# How to optimize convolution using TensorCores¶

Author: Siyuan Feng

In this tutorial, we will demonstrate how to write a high performance convolution schedule using TensorCores in TVM. In this example, we assume the input to convolution has a large batch. We strongly recommend covering the How to optimize convolution on GPU tutorial first.

## TensorCore Introduction¶

Each Tensor Core provides a 4x4x4 matrix processing array that operates `D = A * B + C`, where A, B, C and D are 4x4 matrices as Figure shows. The matrix multiplication inputs A and B are FP16 matrices, while the accumulation matrices C and D may be FP16 or FP32 matrices.

However, CUDA programmers can only use warp-level primitive `wmma::mma_sync(acc_frag, a_frag, b_frag, acc_frag)` to perform 16x16x16 half-precision matrix multiplication on tensor cores. Before invoking the matrix multiplication, programmers must load data from memory into registers with primitive `wmma::load_matrix_sync`, explicitly. The NVCC compiler translates that primitive into multiple memory load instructions. At run time, every thread loads 16 elements from matrix A and 16 elements from B.

## Preparation and Algorithm¶

We use the fixed size for input tensors with 256 channels and 14 x 14 dimensions. The batch size is 256. Convolution filters contain 512 filters of size 3 x 3. We use stride size 1 and padding size 1 for the convolution. In the example, we use NHWCnc memory layout.The following code defines the convolution algorithm in TVM.

```import tvm
from tvm import te
import numpy as np
from tvm.contrib import nvcc

# The sizes of inputs and filters
batch_size = 256
height = 14
width = 14
in_channels = 256
out_channels = 512
kernel_h = 3
kernel_w = 3
stride_h = 1
stride_w = 1

# TensorCore shape
block_size = 16

assert batch_size % block_size == 0
assert in_channels % block_size == 0
assert out_channels % block_size == 0

# Input feature map: (N, H, W, IC, n, ic)
data_shape = (
batch_size // block_size,
height,
width,
in_channels // block_size,
block_size,
block_size,
)
# Kernel: (H, W, IC, OC, ic, oc)
kernel_shape = (
kernel_h,
kernel_w,
in_channels // block_size,
out_channels // block_size,
block_size,
block_size,
)
# Output feature map: (N, H, W, OC, n, oc)
output_shape = (
batch_size // block_size,
height,
width,
out_channels // block_size,
block_size,
block_size,
)

# Reduction axes
kh = te.reduce_axis((0, kernel_h), name="kh")
kw = te.reduce_axis((0, kernel_w), name="kw")
ic = te.reduce_axis((0, in_channels // block_size), name="ic")
ii = te.reduce_axis((0, block_size), name="ii")

# Algorithm
A = te.placeholder(data_shape, name="A", dtype="float16")
W = te.placeholder(kernel_shape, name="W", dtype="float16")
(
batch_size // block_size,
height + 2 * pad_h,
width + 2 * pad_w,
in_channels // block_size,
block_size,
block_size,
),
lambda n, h, w, i, nn, ii: tvm.tir.if_then_else(
tvm.tir.all(h >= pad_h, h - pad_h < height, w >= pad_w, w - pad_w < width),
A[n, h - pad_h, w - pad_w, i, nn, ii],
tvm.tir.const(0.0, "float16"),
),
)
Conv = te.compute(
output_shape,
lambda n, h, w, o, nn, oo: te.sum(
Apad[n, h * stride_h + kh, w * stride_w + kw, ic, nn, ii].astype("float32")
* W[kh, kw, ic, o, ii, oo].astype("float32"),
axis=[ic, kh, kw, ii],
),
name="Conv",
)

s = te.create_schedule(Conv.op)
```

## Memory Scope¶

In traditional GPU schedule, we have global, shared and local memory scope. To support TensorCores, we add another three special memory scope: `wmma.matrix_a`, `wmma.matrix_b` and `wmma.accumulator`. On hardware, all fragments scope stores at the on-chip registers level, the same place with local memory.

```# Designate the memory hierarchy
WS = s.cache_read(W, "shared", [Conv])
AF = s.cache_read(AS, "wmma.matrix_a", [Conv])
WF = s.cache_read(WS, "wmma.matrix_b", [Conv])
ConvF = s.cache_write(Conv, "wmma.accumulator")
```

## Define Tensor Intrinsic¶

In fact, TensorCore is a special hardware operation. So, we can just use tensorize to replace a unit of computation with the TensorCore instruction. The first thing is that we need to define tensor intrinsic.

