Note

Click here to download the full example code

# Fusing Convolution and Batch Norm using Custom Function¶

Fusing adjacent convolution and batch norm layers together is typically an inference-time optimization to improve run-time. It is usually achieved by eliminating the batch norm layer entirely and updating the weight and bias of the preceding convolution [0]. However, this technique is not applicable for training models.

In this tutorial, we will show a different technique to fuse the two layers that can be applied during training. Rather than improved runtime, the objective of this optimization is to reduce memory usage.

The idea behind this optimization is to see that both convolution and batch norm (as well as many other ops) need to save a copy of their input during forward for the backward pass. For large batch sizes, these saved inputs are responsible for most of your memory usage, so being able to avoid allocating another input tensor for every convolution batch norm pair can be a significant reduction.

In this tutorial, we avoid this extra allocation by combining convolution and batch norm into a single layer (as a custom function). In the forward of this combined layer, we perform normal convolution and batch norm as-is, with the only difference being that we will only save the inputs to the convolution. To obtain the input of batch norm, which is necessary to backward through it, we recompute convolution forward again during the backward pass.

It is important to note that the usage of this optimization is situational.
Though (by avoiding one buffer saved) we always reduce the memory allocated at
the end of the forward pass, there are cases when the *peak* memory allocated
may not actually be reduced. See the final section for more details.

For simplicity, in this tutorial we hardcode bias=False, stride=1, padding=0, dilation=1, and groups=1 for Conv2D. For BatchNorm2D, we hardcode eps=1e-3, momentum=0.1, affine=False, and track_running_statistics=False. Another small difference is that we add epsilon in the denominator outside of the square root in the computation of batch norm.

[0] https://nenadmarkus.com/p/fusing-batchnorm-and-conv/

## Backward Formula Implementation for Convolution¶

Implementing a custom function requires us to implement the backward ourselves. In this case, we need both the backward formulas for Conv2D and BatchNorm2D. Eventually we’d chain them together in our unified backward function, but below we first implement them as their own custom functions so we can validate their correctness individually

```
import torch
from torch.autograd.function import once_differentiable
import torch.nn.functional as F
def convolution_backward(grad_out, X, weight):
grad_input = F.conv2d(X.transpose(0, 1), grad_out.transpose(0, 1)).transpose(0, 1)
grad_X = F.conv_transpose2d(grad_out, weight)
return grad_X, grad_input
class Conv2D(torch.autograd.Function):
@staticmethod
def forward(ctx, X, weight):
ctx.save_for_backward(X, weight)
return F.conv2d(X, weight)
# Use @once_differentiable by default unless we intend to double backward
@staticmethod
@once_differentiable
def backward(ctx, grad_out):
X, weight = ctx.saved_tensors
return convolution_backward(grad_out, X, weight)
```

When testing with `gradcheck`

, it is important to use double precision

```
weight = torch.rand(5, 3, 3, 3, requires_grad=True, dtype=torch.double)
X = torch.rand(10, 3, 7, 7, requires_grad=True, dtype=torch.double)
torch.autograd.gradcheck(Conv2D.apply, (X, weight))
```

```
True
```

## Backward Formula Implementation for Batch Norm¶

Batch Norm has two modes: training and `eval`

mode. In training mode
the sample statistics are a function of the inputs. In `eval`

