" * Total bytes = (2 reads + 1 write) × elements × sizeof(dtype)\n",
" * Total bytes = (2 reads + 1 write) \u00d7 elements \u00d7 sizeof(dtype)\n",
"\n",
"Below is our benchmarking utility that measures these metrics:"
]
@@ -368,7 +368,7 @@
"\n",
"According to *Little's Law*, naive implementation has\n",
" - 1 element (4 bytes load + 2 bytes store) per thread\n",
" - 256 threads/block × N blocks\n",
" - 256 threads/block \u00d7 N blocks\n",
" - Limited in-flight operations\n",
"\n",
"In some GPUs, it's insufficient parallelism to saturate memory bandwidth.\n",
@@ -385,7 +385,35 @@
{
"cell_type": "markdown",
"metadata": {},
"source": "## Vectorized Load and Store\n\nTo improve performance according to Little's Law, we need to increase the number\nof in-flight requests. We can do this by increasing the number of bytes handled\nin each load & store operation per thread through vectorized memory access.\n\nSince Ampere GPUs support up to 128-bit per load/store and each element is 16-bit,\nwe can load 8 elements per vectorized operation on contiguous rows.\nCuTe tiling operations make this vectorization straightforward.\n\nUsing ``tiled_tensor = cute.zipped_divide(tensor, tiler)``, we can partition the input\n``tensor`` into groups of ``tiler`` blocks. For vectorization, we specify ``tiler``\nas the block of data each thread accesses (8 contiguous elements in the same row, or ``(1,8)``).\nDifferent threads can then access different blocks by indexing into the 2nd mode of ``tiled_tensor``.\n\n```python\nmA : cute.Tensor # (2048,2048):(2048,1)\ngA = cute.zipped_divide(a, tiler=(1, 8)) # tiled/vectorized => ((1,8),(2048,256)):((0,1),(2048,8))\n```\n\n$\n \\begin{array}{ccccc}\n & ((1,8) & , & (2048,256)) & : ((0,1),(2048,8)) \\\\\n & \\underbrace{\\phantom{(1,8)}}_{tiler} & & \\underbrace{\\phantom{(2048,256)}}_{threads} & \\\\\n & \\text{\\scriptsize per-thread} & & \\text{\\scriptsize num of tiles}\n \\end{array}\n$"
"source": [
"## Vectorized Load and Store\n",
"\n",
"To improve performance according to Little's Law, we need to increase the number\n",
"of in-flight requests. We can do this by increasing the number of bytes handled\n",
"in each load & store operation per thread through vectorized memory access.\n",
"\n",
"Since Ampere GPUs support up to 128-bit per load/store and each element is 16-bit,\n",
"we can load 8 elements per vectorized operation on contiguous rows.\n",
"CuTe tiling operations make this vectorization straightforward.\n",
"\n",
"Using ``tiled_tensor = cute.zipped_divide(tensor, tiler)``, we can partition the input\n",
"``tensor`` into groups of ``tiler`` blocks. For vectorization, we specify ``tiler``\n",
"as the block of data each thread accesses (8 contiguous elements in the same row, or ``(1,8)``).\n",
"Different threads can then access different blocks by indexing into the 2nd mode of ``tiled_tensor``.\n",
" & \\text{\\scriptsize per-thread} & & \\text{\\scriptsize num of tiles}\n",
" \\end{array}\n",
"$"
]
},
{
"cell_type": "code",
@@ -423,14 +451,91 @@
{
"cell_type": "markdown",
"metadata": {},
