matrix Multiplication ====================== In this tutorial, you will write a 25-lines high-performance FP16 matrix multiplication kernel that achieves performance on par with cuBLAS. You will specifically learn about: - Block-level matrix multiplications - Multi-dimensional pointer arithmetic - Program re-ordering for improved L2 cache hit rate - Automatic performance tuning .. GENERATED FROM PYTHON SOURCE LINES 15-42 Motivations ------------- Matrix multiplications are a key building block of most modern high-performance computing systems. They are notoriously hard to optimize, hence their implementation is generally done by hardware vendors themselves as part of so-called "kernel libraries" (e.g., cuBLAS). Unfortunately, these libraries are often proprietary and cannot be easily customized to accomodate the needs of modern deep learning workloads (e.g., fused activation functions). In this tutorial, you will learn how to implement efficient matrix multiplications by yourself with Triton, in a way that is easy to customize and extend. Roughly speaking, the kernel that we will write will implement the following blocked algorithm to multiply a (M, K) by a (K, N) matrix: .. code-block:: python # do in parallel for m in range(0, M, BLOCK_SIZE_M): # do in parallel for n in range(0, N, BLOCK_SIZE_N): acc = zeros((BLOCK_SIZE_M, BLOCK_SIZE_N), dtype=float32) for k in range(0, K, BLOCK_SIZE_K): a = A[m : m+BLOCK_SIZE_M, k : k+BLOCK_SIZE_K] b = B[k : k+BLOCK_SIZE_K, n : n+BLOCK_SIZE_N] acc += dot(a, b) C[m : m+BLOCK_SIZE_M, n : n+BLOCK_SIZE_N] = acc; where each iteration of the doubly-nested for-loop is performed by a dedicated Triton program instance. .. GENERATED FROM PYTHON SOURCE LINES 44-137 Compute Kernel ---------------- The above algorithm is, actually, fairly straightforward to implement in Triton. The main difficulty comes from the computation of the memory locations at which blocks of :code:`A` and :code:`B` must be read in the inner loop. For that, we need multi-dimensional pointer arithmetics. Pointer Arithmetics ~~~~~~~~~~~~~~~~~~~~ For a row-major 2D tensor :code:`X`, the memory location of :code:`X[i, j]` is given b y :code:`&X[i, j] = X + i*stride_xi + j*stride_xj`. Therefore, blocks of pointers for :code:`A[m : m+BLOCK_SIZE_M, k:k+BLOCK_SIZE_K]` and :code:`B[k : k+BLOCK_SIZE_K, n : n+BLOCK_SIZE_N]` can be defined in pseudo-code as: .. code-block:: python &A[m : m+BLOCK_SIZE_M, k:k+BLOCK_SIZE_K] = a_ptr + (m : m+BLOCK_SIZE_M)[:, None]*A.stride(0) + (k : k+BLOCK_SIZE_K)[None, :]*A.stride(1); &B[k : k+BLOCK_SIZE_K, n:n+BLOCK_SIZE_N] = b_ptr + (k : k+BLOCK_SIZE_K)[:, None]*B.stride(0) + (n : n+BLOCK_SIZE_N)[None, :]*B.stride(1); Which means that pointers for blocks of A and B can be initialized (i.e., :code:`k=0`) in Triton as: .. code-block:: python offs_am = pid_m * BLOCK_SIZE_M + tl.arange(0, BLOCK_SIZE_M) offs_bn = pid_n * BLOCK_SIZE_N + tl.arange(0, BLOCK_SIZE_N) offs_k = tl.arange(0, BLOCK_SIZE_K) a_ptrs = a_ptr + (offs_am[:, None]*stride_am + offs_k [None, :]*stride_ak) b_ptrs = b_ptr + (offs_k [:, None]*stride_bk + offs_bn[None, :]*stride_bn) And then updated in the inner loop as follows: .. code-block:: python pa += BLOCK_SIZE_K * stride_ak; pb += BLOCK_SIZE_K * stride_bk; L2 Cache Optimizations ~~~~~~~~~~~~~~~~~~~~~~~~ As mentioned above, each program instance computes a :code:`[BLOCK_SIZE_M, BLOCK_SIZE_N]` block of :code:`C`. It is important to remember that the order in which these blocks are computed does matter, since it