443 lines
17 KiB
ReStructuredText
443 lines
17 KiB
ReStructuredText
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.. DO NOT EDIT.
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.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
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.. "getting-started/tutorials/03-matrix-multiplication.py"
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.. LINE NUMBERS ARE GIVEN BELOW.
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.. only:: html
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.. note::
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:class: sphx-glr-download-link-note
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Click :ref:`here <sphx_glr_download_getting-started_tutorials_03-matrix-multiplication.py>`
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to download the full example code
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.. rst-class:: sphx-glr-example-title
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.. _sphx_glr_getting-started_tutorials_03-matrix-multiplication.py:
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Matrix Multiplication
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======================
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In this tutorial, you will write a 25-lines high-performance FP16 matrix multiplication kernel that achieves performance on par with cuBLAS.
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You will specifically learn about:
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- Block-level matrix multiplications
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- Multi-dimensional pointer arithmetic
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- Program re-ordering for improved L2 cache hit rate
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- Automatic performance tuning
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.. GENERATED FROM PYTHON SOURCE LINES 14-37
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Motivations
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-------------
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Matrix multiplications are a key building block of most modern high-performance computing systems.
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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).
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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).
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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.
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Roughly speaking, the kernel that we will write will implement the following blocked algorithm:
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.. code-block:: python
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# do in parallel
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for m in range(0, M, BLOCK_M):
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# do in parallel
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for n in range(0, N, BLOCK_N):
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acc = zeros((BLOCK_M, BLOCK_N), dtype=float32)
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for k in range(0, K, BLOCK_K):
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a = A[m : m+BLOCK_M, k : k+BLOCK_K]
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b = B[k : k+BLOCK_K, n : n+BLOCK_N]
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acc += dot(a, b)
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C[m : m+BLOCK_M, n : n+BLOCK_N] = acc;
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where each iteration of the doubly-nested for-loop corresponds to a Triton program instance.
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.. GENERATED FROM PYTHON SOURCE LINES 39-110
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Compute Kernel
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----------------
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The above algorithm is, actually, fairly straightforward to implement in Triton.
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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.
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Pointer Arithmetics
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~~~~~~~~~~~~~~~~~~~~
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For a row-major 2D tensor :code:`X`, the memory location of :code:`X[i, j]` is given by :code:`&X[i, j] = X + i*stride_x_0 + j*stride_x_1`.
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Therefore, blocks of pointers for :code:`A[m : m+BLOCK_M, k:k+BLOCK_K]` and :code:`B[k : k+BLOCK_K, n : n+BLOCK_N]` can be defined in pseudo-code as:
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.. code-block:: python
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&A[m : m+BLOCK_M, k:k+BLOCK_K] = A + (m : m+BLOCK_M)[:, None]*A.stride(0) + (k : k+BLOCK_K)[None, :]*A.stride(1);
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&B[k : k+BLOCK_K, n:n+BLOCK_N] = B + (k : k+BLOCK_K)[:, None]*B.stride(0) + (n : n+BLOCK_N)[None, :]*B.stride(1);
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Which means that pointers for blocks of A and B can be initialized (i.e., :code:`k=0`) in Triton as:
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.. code-block:: python
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pid_m = triton.program_id(0)
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pid_n = triton.program_id(1)
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rm = pid_m * BLOCK_M + triton.arange(0, BLOCK_M)
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rn = pid_n * BLOCK_N + triton.arange(0, BLOCK_N)
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rk = triton.arange(0, BLOCK_K)
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// pointer for A operand
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pa = A + (rm[:, None] * stride_a_0 + rk[None, :] * stride_a_1);
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// pointer for B operand
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pb = B + (rk[:, None] * stride_b_0 + rn[None, :] * stride_b_1);
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And then updated in the inner loop as follows:
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.. code-block:: python
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pa += BLOCK_K * stride_a_1;
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pb += BLOCK_K * stride_b_0;
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L2 Cache Optimizations
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~~~~~~~~~~~~~~~~~~~~~~~~
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As mentioned above, each program instance computes an :code:`[BLOCK_M, BLOCK_N]` block of :code:`C`.
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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.
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And unfortunately, a simple row-major ordering
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.. code-block:: Python
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pid = triton.program_id(0);
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grid_m = (M + BLOCK_M - 1) // BLOCK_M;
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grid_n = (N + BLOCK_N - 1) // BLOCK_N;
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pid_m = pid / grid_n;
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pid_n = pid % grid_n;
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is just not going to cut it.
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One possible solution is to launch blocks in an order that promotes data reuse.