mode,
we use the saved running statistics, which are not a function of the inputs.
This makes non-training mode’s backward significantly simpler. Below
we implement and test only the training mode case.

```
def unsqueeze_all(t):
# Helper function to ``unsqueeze`` all the dimensions that we reduce over
return t[None, :, None, None]
def batch_norm_backward(grad_out, X, sum, sqrt_var, N, eps):
# We use the formula: ``out = (X - mean(X)) / (sqrt(var(X)) + eps)``
# in batch norm 2D forward. To simplify our derivation, we follow the
# chain rule and compute the gradients as follows before accumulating
# them all into a final grad_input.
# 1) ``grad of out wrt var(X)`` * ``grad of var(X) wrt X``
# 2) ``grad of out wrt mean(X)`` * ``grad of mean(X) wrt X``
# 3) ``grad of out wrt X in the numerator`` * ``grad of X wrt X``
# We then rewrite the formulas to use as few extra buffers as possible
tmp = ((X - unsqueeze_all(sum) / N) * grad_out).sum(dim=(0, 2, 3))
tmp *= -1
d_denom = tmp / (sqrt_var + eps)**2 # ``d_denom = -num / denom**2``
# It is useful to delete tensors when you no longer need them with ``del``
# For example, we could've done ``del tmp`` here because we won't use it later
# In this case, it's not a big difference because ``tmp`` only has size of (C,)
# The important thing is avoid allocating NCHW-sized tensors unnecessarily
d_var = d_denom / (2 * sqrt_var) # ``denom = torch.sqrt(var) + eps``
# Compute ``d_mean_dx`` before allocating the final NCHW-sized grad_input buffer
d_mean_dx = grad_out / unsqueeze_all(sqrt_var + eps)
d_mean_dx = unsqueeze_all(-d_mean_dx.sum(dim=(0, 2, 3)) / N)
# ``d_mean_dx`` has already been reassigned to a C-sized buffer so no need to worry
# ``(1) unbiased_var(x) = ((X - unsqueeze_all(mean))**2).sum(dim=(0, 2, 3)) / (N - 1)``
grad_input = X * unsqueeze_all(d_var * N)
grad_input += unsqueeze_all(-d_var * sum)
grad_input *= 2 / ((N - 1) * N)
# (2) mean (see above)
grad_input += d_mean_dx
# (3) Add 'grad_out / <factor>' without allocating an extra buffer
grad_input *= unsqueeze_all(sqrt_var + eps)
grad_input += grad_out
grad_input /= unsqueeze_all(sqrt_var + eps) # ``sqrt_var + eps > 0!``
return grad_input
class BatchNorm(torch.autograd.Function):
@staticmethod
def forward(ctx, X, eps=1e-3):
# Don't save ``keepdim`` values for backward
sum = X.sum(dim=(0, 2, 3))
var = X.var(unbiased=True, dim=(0, 2, 3))
N = X.numel() / X.size(1)
sqrt_var = torch.sqrt(var)
ctx.save_for_backward(X)
ctx.eps = eps
ctx.sum = sum
ctx.N = N
ctx.sqrt_var = sqrt_var
mean = sum / N
denom = sqrt_var + eps
out = X - unsqueeze_all(mean)
out /= unsqueeze_all(denom)
return out
@staticmethod
@once_differentiable
def backward(ctx, grad_out):
X, = ctx.saved_tensors
return batch_norm_backward(grad_out, X, ctx.sum, ctx.sqrt_var, ctx.N, ctx.eps)
```

Testing with `gradcheck`

```
a = torch.rand(1, 2, 3, 4, requires_grad=True, dtype=torch.double)
torch.autograd.gradcheck(BatchNorm.apply, (a,), fast_mode=False)
```

```
True
```

## Fusing Convolution and BatchNorm¶

Now that the bulk of the work has been done, we can combine them together. Note that in (1) we only save a single buffer for backward, but this also means we recompute convolution forward in (5). Also see that in (2), (3), (4), and (6), it’s the same exact code as the examples above.

```
class FusedConvBN2DFunction(torch.autograd.Function):
@staticmethod
def forward(ctx, X, conv_weight, eps=1e-3):
assert X.ndim == 4 # N, C, H, W
# (1) Only need to save this single buffer for backward!
ctx.save_for_backward(X, conv_weight)
# (2) Exact same Conv2D forward from example above
X = F.conv2d(X, conv_weight)
# (3) Exact same BatchNorm2D forward from example above
sum = X.sum(dim=(0, 2, 3))
var = X.var(unbiased=True, dim=(0, 2, 3))
N = X.numel() / X.size(1)
sqrt_var = torch.sqrt(var)
ctx.eps = eps
ctx.sum = sum
ctx.N = N
ctx.sqrt_var = sqrt_var
mean = sum / N
denom = sqrt_var + eps
# Try to do as many things in-place as possible
# Instead of `out = (X - a) / b`, doing `out = X - a; out /= b`
# avoids allocating one extra NCHW-sized buffer here
out = X - unsqueeze_all(mean)
out /= unsqueeze_all(denom)
return out
@staticmethod
def backward(ctx, grad_out):
X, conv_weight, = ctx.saved_tensors
# (4) Batch norm backward
# (5) We need to recompute conv
X_conv_out = F.conv2d(X, conv_weight)
grad_out = batch_norm_backward(grad_out, X_conv_out, ctx.sum, ctx.sqrt_var,
ctx.N, ctx.eps)
# (6) Conv2d backward
grad_X, grad_input = convolution_backward(grad_out, X, conv_weight)
return grad_X, grad_input, None, None, None, None, None
```

