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 481.882344
1 1536.0 630.153868 323.368435 511.999982
2 2048.0 668.734716 337.814445 520.126988
3 2560.0 694.237267 362.477870 512.000013
4 3072.0 702.171410 378.092307 501.551037
5 3584.0 725.873439 384.859062 458.751978
6 4096.0 728.177767 383.251446 451.972420
7 4608.0 670.254540 396.387087 428.651163
8 5120.0 688.403381 397.669909 424.455959
9 5632.0 698.542675 398.725657 411.470331
10 6144.0 697.191505 404.543206 409.600010
11 6656.0 700.631610 400.360920 400.360920
12 7168.0 686.754468 383.571898 383.571898
13 7680.0 682.666656 392.587863 386.415087
14 8192.0 639.375598 392.431125 373.424507
15 8704.0 624.502255 391.191007 380.502740
16 9216.0 604.327881 403.989025 382.010363
17 9728.0 585.142883 409.599987 383.369452
18 10240.0 564.965524 409.600010 382.803739
19 10752.0 546.133312 410.577576 380.601764
20 11264.0 530.070590 396.969169 367.804077
21 11776.0 519.052343 409.599991 377.587162
22 12288.0 513.336807 413.911572 383.251457
23 12800.0 504.433489 410.420828 377.163903
24 13312.0 494.180982 408.030638 376.976995
25 13824.0 481.882350 412.656711 379.389355
26 14336.0 471.967074 398.914774 371.158581
27 14848.0 461.297068 406.794504 375.304904
28 15360.0 454.269882 406.887417 377.511515
29 15872.0 447.887117 408.282944 376.225175
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 37.085 seconds)