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This can be done by 'super-grouping' blocks in groups of :code:`GROUP_M` rows before switching to the next column:
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.. code-block:: python
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pid = triton.program_id(0);
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width = GROUP_M * grid_n;
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group_id = pid // width;
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# we need to handle the case where M % (GROUP_M*BLOCK_M) != 0
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group_size = min(grid_m - group_id * GROUP_M, GROUP_M);
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pid_m = group_id * GROUP_M + (pid % group_size);
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pid_n = (pid % width) // (group_size);
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In practice, this can improve the performance of our matrix multiplication kernel by >10\% on some hardware architecture (e.g., 220 to 245 TFLOPS on A100).
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.. GENERATED FROM PYTHON SOURCE LINES 112-115
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Final Result
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-------------
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.. GENERATED FROM PYTHON SOURCE LINES 115-190
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.. code-block:: default
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import torch
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import triton
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import triton.language as tl
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# %
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# :code:`triton.jit`'ed functions can be auto-tuned by using the `triton.autotune` decorator, which consumes:
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# - A list of :code:`triton.Config` objects that define different configurations of meta-parameters (e.g., BLOCK_M) and compilation options (e.g., num_warps) to try
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# - A autotuning *key* whose change in values will trigger evaluation of all the provided configs
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@triton.autotune(
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configs=[
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triton.Config({'BLOCK_M': 128, 'BLOCK_N': 256, 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=3, num_warps=8),
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triton.Config({'BLOCK_M': 256, 'BLOCK_N': 128, 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=3, num_warps=8),
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triton.Config({'BLOCK_M': 256, 'BLOCK_N': 64, 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=4, num_warps=4),
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triton.Config({'BLOCK_M': 64 , 'BLOCK_N': 256, 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=4, num_warps=4),\
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triton.Config({'BLOCK_M': 128, 'BLOCK_N': 128, 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=4, num_warps=4),\
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triton.Config({'BLOCK_M': 128, 'BLOCK_N': 64 , 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=4, num_warps=4),\
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triton.Config({'BLOCK_M': 64 , 'BLOCK_N': 128, 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=4, num_warps=4),
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triton.Config({'BLOCK_M': 128, 'BLOCK_N': 32 , 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=4, num_warps=4),\
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triton.Config({'BLOCK_M': 64 , 'BLOCK_N': 32 , 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=5, num_warps=2),\
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triton.Config({'BLOCK_M': 32 , 'BLOCK_N': 64 , 'BLOCK_K': 32, 'GROUP_M': 8}, num_stages=5, num_warps=2),
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#triton.Config({'BLOCK_M': 64, 'BLOCK_N': 128, 'BLOCK_K': 32, 'GROUP_M': 8}, num_warps=4),
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],
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key=['M', 'N', 'K'],
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)
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# %
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# We can now define our kernel as normal, using all the techniques presented above
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@triton.jit
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def _matmul(A, B, C, M, N, K, stride_am, stride_ak, stride_bk, stride_bn, stride_cm, stride_cn, **META):
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# extract meta-parameters
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BLOCK_M = META['BLOCK_M']
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BLOCK_N = META['BLOCK_N']
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BLOCK_K = META['BLOCK_K']
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GROUP_M = 8
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# matrix multiplication
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pid = tl.program_id(0)
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grid_m = (M + BLOCK_M - 1) // BLOCK_M
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grid_n = (N + BLOCK_N - 1) // BLOCK_N
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# re-order program ID for better L2 performance
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width = GROUP_M * grid_n
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group_id = pid // width
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group_size = min(grid_m - group_id * GROUP_M, GROUP_M)
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pid_m = group_id * GROUP_M + (pid % group_size)
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pid_n = (pid % width) // (group_size)
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# do matrix multiplication
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rm = pid_m * BLOCK_M + tl.arange(0, BLOCK_M)
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rn = pid_n * BLOCK_N + tl.arange(0, BLOCK_N)
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rk = tl.arange(0, BLOCK_K)
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A = A + (rm[:, None] * stride_am + rk[None, :] * stride_ak)
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B = B + (rk[:, None] * stride_bk + rn[None, :] * stride_bn)
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acc = tl.zeros((BLOCK_M, BLOCK_N), dtype=tl.float32)
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for k in range(K, 0, -BLOCK_K):
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a = tl.load(A)
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b = tl.load(B)
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acc += tl.dot(a, b)
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A += BLOCK_K * stride_ak
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B += BLOCK_K * stride_bk
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# triton can accept arbitrary activation function
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# via metaparameters!