The next step is to wrap our functional variant in a stateful nn.Module

```
import torch.nn as nn
import math
class FusedConvBN(nn.Module):
def __init__(self, in_channels, out_channels, kernel_size, exp_avg_factor=0.1,
eps=1e-3, device=None, dtype=None):
super(FusedConvBN, self).__init__()
factory_kwargs = {'device': device, 'dtype': dtype}
# Conv parameters
weight_shape = (out_channels, in_channels, kernel_size, kernel_size)
self.conv_weight = nn.Parameter(torch.empty(*weight_shape, **factory_kwargs))
# Batch norm parameters
num_features = out_channels
self.num_features = num_features
self.eps = eps
# Initialize
self.reset_parameters()
def forward(self, X):
return FusedConvBN2DFunction.apply(X, self.conv_weight, self.eps)
def reset_parameters(self) -> None:
nn.init.kaiming_uniform_(self.conv_weight, a=math.sqrt(5))
```

Use `gradcheck`

to validate the correctness of our backward formula

```
weight = torch.rand(5, 3, 3, 3, requires_grad=True, dtype=torch.double)
X = torch.rand(2, 3, 4, 4, requires_grad=True, dtype=torch.double)
torch.autograd.gradcheck(FusedConvBN2DFunction.apply, (X, weight))
```

```
True
```