affects the L2 cache hit rate of our program. and unfortunately, a a simple row-major ordering .. code-block:: Python pid = triton.program_id(0); grid_m = (M + BLOCK_SIZE_M - 1) // BLOCK_SIZE_M; grid_n = (N + BLOCK_SIZE_N - 1) // BLOCK_SIZE_N; pid_m = pid / grid_n; pid_n = pid % grid_n; is just not going to cut it. One possible solution is to launch blocks in an order that promotes data reuse. This can be done by 'super-grouping' blocks in groups of :code:`GROUP_M` rows before switching to the next column: .. code-block:: python # program ID pid = tl.program_id(axis=0) # number of program ids along the M axis num_pid_m = tl.cdiv(M, BLOCK_SIZE_M) # number of programs ids along the N axis num_pid_n = tl.cdiv(N, BLOCK_SIZE_N) # number of programs in group num_pid_in_group = GROUP_SIZE_M * num_pid_n # id of the group this program is in group_id = pid // num_pid_in_group # row-id of the first program in the group first_pid_m = group_id * GROUP_SIZE_M # if `num_pid_m` isn't divisible by `GROUP_SIZE_M`, the last group is smaller group_size_m = min(num_pid_m - first_pid_m, GROUP_SIZE_M) # *within groups*, programs are ordered in a column-major order # row-id of the program in the *launch grid* pid_m = first_pid_m + (pid % group_size_m) # col-id of the program in the *launch grid* pid_n = (pid % num_pid_in_group) // group_size_m For example, in the following matmul where each matrix is 9 blocks by 9 blocks, we can see that if we compute the output in row-major ordering, we need to load 90 blocks into SRAM to compute the first 9 output blocks, but if we do it in grouped ordering, we only need to load 54 blocks. .. image:: grouped_vs_row_major_ordering.png In practice, this can improve the performance of our matrix multiplication kernel by more than 10\% on some hardware architecture (e.g., 220 to 245 TFLOPS on A100). .. GENERATED FROM PYTHON SOURCE LINES 139-142 Final Result ------------- .. GENERATED FROM PYTHON SOURCE LINES 142-262 .. code-block:: default import torch import triton import triton.language as tl # % # :code:`triton.jit`'ed functions can be auto-tuned by using the `triton.autotune` # decorator, which consumes: # - A list of :code:`triton.Config` objects that define different configurations of # meta-parameters (e.g., BLOCK_SIZE_M) and compilation options (e.g., num_warps) to try # - An autotuning *key* whose change in values will trigger evaluation of all the # provided configs @triton.autotune( configs=[ triton.Config({'BLOCK_SIZE_M': 128, 'BLOCK_SIZE_N': 256, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=3, num_warps=8), triton.Config({'BLOCK_SIZE_M': 256, 'BLOCK_SIZE_N': 128, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=3, num_warps=8), triton.Config({'BLOCK_SIZE_M': 256, 'BLOCK_SIZE_N': 64, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4, num_warps=4), triton.Config({'BLOCK_SIZE_M': 64 , 'BLOCK_SIZE_N': 256, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4, num_warps=4), triton.Config({'BLOCK_SIZE_M': 128, 'BLOCK_SIZE_N': 128, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4, num_warps=4), triton.Config({'BLOCK_SIZE_M': 128, 'BLOCK_SIZE_N': 64 , 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4, num_warps=4), triton.Config({'BLOCK_SIZE_M': 64 , 'BLOCK_SIZE_N': 128, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4, num_warps=4), triton.Config({'BLOCK_SIZE_M': 128, 'BLOCK_SIZE_N': 32 , 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4, num_warps=4), triton.Config({'BLOCK_SIZE_M': 64 , 