Note
Click here to download the full example code
Layer NormalizationΒΆ
Out:
layer-norm:
N Triton Torch Apex
0 1024.0 585.142849 277.694907 468.114273
1 1536.0 630.153868 323.368435 511.999982
2 2048.0 668.734716 337.814445 528.516136
3 2560.0 694.237267 362.477870 512.000013
4 3072.0 712.347810 378.092307 501.551037
5 3584.0 725.873439 384.859062 462.451619
6 4096.0 728.177767 381.023256 458.293714
7 4608.0 670.254540 396.387087 423.724136
8 5120.0 688.403381 397.669909 422.268057
9 5632.0 704.000002 396.969169 411.470331
10 6144.0 707.223041 402.885254 409.600010
11 6656.0 700.631610 400.360920 398.861429
12 7168.0 695.078767 388.772874 386.154893
13 7680.0 682.666656 392.587863 385.203746
14 8192.0 639.375598 392.431125 377.729113
15 8704.0 630.153861 394.515569 380.502740
16 9216.0 609.322328 403.989025 383.002605
17 9728.0 591.817503 408.524944 382.427505
18 10240.0 570.871075 408.578556 381.911416
19 10752.0 551.384634 411.559798 380.601764
20 11264.0 537.982106 404.089694 374.686074
21 11776.0 524.835658 409.599991 378.345375
22 12288.0 518.754611 413.911572 383.251457
23 12800.0 505.679014 410.420828 377.163903
24 13312.0 495.330249 404.927765 376.976995
25 13824.0 482.934503 410.359948 378.739711
26 14336.0 471.967074 397.532043 372.969090
27 14848.0 461.297068 403.341254 375.304904
28 15360.0 454.269882 406.887417 378.092307
29 15872.0 447.098578 409.599996 377.343238
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
@triton.jit
def _layer_norm_fwd_fused(
Out,
A,
Weight,
Bias,
Mean, Rstd,
stride, N, eps,
BLOCK_SIZE: tl.constexpr,
):
# position of elements processed by this program
row = tl.program_id(0)
Out += row * stride
A += row * stride
# compute mean
mean = 0
_mean = tl.zeros([BLOCK_SIZE], dtype=tl.float32)
for off in range(0, N, BLOCK_SIZE):
cols = off + tl.arange(0, BLOCK_SIZE)
a = tl.load(A + cols, mask=cols < N, other=0., eviction_policy="evict_last").to(tl.float32)
_mean += a
mean = tl.sum(_mean, axis=0) / N
# compute variance
_var = tl.zeros([BLOCK_SIZE], dtype=tl.float32)
for off in range(0, N, BLOCK_SIZE):
cols = off + tl.arange(0, BLOCK_SIZE)
a = tl.load(A + cols, mask=cols < N, other=0., eviction_policy="evict_last").to(tl.float32)
a = tl.where(cols < N, a - mean, 0.)
_var += a * a
var = tl.sum(_var, axis=0) / N
rstd = 1 / tl.sqrt(var + eps)
# write-back mean/rstd
tl.store(Mean + row, mean)
tl.store(Rstd + row, rstd)
# multiply by weight and add bias
for off in range(0, N, BLOCK_SIZE):
cols = off + tl.arange(0, BLOCK_SIZE)
mask = cols < N
weight = tl.load(Weight + cols, mask=mask)
bias = tl.load(Bias + cols, mask=mask)
a = tl.load(A + cols, mask=mask, other=0., eviction_policy="evict_first").to(tl.float32)
a_hat = (a - mean) * rstd
out = a_hat * weight + bias
# # write-back
tl.store(Out + cols, out, mask=mask)
# Backward pass (DA + partial DW + partial DB)
@triton.jit
def _layer_norm_bwd_dx_fused(
_DA,
_DOut,
_A,
Weight,
Mean, Rstd,
stride, NumRows, NumCols, eps,
BLOCK_SIZE_N: tl.constexpr,
):
# position of elements processed by this program
pid = tl.program_id(0)
row = pid
A = _A + row * stride
DOut = _DOut + row * stride
DA = _DA + row * stride
mean = tl.load(Mean + row)
rstd = tl.load(Rstd + row)
# load data to SRAM
_mean1 = tl.zeros([BLOCK_SIZE_N], dtype=tl.float32)
_mean2 = tl.zeros([BLOCK_SIZE_N], dtype=tl.float32)
for off in range(0, NumCols, BLOCK_SIZE_N):
cols = off + tl.arange(0, BLOCK_SIZE_N)
mask = cols < NumCols
a = tl.load(A + cols, mask=mask, other=0).to(tl.float32)
dout = tl.load(DOut + cols, mask=mask, other=0).to(tl.float32)
weight = tl.load(Weight + cols, mask=mask, other=0).to(tl.float32)
a_hat = (a - mean) * rstd
wdout = weight * dout
_mean1 += a_hat * wdout
_mean2 += wdout
mean1 = tl.sum(_mean1, axis=0) / NumCols
mean2 = 0.
