Note
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Layer NormalizationΒΆ
Out:
layer-norm-backward:
N Triton Torch Apex
0 1024.0 356.173905 99.902435 315.076934
1 1536.0 405.098894 134.050910 344.523365
2 2048.0 491.520012 159.067963 334.367350
3 2560.0 458.507457 182.314537 330.322572
4 3072.0 519.211251 191.501303 321.956335
5 3584.0 554.941930 207.768111 309.410081
6 4096.0 568.231237 220.907859 299.707322
7 4608.0 502.690905 232.336141 287.251954
8 5120.0 527.381977 243.809526 286.433562
9 5632.0 540.671974 244.426754 291.939522
10 6144.0 550.208948 251.202731 288.000001
11 6656.0 530.710976 255.590406 286.793541
12 7168.0 510.480705 253.734520 277.470965
13 7680.0 487.619051 266.358392 284.884090
14 8192.0 468.114289 258.354805 278.481578
15 8704.0 414.476194 267.472468 285.377055
16 9216.0 431.157889 272.394084 289.887291
17 9728.0 438.033784 279.942444 288.950501
18 10240.0 442.810829 287.102804 290.153487
19 10752.0 427.231788 246.699797 289.941565
20 11264.0 427.071098 245.760001 286.069848
21 11776.0 419.946507 249.447482 288.686414
22 12288.0 415.369018 254.673582 294.617366
23 12800.0 410.695192 253.884294 287.910035
24 13312.0 410.125805 252.559690 289.129403
25 13824.0 404.112047 257.190689 292.056329
26 14336.0 396.844280 256.000002 289.129416
27 14848.0 385.662341 257.479779 288.777966
28 15360.0 378.869469 258.332158 288.225185
29 15872.0 372.000001 261.806182 290.562936
import torch
import triton
import triton.language as tl
try:
# This is https://github.com/NVIDIA/apex, NOT the apex on PyPi, so it
# should not be added to extras_require in setup.py.
import apex
HAS_APEX = True
except ModuleNotFoundError:
HAS_APEX = False
# Forward Pass
@triton.jit
def _layer_norm_fwd_fused(X, Y, W, B, M, V, stride, N, eps,
BLOCK_SIZE: tl.constexpr):
# position of elements processed by this program
row = tl.program_id(0)
cols = tl.arange(0, BLOCK_SIZE)
mask = cols < N
# offset data pointers to start at the row of interest
X += row * stride
Y += row * stride
# load data and cast to float32
x = tl.load(X + cols, mask=mask, other=0).to(tl.float32)
# compute mean
mean = tl.sum(x, axis=0) / N
# compute std
xmean = tl.where(mask, x - mean, 0.)
var = tl.sum(xmean * xmean, axis=0) / N
rstd = 1 / tl.sqrt(var + eps)
xhat = xmean * rstd
# write-back mean/rstd
tl.store(M + row, mean)
tl.store(V + row, rstd)
# multiply by weight and add bias
w = tl.load(W + cols, mask=mask)
b = tl.load(B + cols, mask=mask)
y = xhat * w + b
# write-back
tl.store(Y + cols, y, mask=mask)
# Backward pass (DX + partial DW + partial DB)
@triton.jit
def _layer_norm_bwd_dx_fused(DX, DY, DW, DB, X, W, B, M, V, Lock, stride, N, eps,
GROUP_SIZE_M: tl.constexpr, BLOCK_SIZE_N: tl.constexpr):
# position of elements processed by this program
row = tl.program_id(0)
cols = tl.arange(0, BLOCK_SIZE_N)
mask = cols < N
# offset data pointers to start at the row of interest
X += row * stride
DY += row * stride
DX += row * stride
# offset locks and weight/bias gradient pointer
# each kernel instance accumulates partial sums for
# DW and DB into one of GROUP_SIZE_M independent buffers
# these buffers stay in the L2, which allow this kernel
# to be fast
lock_id = row % GROUP_SIZE_M
Lock += lock_id
Count = Lock + GROUP_SIZE_M
DW = DW + lock_id * N + cols
DB = DB + lock_id * N + cols
# load data to SRAM
x = tl.load(X + cols, mask=mask, other=0).to(tl.float32)
dy = tl.load(DY + cols, mask=mask, other=0).to(tl.float32)
w = tl.load(W + cols, mask=mask).to(tl.float32)
mean = tl.load(M + row)
rstd = tl.load(V + row)
# compute dx
xhat = (x - mean) * rstd
wdy = w * dy
xhat = tl.where(mask, xhat, 0.)
wdy = tl.where(mask, wdy, 0.)