"source": "This vectorized kernel follows a similar structure to its naive non-vectorized counterpart,\nwith one key difference: the tensor slicing pattern. By using `(None, (mi, ni))` as the slice indices,\nwe can extract a `(1,8)` sub-tensor from `gA`, `gB` and `gC` like \n\n$ gA[(None, (mi, ni))]: $\n\n$\n \\begin{array}{ccccc}\n Layout: & ( & (1,8) & , & (2048,256) & ) & : & ((0,1),(2048,8)) & \\xrightarrow{\\text{slice}} & ((1,8)):((0,1)) \\\\\n & & \\underbrace{\\phantom{(1,8)}} & & \\underbrace{\\phantom{(2048,256)}} & & \\\\\n Coord: & ( & None & , & (mi, ni) & ) & &\n \\end{array}\n$\n\nThen tensor data can be loaded into vector via the `gA[(None, (mi, ni))].load()` method. It is equivalent to\n\n```python\nv0 = gA[(0, (mi, ni))] # => mA[(mi, ni * 8 + 0)]\nv1 = gA[(1, (mi, ni))] # => mA[(mi, ni * 8 + 1)]\nv2 = gA[(2, (mi, ni))] # => mA[(mi, ni * 8 + 2)]\nv3 = gA[(3, (mi, ni))] # => mA[(mi, ni * 8 + 3)]\nv4 = gA[(4, (mi, ni))] # => mA[(mi, ni * 8 + 4)]\nv5 = gA[(5, (mi, ni))] # => mA[(mi, ni * 8 + 5)]\nv6 = gA[(6, (mi, ni))] # => mA[(mi, ni * 8 + 6)]\nv7 = gA[(7, (mi, ni))] # => mA[(mi, ni * 8 + 7)]\n```\n\n### Assumed Alignment\n\nIn order to guide compile to use vectorized load/store, we must tell compiler to assume alignment of incoming pointer. \nIt's on users side to guarantee actual pointer at runtime meet the alignment restriction.\n\n```python\na_ = from_dlpack(a, assumed_align=16)\nb_ = from_dlpack(b, assumed_align=16)\nc_ = from_dlpack(c, assumed_align=16)\n\n# Compile kernel with alignment assumption\ncompiled_func = cute.compile(vectorized_elementwise_add, a_, b_, c_)\n```\n\nIt's worth to note that partitioned or tiled tensor could have different alignment of its base pointer because of offset\nduring sub-slice."
"source": [
"This vectorized kernel follows a similar structure to its naive non-vectorized counterpart,\n",
"with one key difference: the tensor slicing pattern. By using `(None, (mi, ni))` as the slice indices,\n",
"we can extract a `(1,8)` sub-tensor from `gA`, `gB` and `gC` like \n",
"source": "## TV Layout\n\nBoth the naive and vectorized kernels follow a common pattern to map thread indices\nto physical addresses in two steps:\n\nStep 1: Map thread index to logical coordinates in `(M, N)`\n\n* `mi = thread_idx // n`\n* `ni = thread_idx % n`\n\nIn native version, each thread process 1 element, in this case, `mi` and `ni` is logical\ncoordinate into data tensor `mA`, `mB` and `mC`.\n\nInt vectorized version, each thread process multiple values of input and output tensor.\nlogical coordinate should be computed with both thread and value index.\n\n* `thread_idx // n`\n* `(thread_idx % n) * 8 + value_idx`\n\n\nStep 2: Map logical coordinates in `(M, N)` to physical addresses using the tensor layout\n\n* Vectorized Load\n\n```python\n frgA = gA[(None, (mi, ni))].load()\n```\n\n* Elementwise Load (less efficient)\n\n```python\n frgA0 = mA[(mi, ni * 8 + 0)]\n frgA1 = mA[(mi, ni * 8 + 1)]\n frgA2 = mA[(mi, ni * 8 + 2)]\n frgA3 = mA[(mi, ni * 8 + 3)]\n frgA4 = mA[(mi, ni * 8 + 4)]\n frgA5 = mA[(mi, ni * 8 + 5)]\n frgA6 = mA[(mi, ni * 8 + 6)]\n frgA7 = mA[(mi, ni * 8 + 7)]\n\n # Or use divided layout\n\n frgA0 = gA[(0, (mi, ni))]\n frgA1 = gA[(1, (mi, ni))]\n frgA2 = gA[(2, (mi, ni))]\n frgA3 = gA[(3, (mi, ni))]\n frgA4 = gA[(4, (mi, ni))]\n frgA5 = gA[(5, (mi, ni))]\n frgA6 = gA[(6, (mi, ni))]\n frgA7 = gA[(7, (mi, ni))]\n```\n\nCuTe introduces TV layout to represent this mapping from thread index and value index\n(i.e., the 8 elements loaded per thread) to the logical coordinate space of a tensor.\nBy configuring different TV layouts, we can experiment with different memory access\npatterns with minimal code changes.\n\n**Definition:** *TV Layout* is rank-2 layout which maps `(thread_index, value_index)` \nto logical coordinate of tensor. \n\nWe always have *TV Layout* with canonical form as `(thread_domain, value_domain):(..., ...)`.\n\nWith *TV Layout*, each thread can find logical coordinates or indices of data partitioned\nto current thread."