There are four basic operation in TensorCore: `fill_fragment`, `load_matrix`, `mma_sync` and `store_matrix`. Since `fill_fragment` and `mma_sync` are both used in matrix multiplication, so we can just write following three intrinsics.

```def intrin_wmma_load_matrix(scope):
n = 16
A = te.placeholder((n, n), name="A", dtype="float16")
BA = tvm.tir.decl_buffer(A.shape, A.dtype, scope="shared", data_alignment=32, offset_factor=256)
C = te.compute((n, n), lambda i, j: A[i, j], name="C")
BC = tvm.tir.decl_buffer(C.shape, C.dtype, scope=scope, data_alignment=32, offset_factor=256)

def intrin_func(ins, outs):
ib = tvm.tir.ir_builder.create()

BA = ins[0]
BC = outs[0]
ib.emit(
tvm.tir.call_intrin(
"handle",
BC.data,
n,
n,
n,
BC.elem_offset // 256,
BA.access_ptr("r"),
n,
"row_major",
)
)
return ib.get()

return te.decl_tensor_intrin(C.op, intrin_func, binds={A: BA, C: BC})

def intrin_wmma_gemm():
n = 16
A = te.placeholder((n, n), name="A", dtype="float16")
B = te.placeholder((n, n), name="B", dtype="float16")
k = te.reduce_axis((0, n), name="k")
C = te.compute(
(n, n),
lambda ii, jj: te.sum(A[ii, k].astype("float") * B[k, jj].astype("float"), axis=k),
name="C",
)
BA = tvm.tir.decl_buffer(
A.shape, A.dtype, name="BA", scope="wmma.matrix_a", data_alignment=32, offset_factor=256
)
BB = tvm.tir.decl_buffer(
B.shape, B.dtype, name="BB", scope="wmma.matrix_b", data_alignment=32, offset_factor=256
)
BC = tvm.tir.decl_buffer(
C.shape, C.dtype, name="BC", scope="wmma.accumulator", data_alignment=32, offset_factor=256
)

def intrin_func(ins, outs):
BA, BB = ins
(BC,) = outs

def init():
ib = tvm.tir.ir_builder.create()
ib.emit(
tvm.tir.call_intrin(
"handle", "tir.tvm_fill_fragment", BC.data, n, n, n, BC.elem_offset // 256, 0.0
)
)
return ib.get()

def update():
ib = tvm.tir.ir_builder.create()
ib.emit(
tvm.tir.call_intrin(
"handle",
"tir.tvm_mma_sync",
BC.data,
BC.elem_offset // 256,
BA.data,
BA.elem_offset // 256,
BB.data,
BB.elem_offset // 256,
BC.data,
BC.elem_offset // 256,
)
)
return ib.get()

return update(), init(), update()

return te.decl_tensor_intrin(C.op, intrin_func, binds={A: BA, B: BB, C: BC})

def intrin_wmma_store_matrix():
n = 16
A = te.placeholder((n, n), name="A", dtype="float32")
BA = tvm.tir.decl_buffer(
A.shape, A.dtype, scope="wmma.accumulator", data_alignment=32, offset_factor=256
)
C = te.compute((n, n), lambda i, j: A[i, j], name="C")
BC = tvm.tir.decl_buffer(C.shape, C.dtype, scope="global", data_alignment=32, offset_factor=256)

def intrin_func(ins, outs):
ib = tvm.tir.ir_builder.create()
BA = ins[0]
BC = outs[0]
ib.emit(
tvm.tir.call_intrin(
"handle",
"tir.tvm_store_matrix_sync",
BA.data,
n,
n,
n,
BA.elem_offset // 256,
BC.access_ptr("w"),
n,
"row_major",
)
)
return ib.get()

return te.decl_tensor_intrin(C.op, intrin_func, binds={A: BA, C: BC})
```

## Scheduling the Computation¶

To use TensorCores in TVM, we must schedule the computation into specific structure to match the tensor intrinsic. The same as traditional GPU programs, we can also use shared memory to boost the speed. If you have any questions about blocking and shared memory, please refer How to optimize convolution on GPU.