mean1 = tl.sum(xhat * wdy, axis=0) / N
mean2 = tl.sum(wdy, axis=0) / N
dx = (wdy - (xhat * mean1 + mean2)) * rstd
# write-back dx
tl.store(DX + cols, dx, mask=mask)
# accumulate partial sums for dw/db
partial_dw = (dy * xhat).to(w.dtype)
partial_db = (dy).to(w.dtype)
while tl.atomic_cas(Lock, 0, 1) == 1:
pass
count = tl.load(Count)
# first store doesn't accumulate
if count == 0:
tl.atomic_xchg(Count, 1)
else:
partial_dw += tl.load(DW, mask=mask)
partial_db += tl.load(DB, mask=mask)
tl.store(DW, partial_dw, mask=mask)
tl.store(DB, partial_db, mask=mask)
# release lock
tl.atomic_xchg(Lock, 0)
# Backward pass (total DW + total DB)
@triton.jit
def _layer_norm_bwd_dwdb(DW, DB, FINAL_DW, FINAL_DB, M, N,
BLOCK_SIZE_M: tl.constexpr, BLOCK_SIZE_N: tl.constexpr):
pid = tl.program_id(0)
cols = pid * BLOCK_SIZE_N + tl.arange(0, BLOCK_SIZE_N)
dw = tl.zeros((BLOCK_SIZE_M, BLOCK_SIZE_N), dtype=tl.float32)
db = tl.zeros((BLOCK_SIZE_M, BLOCK_SIZE_N), dtype=tl.float32)
for i in range(0, M, BLOCK_SIZE_M):
rows = i + tl.arange(0, BLOCK_SIZE_M)
mask = (rows[:, None] < M) & (cols[None, :] < N)
offs = rows[:, None] * N + cols[None, :]
dw += tl.load(DW + offs, mask=mask, other=0.)
db += tl.load(DB + offs, mask=mask, other=0.)
sum_dw = tl.sum(dw, axis=0)
sum_db = tl.sum(db, axis=0)
tl.store(FINAL_DW + cols, sum_dw, mask=cols < N)
tl.store(FINAL_DB + cols, sum_db, mask=cols < N)
class LayerNorm(torch.autograd.Function):
@staticmethod
def forward(ctx, x, normalized_shape, weight, bias, eps):
# allocate output
y = torch.empty_like(x)
# reshape input data into 2D tensor
x_arg = x.reshape(-1, x.shape[-1])
M, N = x_arg.shape
mean = torch.empty((M, ), dtype=torch.float32, device='cuda')
rstd = torch.empty((M, ), dtype=torch.float32, device='cuda')
# Less than 64KB per feature: enqueue fused kernel
MAX_FUSED_SIZE = 65536 // x.element_size()
BLOCK_SIZE = min(MAX_FUSED_SIZE, triton.next_power_of_2(N))
if N > BLOCK_SIZE:
raise RuntimeError("This layer norm doesn't support feature dim >= 64KB.")
# heuristics for number of warps
num_warps = min(max(BLOCK_SIZE // 256, 1), 8)
# enqueue kernel
_layer_norm_fwd_fused[(M,)](x_arg, y, weight, bias, mean, rstd,
x_arg.stride(0), N, eps,
BLOCK_SIZE=BLOCK_SIZE, num_warps=num_warps)
ctx.save_for_backward(x, weight, bias, mean, rstd)
ctx.BLOCK_SIZE = BLOCK_SIZE
ctx.num_warps = num_warps
ctx.eps = eps
return y
@staticmethod
def backward(ctx, dy):
x, w, b, m, v = ctx.saved_tensors
# heuristics for amount of parallel reduction stream for DG/DB
N = w.shape[0]
GROUP_SIZE_M = 64
if N <= 8192: GROUP_SIZE_M = 96
if N <= 4096: GROUP_SIZE_M = 128
if N <= 1024: GROUP_SIZE_M = 256
# allocate output
locks = torch.zeros(2 * GROUP_SIZE_M, dtype=torch.int32, device='cuda')
_dw = torch.empty((GROUP_SIZE_M, w.shape[0]), dtype=x.dtype, device=w.device)
_db = torch.empty((GROUP_SIZE_M, w.shape[0]), dtype=x.dtype, device=w.device)
dw = torch.empty((w.shape[0],), dtype=w.dtype, device=w.device)
db = torch.empty((w.shape[0],), dtype=w.dtype, device=w.device)
dx = torch.empty_like(dy)
# enqueue kernel using forward pass heuristics
# also compute partial sums for DW and DB
x_arg = x.reshape(-1, x.shape[-1])
M, N = x_arg.shape
_layer_norm_bwd_dx_fused[(M,)](dx, dy, _dw, _db, x, w, b, m, v, locks,