mean2 = tl.sum(_mean2, axis=0) / NumCols
for off in range(0, NumCols, BLOCK_SIZE_N):
cols = off + tl.arange(0, BLOCK_SIZE_N)
mask = cols < NumCols
a = tl.load(A + cols, mask=mask, other=0).to(tl.float32)
dout = tl.load(DOut + cols, mask=mask, other=0).to(tl.float32)
weight = tl.load(Weight + cols, mask=mask, other=0).to(tl.float32)
a_hat = (a - mean) * rstd
wdout = weight * dout
da = (wdout - (a_hat * mean1 + mean2)) * rstd
# write-back dx
tl.store(DA + cols, da, mask=mask)
# Backward pass (total DW + total DB)
@triton.jit
def _layer_norm_bwd_dwdb(
A, DOut,
Mean, Var,
DW,
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, :]
a = tl.load(A + offs, mask=mask, other=0.).to(tl.float32)
dout = tl.load(DOut + offs, mask=mask, other=0.).to(tl.float32)
mean = tl.load(Mean + rows, mask=rows < M, other=0.)
rstd = tl.load(Var + rows, mask=rows < M, other=0.)
a_hat = (a - mean[:, None]) * rstd[:, None]
dw += dout * a_hat
db += dout
sum_dw = tl.sum(dw, axis=0)
sum_db = tl.sum(db, axis=0)
tl.store(DW + cols, sum_dw, mask=cols < N)
tl.store(DB + cols, sum_db, mask=cols < N)
class LayerNorm(torch.autograd.Function):
@staticmethod
def forward(ctx, a, normalized_shape, weight, bias, eps):
# allocate output
out = torch.empty_like(a)
# reshape input data into 2D tensor
a_arg = a.reshape(-1, a.shape[-1])
M, N = a_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 // a.element_size()
BLOCK_SIZE = min(MAX_FUSED_SIZE, triton.next_power_of_2(N))
BLOCK_SIZE = max(BLOCK_SIZE, 128)
BLOCK_SIZE = min(BLOCK_SIZE, 4096)
# heuristics for number of warps
num_warps = min(max(BLOCK_SIZE // 256, 1), 8)
_layer_norm_fwd_fused[(M,)](
out,
a_arg,
weight,
bias,
mean, rstd,
a_arg.stride(0), N, eps,
BLOCK_SIZE=BLOCK_SIZE,
num_warps=num_warps,
)
ctx.save_for_backward(
a, weight, bias, mean, rstd,
)
ctx.BLOCK_SIZE = BLOCK_SIZE
ctx.num_warps = num_warps
ctx.eps = eps
if hasattr(bias, "config"):
assert bias.config.grad_scale_name == weight.config.grad_scale_name
grad_scale_name = bias.config.grad_scale_name
else:
grad_scale_name = None
ctx.grad_scale_gain_bias_name = grad_scale_name
return out
@staticmethod
def backward(ctx, dout):
assert dout.is_contiguous()
a, weight, bias, mean, var = ctx.saved_tensors
# heuristics for amount of parallel reduction stream for DG/DB
N = weight.shape[0]
# allocate output
da = torch.empty_like(dout)
# enqueue kernel using forward pass heuristics
# also compute partial sums for DW and DB
x_arg = a.reshape(-1, a.shape[-1])
M, N = x_arg.shape
dweight = torch.empty((weight.shape[0],), dtype=weight.dtype, device=weight.device)
dbias = torch.empty((weight.shape[0],), dtype=weight.dtype, device=weight.device)
_layer_norm_bwd_dx_fused[(M,)](
da,
dout,
a,
weight,
mean, var,
x_arg.stride(0), M, N,
ctx.eps,
BLOCK_SIZE_N=ctx.BLOCK_SIZE,
num_warps=ctx.num_warps,
)
# accumulate partial sums in separate kernel
grid = lambda meta: [triton.cdiv(N, meta["BLOCK_SIZE_N"])]
_layer_norm_bwd_dwdb[grid](
a, dout,
mean, var,
dweight,
dbias,
M,
N,
BLOCK_SIZE_M=32,
BLOCK_SIZE_N=128,
)
return (da, None, dweight, dbias, None, None,
None, None, None, None,
None,
None, None, None,
None,
None, None, None,
None, None, None,
None, None, None)
def layer_norm(a, normalized_shape, weight, bias, eps):
return LayerNorm.apply(a, normalized_shape, weight, bias, eps)
def test_layer_norm(M, N, dtype, eps=1e-5, device='cuda'):
torch.manual_seed(0)
# 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',
args={'M': 4096, 'dtype': torch.float16, 'mode': 'forward'}
)
)
def bench_layer_norm(M, N, dtype, provider, mode, 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)
# test_layer_norm(1151, 8192, torch.float16)
bench_layer_norm.run(save_path='.', print_data=True)
Total running time of the script: ( 5 minutes 23.495 seconds)