.. 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 (MxK) by a (KxN) 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 corresponds to a Triton program instance. .. GENERATED FROM PYTHON SOURCE LINES 44-129 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_x_0 + j*stride_x_1`. 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 + (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 + (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 pid_m = triton.program_id(0) pid_n = triton.program_id(1) rm = pid_m * BLOCK_SIZE_M + triton.arange(0, BLOCK_SIZE_M) rn = pid_n * BLOCK_SIZE_N + triton.arange(0, BLOCK_SIZE_N) rk = triton.arange(0, BLOCK_SIZE_K) // pointer for A operand pa = A + (rm[:, None] * stride_a_0 + rk[None, :] * stride_a_1); // pointer for B operand pb = B + (rk[:, None] * stride_b_0 + rn[None, :] * stride_b_1); And then updated in the inner loop as follows: .. code-block:: python pa += BLOCK_SIZE_K * stride_a_1; pb += BLOCK_SIZE_K * stride_b_0; 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 pid = triton.program_id(0); width = GROUP_M * grid_n; group_id = pid // width; # we need to handle the case where M % (GROUP_M*BLOCK_SIZE_M) != 0 group_size = min(grid_m - group_id * GROUP_M, GROUP_M); pid_m = group_id * GROUP_M + (pid % group_size); pid_n = (pid % width) // (group_size); 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 131-134 Final Result ------------- .. GENERATED FROM PYTHON SOURCE LINES 134-263 .. 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, ): """Kernel for computing the matmul AB = C 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 pid = tl.program_id(axis=0) # the number of blocks is the ceil(M / BLOCK_SIZE_M) since we need an extra block # Note that this will lead to some quantization in performance where time-taken jumps # when you need to add a new block n_blocks_m = (M + BLOCK_SIZE_M - 1) // BLOCK_SIZE_M n_blocks_n = (N + BLOCK_SIZE_N - 1) // BLOCK_SIZE_N # Map PIDs to the block they should compute. This is done in a grouped ordering # to promote L2 cache reuse. n_output_blocks_in_group = GROUP_SIZE_M * n_blocks_n group_id = pid // n_output_blocks_in_group first_m_block_in_group = group_id * GROUP_SIZE_M # If the number of blocks is not divisible by the group size, the last group is smaller group_size_m = min(n_blocks_m - first_m_block_in_group, GROUP_SIZE_M) # Within a group, we compute in col-major ordering, block_m and block_n are the # output row and col that this program is computing in terms of blocks block_m = first_m_block_in_group + (pid % group_size_m) block_n = (pid % n_output_blocks_in_group) // group_size_m # Convert from block indices back to element indices m_start = block_m * BLOCK_SIZE_M n_start = block_n * BLOCK_SIZE_N # Expand out to all the offsets for each of the elements in this block. m_offsets_a = (m_start + tl.arange(0, BLOCK_SIZE_M))[:, None] n_offsets_b = (n_start + tl.arange(0, BLOCK_SIZE_N))[None, :] k_offsets = tl.arange(0, BLOCK_SIZE_K) # Get the pointers for the first block of each. We will advance this pointer # as we move in the K direction and accumulate. # a_ptrs should contain BLOCK_SIZE_M * BLOCK_SIZE_K pointers a_ptrs = a_ptr + (stride_am * m_offsets_a + stride_ak * k_offsets[None, :]) # b_ptrs should contain BLOCK_SIZE_K * BLOCK_SIZE_N pointers b_ptrs = b_ptr + (stride_bk * k_offsets[:, None] + stride_bn * n_offsets_b) # We accumulate internally in fp32, but the output is written out in the dtype # of the tensor when it is stored 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 # accumulate it incorrectly. 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 # triton can accept arbitrary activation function via metaparameters! if meta['ACTIVATION']: accumulator = meta['ACTIVATION'](accumulator) m_offsets_c = (m_start + tl.arange(0, BLOCK_SIZE_M))[:, None] n_offsets_c = (n_start + tl.arange(0, BLOCK_SIZE_N))[None, :] c_ptrs = c_ptr + stride_cm * m_offsets_c + stride_cn * n_offsets_c mask = (m_offsets_c < M) & (n_offsets_c < N) tl.store(c_ptrs, accumulator, mask=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 264-266 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 266-302 .. 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 303-307 Unit Test ----------- We can test our custom matrix multiplication operation against a native torch implementation (i.e., cuBLAS) .. GENERATED FROM PYTHON SOURCE LINES 307-320 .. 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 321-327 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 327-368 .. 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(1, 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) 0 128.0 0.455111 ... 0.512000 0.512000 1 256.0 2.730667 ... 3.276800 2.978909 2 384.0 7.372800 ... 7.899428 7.899428 3 512.0 14.563555 ... 16.384000 16.384000 4 640.0 22.260869 ... 24.380953 24.380953 5 768.0 32.768000 ... 34.028308 34.028308 6 896.0 39.025776 ... 40.140799 35.150663 7 1024.0 49.932191 ... 52.428801 52.428801 8 1152.0 44.566925 ... 46.656000 46.656000 9 1280.0 51.200001 ... 56.109587 56.109587 10 1408.0 64.138541 ... 64.902096 64.138541 11 1536.0 80.430545 ... 76.106321 75.296679 12 1664.0 62.929456 ... 62.061463 62.061463 13 1792.0 72.983276 ... 69.810085 69.379162 14 1920.0 69.120002 ... 70.892307 69.120002 15 2048.0 73.584279 ... 74.898285 74.565406 16 2176.0 83.155572 ... 80.817862 79.855747 17 2304.0 68.446623 ... 72.828879 73.275679 18 2432.0 71.305746 ... 82.388456 81.908060 19 2560.0 78.019048 ... 77.283019 75.676673 20 2688.0 83.552988 ... 83.552988 83.922689 21 2816.0 81.827785 ... 77.330158 79.154642 22 2944.0 81.166173 ... 77.747321 79.483304 23 3072.0 79.863336 ... 82.661468 82.420822 24 3200.0 83.660130 ... 90.395483 85.906037 25 3328.0 83.226931 ... 87.368079 83.613586 26 3456.0 80.220468 ... 81.600781 83.459178 27 3584.0 87.466332 ... 92.887804 84.983685 28 3712.0 84.159518 ... 83.178475 83.666116 29 3840.0 83.591840 ... 84.228485 85.663823 30 3968.0 91.885495 ... 84.680037 84.154440 31 4096.0 89.181212 ... 90.260743 90.200084 [32 rows x 5 columns] .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 2 minutes 29.710 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 `_