x_arg.stride(0), N, ctx.eps,
BLOCK_SIZE_N=ctx.BLOCK_SIZE,
GROUP_SIZE_M=GROUP_SIZE_M,
num_warps=ctx.num_warps)
grid = lambda meta: [triton.cdiv(N, meta['BLOCK_SIZE_N'])]
# accumulate partial sums in separate kernel
_layer_norm_bwd_dwdb[grid](_dw, _db, dw, db, GROUP_SIZE_M, N,
BLOCK_SIZE_M=32,
BLOCK_SIZE_N=128)
return dx, None, dw, db, None
layer_norm = LayerNorm.apply
def test_layer_norm(M, N, dtype, eps=1e-5, device='cuda'):
# create data
x_shape = (M, N)
w_shape = (x_shape[-1], )
weight = torch.rand(w_shape, dtype=dtype, device='cuda', requires_grad=True)
bias = torch.rand(w_shape, dtype=dtype, device='cuda', requires_grad=True)
x = -2.3 + 0.5 * torch.randn(x_shape, dtype=dtype, device='cuda')
dy = .1 * torch.randn_like(x)
x.requires_grad_(True)
# forward pass
y_tri = layer_norm(x, w_shape, weight, bias, eps)
y_ref = torch.nn.functional.layer_norm(x, w_shape, weight, bias, eps).to(dtype)
# backward pass (triton)
y_tri.backward(dy, retain_graph=True)
dx_tri, dw_tri, db_tri = [_.grad.clone() for _ in [x, weight, bias]]
x.grad, weight.grad, bias.grad = None, None, None
# backward pass (torch)
y_ref.backward(dy, retain_graph=True)
dx_ref, dw_ref, db_ref = [_.grad.clone() for _ in [x, weight, bias]]
# compare
triton.testing.assert_almost_equal(y_tri, y_ref)
triton.testing.assert_almost_equal(dx_tri, dx_ref)
triton.testing.assert_almost_equal(db_tri, db_ref, decimal=1)
triton.testing.assert_almost_equal(dw_tri, dw_ref, decimal=1)
@triton.testing.perf_report(
triton.testing.Benchmark(
x_names=['N'],
x_vals=[512 * i for i in range(2, 32)],
line_arg='provider',
line_vals=['triton', 'torch'] + (['apex'] if HAS_APEX else []),
line_names=['Triton', 'Torch'] + (['Apex'] if HAS_APEX else []),
styles=[('blue', '-'), ('green', '-'), ('orange', '-')],
ylabel='GB/s',
plot_name='layer-norm-backward',
args={'M': 4096, 'dtype': torch.float16, 'mode': 'backward'}
)
)
def bench_layer_norm(M, N, dtype, provider, mode='backward', eps=1e-5, device='cuda'):
# create data
x_shape = (M, N)
w_shape = (x_shape[-1], )
weight = torch.rand(w_shape, dtype=dtype, device='cuda', requires_grad=True)
bias = torch.rand(w_shape, dtype=dtype, device='cuda', requires_grad=True)
x = -2.3 + 0.5 * torch.randn(x_shape, dtype=dtype, device='cuda')
dy = .1 * torch.randn_like(x)
x.requires_grad_(True)
# utility functions
if provider == 'triton':
y_fwd = lambda: layer_norm(x, w_shape, weight, bias, eps)
if provider == 'torch':
y_fwd = lambda: torch.nn.functional.layer_norm(x, w_shape, weight, bias, eps)
if provider == 'apex':
apex_layer_norm = apex.normalization.FusedLayerNorm(w_shape).to(x.device).to(x.dtype)
y_fwd = lambda: apex_layer_norm(x)
# forward pass
if mode == 'forward':
gbps = lambda ms: 2 * x.numel() * x.element_size() / ms * 1e-6
ms, min_ms, max_ms = triton.testing.do_bench(y_fwd, rep=500)
# backward pass
if mode == 'backward':
gbps = lambda ms: 3 * x.numel() * x.element_size() / ms * 1e-6
y = y_fwd()
ms, min_ms, max_ms = triton.testing.do_bench(lambda: y.backward(dy, retain_graph=True),
grad_to_none=[x], rep=500)
return gbps(ms), gbps(max_ms), gbps(min_ms)
bench_layer_norm.run(save_path='.', print_data=True)
Total running time of the script: ( 2 minutes 12.415 seconds)