In this example, each block contains 2x4 warps, and each warp calls 4x2 TensorCore instructions. Thus, the output shape of each warp is 64x32 and each block outputs 128x128 titles. Due to the limit of shared memory space, we only load 2 blocks (2x128x128 tiles) one time.

Note

Warp-level Operation

Note that all TensorCore instructions are warp-level instructions, which means all 32 threads in a warp should do this instruction simultaneously. Making theadIdx.x extent=32 is one of the easiest way to solve this. Then We can bind threadIdx.x to any loops except those contain TensorCore intrinsics directly or indirectly. Also note that it is not the unique solution. The only thing we should do is to make sure all threads in a warp can call TensorCore at the same time.

```# Define tiling sizes
block_row_warps = 4
block_col_warps = 2
warp_row_tiles = 2
warp_col_tiles = 4
warp_size = 32
chunk = 2

nc, hc, wc, oc, nnc, ooc = Conv.op.axis
block_k = s[Conv].fuse(hc, wc)
s[Conv].bind(block_k, block_z)
nc, nci = s[Conv].split(nc, factor=warp_row_tiles)
block_i, nc = s[Conv].split(nc, factor=block_row_warps)
oc, oci = s[Conv].split(oc, factor=warp_col_tiles)
block_j, oc = s[Conv].split(oc, factor=block_col_warps)
s[Conv].reorder(block_k, block_i, block_j, nc, oc, nci, oci, nnc, ooc)
s[Conv].bind(block_i, block_x)
s[Conv].bind(block_j, block_y)

# Schedule local computation
s[ConvF].compute_at(s[Conv], oc)
n, h, w, o, nnf, oof = ConvF.op.axis
ko, ki = s[ConvF].split(ic, factor=chunk)
s[ConvF].reorder(ko, kh, ki, kw, n, o, nnf, oof, ii)

# Move intermediate computation into each output compute tile
s[AF].compute_at(s[ConvF], kw)
s[WF].compute_at(s[ConvF], kw)

# Schedule for A's share memory
s[AS].compute_at(s[ConvF], kh)
n, h, w, i, nn, ii = AS.op.axis
tx, xo = s[AS].split(n, nparts=block_row_warps)
ty, yo = s[AS].split(xo, nparts=block_col_warps)
t = s[AS].fuse(nn, ii)
to, ti = s[AS].split(t, factor=warp_size)

# Schedule for W's share memory
s[WS].compute_at(s[ConvF], kh)
kh, kw, ic, o, ii, oo = WS.op.axis
tx, xo = s[WS].split(o, nparts=block_row_warps)
ty, yo = s[WS].split(xo, nparts=block_col_warps)
t = s[WS].fuse(ii, oo)
to, ti = s[WS].split(t, nparts=warp_size)
s[WS].vectorize(ti)
print(tvm.lower(s, [A, W, Conv], simple_mode=True))
```