'BLOCK_SIZE_N': 32 , 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=5, num_warps=2), triton.Config({'BLOCK_SIZE_M': 32 , 'BLOCK_SIZE_N': 64 , 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=5, num_warps=2), ], key=['M', 'N', 'K'], ) # % # We can now define our kernel as normal, using all the techniques presented above @triton.jit def matmul_kernel( # Pointers to matrices a_ptr, b_ptr, c_ptr, # Matrix dimensions M, N, K, # The stride variables represent how much to increase the ptr by when moving by 1 # element in a particular dimension. E.g. stride_am is how much to increase a_ptr # by to get the element one row down (A has M rows) stride_am, stride_ak, stride_bk, stride_bn, stride_cm, stride_cn, # Meta-parameters **meta, ): """Kernel for computing the matmul C = A x B. A has shape (M, K), B has shape (K, N) and C has shape (M, N) """ # extract meta-parameters BLOCK_SIZE_M = meta['BLOCK_SIZE_M'] BLOCK_SIZE_N = meta['BLOCK_SIZE_N'] BLOCK_SIZE_K = meta['BLOCK_SIZE_K'] GROUP_SIZE_M = 8 # ----------------------------------------------------------- # Map program ids `pid` to the block of C it should compute. # This is done in a grouped ordering to promote L2 data reuse # See above `L2 Cache Optimizations` section for details pid = tl.program_id(axis=0) num_pid_m = tl.cdiv(M, BLOCK_SIZE_M) num_pid_n = tl.cdiv(N, BLOCK_SIZE_N) num_pid_in_group = GROUP_SIZE_M * num_pid_n group_id = pid // num_pid_in_group first_pid_m = group_id * GROUP_SIZE_M group_size_m = min(num_pid_m - first_pid_m, GROUP_SIZE_M) pid_m = first_pid_m + (pid % group_size_m) pid_n = (pid % num_pid_in_group) // group_size_m # ---------------------------------------------------------- # Create pointers for the first blocks of A and B. # We will advance this pointer as we move in the K direction # and accumulate # a_ptrs is a block of [BLOCK_SIZE_M, BLOCK_SIZE_K] pointers # b_ptrs is a block of [BLOCK_SIZE_K, BLOCK_SIZE_n] pointers # see above `Pointer Arithmetics` section for details offs_am = pid_m * BLOCK_SIZE_M + tl.arange(0, BLOCK_SIZE_M) offs_bn = pid_n * BLOCK_SIZE_N + tl.arange(0, BLOCK_SIZE_N) offs_k = tl.arange(0, BLOCK_SIZE_K) a_ptrs = a_ptr + (offs_am[:, None]*stride_am + offs_k [None, :]*stride_ak) b_ptrs = b_ptr + (offs_k [:, None]*stride_bk + offs_bn[None, :]*stride_bn) # ----------------------------------------------------------- # Iterate to compute a block of the C matrix # We accumulate into a `[BLOCK_SIZE_M, BLOCK_SIZE_N]` block # of fp32 values for higher accuracy. # `accumulator` will be converted back to fp16 after the loop accumulator = tl.zeros((BLOCK_SIZE_M, BLOCK_SIZE_N), dtype=tl.float32) for k in range(0, K, BLOCK_SIZE_K): # Note that for simplicity, we don't apply a mask here. # This means that if K is not a multiple of BLOCK_SIZE_K, # this will access out-of-bounds memory and produce an # error or (worse!) incorrect results. a = tl.load(a_ptrs) b = tl.load(b_ptrs) # We accumulate along the K dimension accumulator += tl.dot(a, b) # Advance the ptrs to the next K block a_ptrs += BLOCK_SIZE_K * stride_ak b_ptrs += BLOCK_SIZE_K * stride_bk # you can fuse arbitrary activation functions here # while the accumulator is still in FP32 ! if meta['ACTIVATION']: accumulator = meta['ACTIVATION'](accumulator) c = accumulator.to(tl.float16) # ----------------------------------------------------------- # Write back the block of the output matrix C offs_cm = pid_m * BLOCK_SIZE_M + tl.arange(0, BLOCK_SIZE_M) offs_cn = pid_n * BLOCK_SIZE_N + tl.arange(0, BLOCK_SIZE_N) c_ptrs = c_ptr + stride_cm * offs_cm[:, None] + stride_cn * offs_cn[None, :] c_mask = (offs_cm[:, None] < M) & (offs_cn[None, :] < N) tl.store(c_ptrs, c, mask=c_mask) # we can fuse `leaky_relu` by providing it as an `ACTIVATION` meta-parameter in `_matmul` @triton.jit def leaky_relu(x): return tl.where(x >= 0, x, 0.01 * x) .. GENERATED FROM PYTHON SOURCE LINES 263-265 We can now create a convenience wrapper function that only takes two input tensors and (1) checks any shape constraint; (2) allocates the output; (3) launches the above kernel .. GENERATED FROM PYTHON SOURCE LINES 265-294 .. code-block:: default def matmul(a, b, activation=None): # checks constraints assert a.shape[1] == b.shape[0], "incompatible dimensions" assert a.is_contiguous(), "matrix A must be contiguous" assert b.is_contiguous(), "matrix B must be contiguous" M, K = a.shape K, N = b.shape assert ( K % 32 == 0 ), "We don't check memory-out-of-bounds with K so K must be divisible by BLOCK_SIZE_K" # allocates output c = torch.empty((M, N), device=a.device, dtype=a.dtype) # 1D launch kernel where each block gets its own program. grid = lambda META: ( triton.cdiv(M, META['BLOCK_SIZE_M']) * triton.cdiv(N, META['BLOCK_SIZE_N']), ) matmul_kernel[grid]( a, b, c, M, N, K, a.stride(0), a.stride(1), b.stride(0), b.stride(1), c.stride(0), c.stride(1), ACTIVATION=activation, ) return c .. GENERATED FROM PYTHON SOURCE LINES 295-299 Unit Test ----------- We can test our custom matrix multiplication operation against a native torch implementation (i.e., cuBLAS) .. GENERATED FROM PYTHON SOURCE LINES 299-312 .. code-block:: default torch.manual_seed(0) a = torch.randn((512, 512), device='cuda', dtype=torch.float16) b = torch.randn((512, 512), device='cuda', dtype=torch.float16) triton_output = matmul(a, b, activation=None) torch_output = torch.matmul(a, b) print(f"triton_output={triton_output}") print(f"torch_output={torch_output}") if triton.testing.allclose(triton_output, torch_output): print("✅ Triton and Torch match") else: print("❌ Triton and Torch differ") .. rst-class:: sphx-glr-script-out Out: .. code-block:: none triton_output=tensor([[ 1.1045, -36.9688, 31.4688, ..., -11.3984, 24.4531, -32.3438], [ 6.3555, -19.6094, 34.0938, ..., -5.8945, 5.2891, 6.8867], [-32.0625, 5.9492, 15.3984, ..., -21.3906, -23.9844, -10.1328], ..., [ -5.7031, 7.4492, 8.2656, ..., -10.6953, -40.0000, 17.7500], [ 25.5000, 24.3281, -8.4688, ..., -18.9375, 32.5312, -29.9219], [ -5.3477, 4.9844, 11.8906, ..., 5.5898, 6.4023, -17.3125]], device='cuda:0', dtype=torch.float16) torch_output=tensor([[ 1.1045, -36.9688, 31.4688, ..., -11.3906, 24.4531, -32.3438], [ 6.3516, -19.6094, 34.0938, ..., -5.8906, 5.2812, 6.8828], [-32.0625, 5.9531, 15.3984, ..., -21.4062, -23.9844, -10.1328], ..., [ -5.7070, 7.4492, 8.2656, ..., -10.6953, -40.0000, 17.7500], [ 25.5000, 24.3438, -8.4609, ..., -18.9375, 32.5312, -29.9219], [ -5.3477, 4.9805, 11.8828, ..., 5.5859, 6.4023, -17.3125]], device='cuda:0', dtype=torch.float16) ✅ Triton and Torch match .. GENERATED FROM PYTHON SOURCE LINES 313-319 Benchmark -------------- Square Matrix Performance ~~~~~~~~~~~~~~~~~~~~~~~~~~ We can now compare the performance of our kernel against that of cuBLAS. Here we focus on square matrices, but feel free to arrange this script as you wish to benchmark any other matrix shape. .. GENERATED FROM PYTHON SOURCE LINES 319-360 .. code-block:: default @triton.testing.perf_report( triton.testing.Benchmark( x_names=['M', 'N', 'K'], # argument names to