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 334.367358 520.126988
3 2560.0 694.237267 365.714281 518.481028
4 3072.0 712.347810 378.092307 496.484863
5 3584.0 725.873439 384.859062 451.527536
6 4096.0 728.177767 381.023256 455.111095
7 4608.0 670.254540 394.267384 421.302872
8 5120.0 688.403381 397.669909 424.455959
9 5632.0 698.542675 395.228063 411.470331
10 6144.0 702.171410 402.885254 411.313806
11 6656.0 700.631610 398.861429 400.360920
12 7168.0 690.891575 396.844306 387.459443
13 7680.0 678.895043 393.846167 386.415087
14 8192.0 636.271854 393.609605 372.363633
15 8704.0 627.315309 389.005597 380.502740
16 9216.0 606.814809 407.337026 383.999986
17 9728.0 587.350922 409.599987 382.427505
18 10240.0 564.965524 408.578556 382.803739
19 10752.0 547.872604 411.559798 381.445676
20 11264.0 533.207081 406.826188 373.134567
21 11776.0 520.486200 409.599991 377.587162
22 12288.0 514.680630 413.911572 383.251457
23 12800.0 504.433489 410.420828 376.470582
24 13312.0 494.180982 405.699062 376.976995
25 13824.0 482.934503 411.122660 379.389355
26 14336.0 471.967074 406.695045 374.185964
27 14848.0 461.297068 408.192434 375.304904
28 15360.0 454.269882 406.214870 378.092307
29 15872.0 447.098578 406.974373 376.783377
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)
UNROLL: tl.constexpr = 4
for i in range(0, M, BLOCK_SIZE_M * UNROLL):
for j in range(UNROLL):
rows = i + j * BLOCK_SIZE_M + 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,
)
if N > 10240:
BLOCK_SIZE_N = 128
BLOCK_SIZE_M = 32
num_warps = 4
else:
# maximize occupancy for small N
BLOCK_SIZE_N = 16
BLOCK_SIZE_M = 16
num_warps = 8
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=BLOCK_SIZE_M,
BLOCK_SIZE_N=BLOCK_SIZE_N,
num_warps=num_warps
)
return (da, None, dweight, dbias, 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 36.337 seconds)