Out:

```primfn(A_1: handle, W_1: handle, Conv_1: handle) -> ()
attr = {"global_symbol": "main", "tir.noalias": True}
buffers = {A: Buffer(A_2: Pointer(float16), float16, [16, 14, 14, 16, 16, 16], []),
Conv: Buffer(Conv_2: Pointer(float32), float32, [16, 14, 14, 32, 16, 16], []),
W: Buffer(W_2: Pointer(float16), float16, [3, 3, 16, 32, 16, 16], [])}
buffer_map = {A_1: A, W_1: W, Conv_1: Conv} {
attr [IterVar(blockIdx.z: int32, (nullptr), "ThreadIndex", "blockIdx.z")] "thread_extent" = 196;
attr [Conv.wmma.accumulator: Pointer(wmma.accumulator float32)] "storage_scope" = "wmma.accumulator";
allocate(Conv.wmma.accumulator, float32, [2048]);
attr [Apad.shared: Pointer(shared float16)] "storage_scope" = "shared";
attr [W.shared: Pointer(shared float16)] "storage_scope" = "shared";
allocate(W.shared, float16, [12288]);
attr [Apad.shared.wmma.matrix_a: Pointer(wmma.matrix_a float16)] "storage_scope" = "wmma.matrix_a";
attr [W.shared.wmma.matrix_b: Pointer(wmma.matrix_b float16)] "storage_scope" = "wmma.matrix_b";
allocate(W.shared.wmma.matrix_b, float16, [1024]);
attr [IterVar(blockIdx.x: int32, (nullptr), "ThreadIndex", "blockIdx.x")] "thread_extent" = 2;
attr [IterVar(blockIdx.y: int32, (nullptr), "ThreadIndex", "blockIdx.y")] "thread_extent" = 4;
for (n.c.init: int32, 0, 2) {
for (o.c.init: int32, 0, 4) {
for (nn.c.init: int32, 0, 16) {
for (oo.c.init: int32, 0, 16) {
Conv.wmma.accumulator[((((n.c.init*1024) + (o.c.init*256)) + (nn.c.init*16)) + oo.c.init)] = 0f32
}
}
}
}
for (ic.outer: int32, 0, 8) {
for (kh: int32, 0, 3) {
for (ax2: int32, 0, 3) {
for (ax3: int32, 0, 2) {
for (ax4.ax5.fused.outer: int32, 0, 8) {
Apad.shared[((((((threadIdx.y*3072) + (threadIdx.z*1536)) + (ax2*512)) + (ax3*256)) + (ax4.ax5.fused.outer*32)) + threadIdx.x)] = @tir.if_then_else(((((1 <= (floordiv(blockIdx.z, 14) + kh)) && ((floordiv(blockIdx.z, 14) + kh) < 15)) && (1 <= (ax2 + floormod(blockIdx.z, 14)))) && ((ax2 + floormod(blockIdx.z, 14)) < 15)), (float16*)A_2[(((((((((((blockIdx.x*6422528) + (threadIdx.y*1605632)) + (threadIdx.z*802816)) + (kh*57344)) + (blockIdx.z*4096)) + (ax2*4096)) + (ic.outer*512)) + (ax3*256)) + (ax4.ax5.fused.outer*32)) + threadIdx.x) - 61440)], 0f16, dtype=float16)
}
}
}
for (ax1: int32, 0, 3) {
for (ax2_1: int32, 0, 2) {
W.shared[ramp((((((ax1*4096) + (ax2_1*2048)) + (threadIdx.y*512)) + (threadIdx.z*256)) + (threadIdx.x*8)), 1, 8)] = (float16x8*)W_2[ramp(((((((((kh*393216) + (ax1*131072)) + (ic.outer*16384)) + (ax2_1*8192)) + (blockIdx.y*2048)) + (threadIdx.y*512)) + (threadIdx.z*256)) + (threadIdx.x*8)), 1, 8)]
}
}
for (ic.inner: int32, 0, 2) {
for (kw: int32, 0, 3) {
for (ax0: int32, 0, 2) {
for (ax4: int32, 0, 16) {
for (ax5: int32, 0, 16) {
Apad.shared.wmma.matrix_a[(((ax0*256) + (ax4*16)) + ax5)] = (float16*)Apad.shared[((((((threadIdx.y*3072) + (ax0*1536)) + (kw*512)) + (ic.inner*256)) + (ax4*16)) + ax5)]
}
}
}
for (ax3_1: int32, 0, 4) {
for (ax4_1: int32, 0, 16) {
for (ax5_1: int32, 0, 16) {
W.shared.wmma.matrix_b[(((ax3_1*256) + (ax4_1*16)) + ax5_1)] = (float16*)W.shared[((((((kw*4096) + (ic.inner*2048)) + (threadIdx.z*1024)) + (ax3_1*256)) + (ax4_1*16)) + ax5_1)]
}
}
}
for (n.c: int32, 0, 2) {
for (o.c: int32, 0, 4) {
for (nn.c: int32, 0, 16) {
for (oo.c: int32, 0, 16) {
for (ii: int32, 0, 16) {
Conv.wmma.accumulator[((((n.c*1024) + (o.c*256)) + (nn.c*16)) + oo.c)] = ((float32*)Conv.wmma.accumulator[((((n.c*1024) + (o.c*256)) + (nn.c*16)) + oo.c)] + (cast(float32, (float16*)Apad.shared.wmma.matrix_a[(((n.c*256) + (nn.c*16)) + ii)])*cast(float32, (float16*)W.shared.wmma.matrix_b[(((o.c*256) + (ii*16)) + oo.c)])))
}
}
}
}
}
}
}
}
}
for (n.inner: int32, 0, 2) {
for (o.inner: int32, 0, 4) {
for (nn: int32, 0, 16) {
for (oo: int32, 0, 16) {
Conv_2[(((((((((blockIdx.x*12845056) + (threadIdx.y*3211264)) + (n.inner*1605632)) + (blockIdx.z*8192)) + (blockIdx.y*2048)) + (threadIdx.z*1024)) + (o.inner*256)) + (nn*16)) + oo)] = (float32*)Conv.wmma.accumulator[((((n.inner*1024) + (o.inner*256)) + (nn*16)) + oo)]
}
}
}
}
}
}
```