use as an x-axis for the plot x_vals=[ 128 * i for i in range(2, 33) ], # different possible values for `x_name` line_arg='provider', # argument name whose value corresponds to a different line in the plot # possible values for `line_arg`` line_vals=['cublas', 'cublas + relu', 'triton', 'triton + relu'], # label name for the lines line_names=["cuBLAS", "cuBLAS (+ torch.nn.LeakyReLU)", "Triton", "Triton (+ LeakyReLU)"], # line styles styles=[('green', '-'), ('green', '--'), ('blue', '-'), ('blue', '--')], ylabel="TFLOPS", # label name for the y-axis plot_name="matmul-performance", # name for the plot. Used also as a file name for saving the plot. args={}, ) ) def benchmark(M, N, K, provider): a = torch.randn((M, K), device='cuda', dtype=torch.float16) b = torch.randn((K, N), device='cuda', dtype=torch.float16) if provider == 'cublas': ms, min_ms, max_ms = triton.testing.do_bench(lambda: torch.matmul(a, b)) if provider == 'triton': ms, min_ms, max_ms = triton.testing.do_bench(lambda: matmul(a, b)) if provider == 'cublas + relu': torch_relu = torch.nn.ReLU(inplace=True) ms, min_ms, max_ms = triton.testing.do_bench( lambda: torch_relu(torch.matmul(a, b)) ) if provider == 'triton + relu': ms, min_ms, max_ms = triton.testing.do_bench( lambda: matmul(a, b, activation=leaky_relu) ) perf = lambda ms: 2 * M * N * K * 1e-12 / (ms * 1e-3) return perf(ms), perf(max_ms), perf(min_ms) benchmark.run(show_plots=True, print_data=True) .. image:: /getting-started/tutorials/images/sphx_glr_03-matrix-multiplication_001.png :alt: 03 matrix multiplication :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out Out: .. code-block:: none matmul-performance: M cuBLAS ... Triton Triton (+ LeakyReLU) 0 256.0 2.730667 ... 2.978909 2.978909 1 384.0 7.372800 ... 8.507077 8.507077 2 512.0 14.563555 ... 16.384000 15.420235 3 640.0 22.260869 ... 24.380953 24.380953 4 768.0 32.768000 ... 34.028308 34.028308 5 896.0 39.025776 ... 39.025776 39.025776 6 1024.0 49.932191 ... 53.773130 52.428801 7 1152.0 45.242181 ... 47.396572 46.656000 8 1280.0 51.200001 ... 56.888887 56.888887 9 1408.0 64.138541 ... 67.305878 67.305878 10 1536.0 80.430545 ... 79.526831 79.526831 11 1664.0 63.372618 ... 62.492442 62.061463 12 1792.0 72.983276 ... 71.588687 71.588687 13 1920.0 68.776119 ... 70.530615 70.530615 14 2048.0 73.908442 ... 77.314362 76.959706 15 2176.0 83.500614 ... 85.998493 85.269692 16 2304.0 68.251065 ... 76.809875 76.563695 17 2432.0 71.305746 ... 74.719317 84.877538 18 2560.0 78.019048 ... 81.310171 81.108913 19 2688.0 83.552988 ... 89.995386 89.464755 20 2816.0 79.587973 ... 82.916747 82.602666 21 2944.0 82.102191 ... 82.646820 83.060049 22 3072.0 81.707223 ... 87.516392 88.060814 23 3200.0 80.402009 ... 87.551302 87.189747 24 3328.0 80.798314 ... 83.905938 84.596116 25 3456.0 82.688790 ... 91.252485 90.994998 26 3584.0 86.540320 ... 91.563533 95.350361 27 3712.0 82.491612 ... 86.867254 86.791782 28 3840.0 82.592983 ... 92.390975 86.400002 29 3968.0 93.254827 ... 85.093402 91.130650 30 4096.0 86.202781 ... 87.982773 89.210850 [31 rows x 5 columns] .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 5 minutes 26.818 seconds) .. _sphx_glr_download_getting-started_tutorials_03-matrix-multiplication.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: 03-matrix-multiplication.py <03-matrix-multiplication.py>` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: 03-matrix-multiplication.ipynb <03-matrix-multiplication.ipynb>` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_