"source": [
"## TV Layout\n",
"\n",
"Both the naive and vectorized kernels follow a common pattern to map thread indices\n",
"to physical addresses in two steps:\n",
"\n",
"Step 1: Map thread index to logical coordinates in `(M, N)`\n",
"\n",
"* `mi = thread_idx // n`\n",
"* `ni = thread_idx % n`\n",
"\n",
"In native version, each thread process 1 element, in this case, `mi` and `ni` is logical\n",
"coordinate into data tensor `mA`, `mB` and `mC`.\n",
"\n",
"Int vectorized version, each thread process multiple values of input and output tensor.\n",
"logical coordinate should be computed with both thread and value index.\n",
"\n",
"* `thread_idx // n`\n",
"* `(thread_idx % n) * 8 + value_idx`\n",
"\n",
"\n",
"Step 2: Map logical coordinates in `(M, N)` to physical addresses using the tensor layout\n",
"\n",
"* Vectorized Load\n",
"\n",
"```python\n",
" frgA = gA[(None, (mi, ni))].load()\n",
"```\n",
"\n",
"* Elementwise Load (less efficient)\n",
"\n",
"```python\n",
" frgA0 = mA[(mi, ni * 8 + 0)]\n",
" frgA1 = mA[(mi, ni * 8 + 1)]\n",
" frgA2 = mA[(mi, ni * 8 + 2)]\n",
" frgA3 = mA[(mi, ni * 8 + 3)]\n",
" frgA4 = mA[(mi, ni * 8 + 4)]\n",
" frgA5 = mA[(mi, ni * 8 + 5)]\n",
" frgA6 = mA[(mi, ni * 8 + 6)]\n",
" frgA7 = mA[(mi, ni * 8 + 7)]\n",
"\n",
" # Or use divided layout\n",
"\n",
" frgA0 = gA[(0, (mi, ni))]\n",
" frgA1 = gA[(1, (mi, ni))]\n",
" frgA2 = gA[(2, (mi, ni))]\n",
" frgA3 = gA[(3, (mi, ni))]\n",
"```\n",
"\n",
"CuTe introduces TV layout to represent this mapping from thread index and value index\n",
"(i.e., the 4 elements loaded per thread) to the logical coordinate space of a tensor.\n",
"By configuring different TV layouts, we can experiment with different memory access\n",
"patterns with minimal code changes.\n",
"\n",
"**Definition:** *TV Layout* is rank-2 layout which maps `(thread_index, value_index)` \n",
"to logical coordinate of tensor. \n",
"\n",
"We always have *TV Layout* with canonical form as `(thread_domain, value_domain):(..., ...)`.\n",
"\n",
"With *TV Layout*, each thread can find logical coordinates or indices of data partitioned\n",
"to current thread.\n"
]
},
{
"cell_type": "markdown",
@@ -1057,4 +1223,4 @@
},
"nbformat": 4,
"nbformat_minor": 4
}
}
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Junkai-Wu