## Lowering Computation to Intrinsics¶

The last phase is to lower the computation loops down to TensorCore hardware intrinsics by mapping the 2D convolution to tensor intrinsics

```s[AF].tensorize(AF.op.axis[-2], intrin_wmma_load_matrix("wmma.matrix_a"))
s[Conv].tensorize(nnc, intrin_wmma_store_matrix())
s[ConvF].tensorize(nnf, intrin_wmma_gemm())
print(tvm.lower(s, [A, W, Conv], simple_mode=True))
```

Out:

```primfn(A_1: handle, W_1: handle, Conv_1: handle) -> ()
attr = {"global_symbol": "main", "tir.noalias": True}
buffers = {Conv: Buffer(Conv_2: Pointer(float32), float32, [16, 14, 14, 32, 16, 16], []),
A: Buffer(A_2: Pointer(float16), float16, [16, 14, 14, 16, 16, 16], []),
W: Buffer(W_2: Pointer(float16), float16, [3, 3, 16, 32, 16, 16], [])}
buffer_map = {A_1: A, W_1: W, Conv_1: Conv} {
attr [IterVar(blockIdx.z: int32, (nullptr), "ThreadIndex", "blockIdx.z")] "thread_extent" = 196;
attr [Conv.wmma.accumulator: Pointer(wmma.accumulator float32)] "storage_scope" = "wmma.accumulator";
allocate(Conv.wmma.accumulator, float32, [2048]);
attr [Apad.shared: Pointer(shared float16)] "storage_scope" = "shared";
attr [W.shared: Pointer(shared float16)] "storage_scope" = "shared";
allocate(W.shared, float16, [12288]);
attr [Apad.shared.wmma.matrix_a: Pointer(wmma.matrix_a float16)] "storage_scope" = "wmma.matrix_a";
attr [W.shared.wmma.matrix_b: Pointer(wmma.matrix_b float16)] "storage_scope" = "wmma.matrix_b";
allocate(W.shared.wmma.matrix_b, float16, [1024]);
attr [IterVar(blockIdx.x: int32, (nullptr), "ThreadIndex", "blockIdx.x")] "thread_extent" = 2;
attr [IterVar(blockIdx.y: int32, (nullptr), "ThreadIndex", "blockIdx.y")] "thread_extent" = 4;
for (n.c.init: int32, 0, 2) {
for (o.c.init: int32, 0, 4) {
@tir.tvm_fill_fragment(Conv.wmma.accumulator, 16, 16, 16, ((n.c.init*4) + o.c.init), 0f32, dtype=handle)
}
}
for (ic.outer: int32, 0, 8) {
for (kh: int32, 0, 3) {
for (ax2: int32, 0, 3) {
for (ax3: int32, 0, 2) {
for (ax4.ax5.fused.outer: int32, 0, 8) {
Apad.shared[((((((threadIdx.y*3072) + (threadIdx.z*1536)) + (ax2*512)) + (ax3*256)) + (ax4.ax5.fused.outer*32)) + threadIdx.x)] = @tir.if_then_else(((((1 <= (floordiv(blockIdx.z, 14) + kh)) && ((floordiv(blockIdx.z, 14) + kh) < 15)) && (1 <= (ax2 + floormod(blockIdx.z, 14)))) && ((ax2 + floormod(blockIdx.z, 14)) < 15)), (float16*)A_2[(((((((((((blockIdx.x*6422528) + (threadIdx.y*1605632)) + (threadIdx.z*802816)) + (kh*57344)) + (blockIdx.z*4096)) + (ax2*4096)) + (ic.outer*512)) + (ax3*256)) + (ax4.ax5.fused.outer*32)) + threadIdx.x) - 61440)], 0f16, dtype=float16)
}
}
}
for (ax1: int32, 0, 3) {
for (ax2_1: int32, 0, 2) {