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if META['ACTIVATION']:
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acc = META['ACTIVATION'](acc)
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# rematerialize rm and rn to save registers
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rm = pid_m * BLOCK_M + tl.arange(0, BLOCK_M)
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rn = pid_n * BLOCK_N + tl.arange(0, BLOCK_N)
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C = C + (rm[:, None] * stride_cm + rn[None, :] * stride_cn)
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mask = (rm[:, None] < M) & (rn[None, :] < N)
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tl.store(C, acc, mask=mask)
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# we can fuse `leaky_relu` by providing it as an `ACTIVATION` meta-parameter in `_matmul`
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@triton.jit
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def leaky_relu(x):
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return tl.where(x >= 0, x, 0.01*x)
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.. GENERATED FROM PYTHON SOURCE LINES 191-193
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We can now create a convenience wrapper function that only takes two input tensors
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and (1) checks any shape constraint; (2) allocates the output; (3) launches the above kernel
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.. GENERATED FROM PYTHON SOURCE LINES 193-214
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.. code-block:: default
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def matmul(a, b, activation=None):
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# checks constraints
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assert a.shape[1] == b.shape[0], "incompatible dimensions"
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assert a.is_contiguous(), "matrix A must be contiguous"
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assert b.is_contiguous(), "matrix B must be contiguous"
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M, K = a.shape
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_, N = b.shape
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# allocates output
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c = torch.empty((M, N), device=a.device, dtype=a.dtype)
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# launch kernel
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grid = lambda META: (triton.cdiv(M, META['BLOCK_M']) * triton.cdiv(N, META['BLOCK_N']), )
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pgm = _matmul[grid](
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a, b, c, M, N, K, \
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a.stride(0), a.stride(1), b.stride(0), b.stride(1), c.stride(0), c.stride(1),\
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ACTIVATION = activation
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)
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# done; return the output tensor
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return c
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.. GENERATED FROM PYTHON SOURCE LINES 215-219
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Unit Test
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-----------
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We can test our custom matrix multiplication operation against a native torch implementation (i.e., cuBLAS)
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.. GENERATED FROM PYTHON SOURCE LINES 219-229
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.. code-block:: default
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torch.manual_seed(0)
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a = torch.randn((512, 512), device='cuda', dtype=torch.float16)
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b = torch.randn((512, 512), device='cuda', dtype=torch.float16)
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c_0 = matmul(a, b, activation=None)
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c_1 = torch.matmul(a, b)
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print(c_0)
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print(c_1)
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print(triton.testing.allclose(c_0, c_1))
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.. rst-class:: sphx-glr-script-out
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Out:
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.. code-block:: none
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tensor([[ 1.1045, -36.9688, 31.4688, ..., -11.3984, 24.4531, -32.3438],
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[ 6.3555, -19.6094, 34.0938, ..., -5.8945, 5.2891, 6.8867],
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[-32.0625, 5.9492, 15.3984, ..., -21.3906, -23.9844, -10.1328],
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...,
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[ -5.7031, 7.4492, 8.2656, ..., -10.6953, -40.0000, 17.7500],
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[ 25.5000, 24.3281, -8.4688, ..., -18.9375, 32.5312, -29.9219],
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[ -5.3477, 4.9844, 11.8906, ..., 5.5898, 6.4023, -17.3125]],
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device='cuda:0', dtype=torch.float16)
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tensor([[ 1.1045, -36.9688, 31.4688, ..., -11.3906, 24.4531, -32.3438],
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[ 6.3516, -19.6094, 34.0938, ..., -5.8906, 5.2812, 6.8828],
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[-32.0625, 5.9531, 15.3984, ..., -21.4062, -23.9844, -10.1328],
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...,
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[ -5.7070, 7.4492, 8.2656, ..., -10.6953, -40.0000, 17.7500],
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[ 25.5000, 24.3438, -8.4609, ..., -18.9375, 32.5312, -29.9219],
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[ -5.3477, 4.9805, 11.8828, ..., 5.5859, 6.4023, -17.3125]],
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device='cuda:0', dtype=torch.float16)
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tensor(True, device='cuda:0')
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.. GENERATED FROM PYTHON SOURCE LINES 230-236
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Benchmark
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--------------
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Square Matrix Performance
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~~~~~~~~~~~~~~~~~~~~~~~~~~
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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.
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.. GENERATED FROM PYTHON SOURCE LINES 236-268
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.. code-block:: default
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@triton.testing.perf_report(
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triton.testing.Benchmark(
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x_names=['M', 'N', 'K'], # argument names to use as an x-axis for the plot
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x_vals=[128 * i for i in range(1, 33)], # 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=['cublas', 'cublas + relu', 'triton', 'triton + relu'], # possible values for `line_arg``
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line_names=["cuBLAS", "cuBLAS (+ torch.nn.LeakyReLU)", "Triton", "Triton (+ LeakyReLU)"], # label name for the lines
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styles=[('green', '-'), ('green', '--'), ('blue', '-'), ('blue', '--')], # line styles
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ylabel="TFLOPS", # label name for the y-axis
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plot_name="matmul-performance", # name for the plot. Used also as a file name for saving the plot.