W.shared[ramp((((((ax1*4096) + (ax2_1*2048)) + (threadIdx.y*512)) + (threadIdx.z*256)) + (threadIdx.x*8)), 1, 8)] = (float16x8*)W_2[ramp(((((((((kh*393216) + (ax1*131072)) + (ic.outer*16384)) + (ax2_1*8192)) + (blockIdx.y*2048)) + (threadIdx.y*512)) + (threadIdx.z*256)) + (threadIdx.x*8)), 1, 8)]
}
}
for (ic.inner: int32, 0, 2) {
for (kw: int32, 0, 3) {
for (ax0: int32, 0, 2) {
@tir.tvm_load_matrix_sync(Apad.shared.wmma.matrix_a, 16, 16, 16, ax0, @tir.tvm_access_ptr(@tir.type_annotation(, dtype=float16), Apad.shared, ((((threadIdx.y*3072) + (ax0*1536)) + (kw*512)) + (ic.inner*256)), 256, 1, dtype=handle), 16, "row_major", dtype=handle)
}
for (ax3_1: int32, 0, 4) {
@tir.tvm_load_matrix_sync(W.shared.wmma.matrix_b, 16, 16, 16, ax3_1, @tir.tvm_access_ptr(@tir.type_annotation(, dtype=float16), W.shared, ((((kw*4096) + (ic.inner*2048)) + (threadIdx.z*1024)) + (ax3_1*256)), 256, 1, dtype=handle), 16, "row_major", dtype=handle)
}
for (n.c: int32, 0, 2) {
for (o.c: int32, 0, 4) {
@tir.tvm_mma_sync(Conv.wmma.accumulator, ((n.c*4) + o.c), Apad.shared.wmma.matrix_a, n.c, W.shared.wmma.matrix_b, o.c, Conv.wmma.accumulator, ((n.c*4) + o.c), dtype=handle)
}
}
}
}
}
}
for (n.inner: int32, 0, 2) {
for (o.inner: int32, 0, 4) {
@tir.tvm_store_matrix_sync(Conv.wmma.accumulator, 16, 16, 16, ((n.inner*4) + o.inner), @tir.tvm_access_ptr(@tir.type_annotation(, dtype=float32), Conv_2, (((((((blockIdx.x*12845056) + (threadIdx.y*3211264)) + (n.inner*1605632)) + (blockIdx.z*8192)) + (blockIdx.y*2048)) + (threadIdx.z*1024)) + (o.inner*256)), 256, 2, dtype=handle), 16, "row_major", dtype=handle)
}
}
}
}
```

## Generate CUDA Kernel¶

Finally we use TVM to generate and compile the CUDA kernel, and evaluate the latency of convolution. Since TensorCores are only supported in NVIDIA GPU with Compute Capability 7.0 or higher, it may not be able to run on our build server

```dev = tvm.cuda(0)
if nvcc.have_tensorcore(dev.compute_version):
with tvm.transform.PassContext(config={"tir.UnrollLoop": {"auto_max_step": 16}}):
func = tvm.build(s, [A, W, Conv], "cuda")
a_np = np.random.uniform(size=data_shape).astype(A.dtype)
w_np = np.random.uniform(size=kernel_shape).astype(W.dtype)
a = tvm.nd.array(a_np, dev)
w = tvm.nd.array(w_np, dev)
c = tvm.nd.array(np.zeros(output_shape, dtype=Conv.dtype), dev)
evaluator = func.time_evaluator(func.entry_name, dev, number=10)
print("conv2d with tensor core: %f ms" % (evaluator(a, w, c).mean * 1e3))
```

Out:

```conv2d with tensor core: 9.623236 ms
```

## Summary¶

This tutorial demonstrates how TVM scheduling primitives can be used to call TensorCores on specific GPUs.

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