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args={}
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)
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)
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def benchmark(M, N, K, provider):
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a = torch.randn((M, K), device='cuda', dtype=torch.float16)
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b = torch.randn((K, N), device='cuda', dtype=torch.float16)
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if provider == 'cublas':
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ms, min_ms, max_ms = triton.testing.do_bench(lambda: torch.matmul(a, b))
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if provider == 'triton':
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ms, min_ms, max_ms = triton.testing.do_bench(lambda: matmul(a, b))
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if provider == 'cublas + relu':
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torch_relu = torch.nn.ReLU(inplace=True)
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ms, min_ms, max_ms = triton.testing.do_bench(lambda: torch_relu(torch.matmul(a, b)))
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if provider == 'triton + relu':
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ms, min_ms, max_ms = triton.testing.do_bench(lambda: matmul(a, b, activation=leaky_relu))
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perf = lambda ms: 2 * M * N * K * 1e-12 / (ms * 1e-3)
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return perf(ms), perf(max_ms), perf(min_ms)
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benchmark.run(show_plots=True, print_data=True)
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.. image:: /getting-started/tutorials/images/sphx_glr_03-matrix-multiplication_001.png
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:alt: 03 matrix multiplication
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:class: sphx-glr-single-img
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.. rst-class:: sphx-glr-script-out
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Out:
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.. code-block:: none
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matmul-performance:
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M cuBLAS ... Triton Triton (+ LeakyReLU)
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0 128.0 0.455111 ... 0.512000 0.512000
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1 256.0 2.978909 ... 2.978909 2.978909
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2 384.0 7.372800 ... 8.507077 8.507077
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3 512.0 14.563555 ... 16.384000 15.420235
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4 640.0 22.260869 ... 23.272727 23.272727
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5 768.0 32.768000 ... 34.028308 34.028308
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6 896.0 39.025776 ... 39.025776 35.123201
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7 1024.0 49.932191 ... 52.428801 52.428801
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8 1152.0 45.242181 ... 46.656000 45.938215
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9 1280.0 51.200001 ... 56.109587 56.109587
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10 1408.0 64.138541 ... 65.684049 65.684049
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11 1536.0 80.430545 ... 76.106321 76.106321
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12 1664.0 63.372618 ... 61.636381 61.636381
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13 1792.0 72.983276 ... 68.953520 68.953520
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14 1920.0 69.467336 ... 70.172588 68.776119
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15 2048.0 73.908442 ... 75.573044 75.573044
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16 2176.0 83.500614 ... 79.855747 79.226957
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17 2304.0 68.251065 ... 72.607513 73.051599
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18 2432.0 71.305746 ... 79.813818 78.917033
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19 2560.0 77.833728 ... 76.382283 76.740048
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20 2688.0 82.823267 ... 80.880718 82.642823
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21 2816.0 83.873477 ... 77.882512 77.330158
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22 2944.0 82.102191 ... 80.122235 79.610276
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23 3072.0 78.643199 ... 81.707223 83.514905
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24 3200.0 84.321474 ... 89.012517 85.219705
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25 3328.0 83.226931 ... 86.320498 82.464255
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26 3456.0 81.026701 ... 84.508982 86.689860
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27 3584.0 87.211821 ... 91.750399 87.042978
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28 3712.0 84.230479 ... 83.526206 82.902362
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29 3840.0 80.197243 ... 82.102449 80.667046
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30 3968.0 88.231331 ... 81.079024 83.807647
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31 4096.0 93.596744 ... 90.504200 89.777746
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[32 rows x 5 columns]
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.. rst-class:: sphx-glr-timing
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**Total running time of the script:** ( 2 minutes 26.808 seconds)
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.. _sphx_glr_download_getting-started_tutorials_03-matrix-multiplication.py:
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.. only :: html
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.. container:: sphx-glr-footer
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:class: sphx-glr-footer-example
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.. container:: sphx-glr-download sphx-glr-download-python
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:download:`Download Python source code: 03-matrix-multiplication.py <03-matrix-multiplication.py>`
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.. container:: sphx-glr-download sphx-glr-download-jupyter
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:download:`Download Jupyter notebook: 03-matrix-multiplication.ipynb <03-matrix-multiplication.ipynb>`
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.. only:: html
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.. rst-class:: sphx-glr-signature
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`Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
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