## Testing out our new Layer¶

Use `FusedConvBN`

to train a basic network
The code below is after some light modifications to the example here:
https://github.com/pytorch/examples/tree/master/mnist

```
import torch.optim as optim
from torchvision import datasets, transforms
from torch.optim.lr_scheduler import StepLR
# Record memory allocated at the end of the forward pass
memory_allocated = [[],[]]
class Net(nn.Module):
def __init__(self, fused=True):
super(Net, self).__init__()
self.fused = fused
if fused:
self.convbn1 = FusedConvBN(1, 32, 3)
self.convbn2 = FusedConvBN(32, 64, 3)
else:
self.conv1 = nn.Conv2d(1, 32, 3, 1, bias=False)
self.bn1 = nn.BatchNorm2d(32, affine=False, track_running_stats=False)
self.conv2 = nn.Conv2d(32, 64, 3, 1, bias=False)
self.bn2 = nn.BatchNorm2d(64, affine=False, track_running_stats=False)
self.fc1 = nn.Linear(9216, 128)
self.dropout = nn.Dropout(0.5)
self.fc2 = nn.Linear(128, 10)
def forward(self, x):
if self.fused:
x = self.convbn1(x)
else:
x = self.conv1(x)
x = self.bn1(x)
F.relu_(x)
if self.fused:
x = self.convbn2(x)
else:
x = self.conv2(x)
x = self.bn2(x)
F.relu_(x)
x = F.max_pool2d(x, 2)
F.relu_(x)
x = x.flatten(1)
x = self.fc1(x)
x = self.dropout(x)
F.relu_(x)
x = self.fc2(x)
output = F.log_softmax(x, dim=1)
if fused:
memory_allocated[0].append(torch.cuda.memory_allocated())
else:
memory_allocated[1].append(torch.cuda.memory_allocated())
return output
def train(model, device, train_loader, optimizer, epoch):
model.train()
for batch_idx, (data, target) in enumerate(train_loader):
data, target = data.to(device), target.to(device)
optimizer.zero_grad()
output = model(data)
loss = F.nll_loss(output, target)
loss.backward()
optimizer.step()
if batch_idx % 2 == 0:
print('Train Epoch: {} [{}/{} ({:.0f}%)]\tLoss: {:.6f}'.format(
epoch, batch_idx * len(data), len(train_loader.dataset),
100. * batch_idx / len(train_loader), loss.item()))
def test(model, device, test_loader):
model.eval()
test_loss = 0
correct = 0
# Use inference mode instead of no_grad, for free improved test-time performance
with torch.inference_mode():
for data, target in test_loader:
data, target = data.to(device), target.to(device)
output = model(data)
# sum up batch loss
test_loss += F.nll_loss(output, target, reduction='sum').item()
# get the index of the max log-probability
pred = output.argmax(dim=1, keepdim=True)
correct += pred.eq(target.view_as(pred)).sum().item()
test_loss /= len(test_loader.dataset)
print('\nTest set: Average loss: {:.4f}, Accuracy: {}/{} ({:.0f}%)\n'.format(
test_loss, correct, len(test_loader.dataset),
100. * correct / len(test_loader.dataset)))
use_cuda = torch.cuda.is_available()
device = torch.device("cuda" if use_cuda else "cpu")
train_kwargs = {'batch_size': 2048}
test_kwargs = {'batch_size': 2048}
if use_cuda:
cuda_kwargs = {'num_workers': 1,
'pin_memory': True,
'shuffle': True}
train_kwargs.update(cuda_kwargs)
test_kwargs.update(cuda_kwargs)
transform = transforms.Compose([
transforms.ToTensor(),
transforms.Normalize((0.1307,), (0.3081,))
])
dataset1 = datasets.MNIST('../data', train=True, download=True,
transform=transform)
dataset2 = datasets.MNIST('../data', train=False,
transform=transform)
train_loader = torch.utils.data.DataLoader(dataset1, **train_kwargs)
test_loader = torch.utils.data.DataLoader(dataset2, **test_kwargs)
```

```
Downloading http://yann.lecun.com/exdb/mnist/train-images-idx3-ubyte.gz
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Downloading http://yann.lecun.com/exdb/mnist/train-labels-idx1-ubyte.gz
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Extracting ../data/MNIST/raw/train-labels-idx1-ubyte.gz to ../data/MNIST/raw
Downloading http://yann.lecun.com/exdb/mnist/t10k-images-idx3-ubyte.gz
Downloading http://yann.lecun.com/exdb/mnist/t10k-images-idx3-ubyte.gz to ../data/MNIST/raw/t10k-images-idx3-ubyte.gz
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Extracting ../data/MNIST/raw/t10k-images-idx3-ubyte.gz to ../data/MNIST/raw
Downloading http://yann.lecun.com/exdb/mnist/t10k-labels-idx1-ubyte.gz
Downloading http://yann.lecun.com/exdb/mnist/t10k-labels-idx1-ubyte.gz to ../data/MNIST/raw/t10k-labels-idx1-ubyte.gz
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Extracting ../data/MNIST/raw/t10k-labels-idx1-ubyte.gz to ../data/MNIST/raw
```

## A Comparison of Memory Usage¶

If CUDA is enabled, print out memory usage for both fused=True and fused=False For an example run on NVIDIA GeForce RTX 3070, NVIDIA CUDA® Deep Neural Network library (cuDNN) 8.0.5: fused peak memory: 1.56GB, unfused peak memory: 2.68GB

It is important to note that the *peak* memory usage for this model may vary depending
the specific cuDNN convolution algorithm used. For shallower models, it
may be possible for the peak memory allocated of the fused model to exceed
that of the unfused model! This is because the memory allocated to compute
certain cuDNN convolution algorithms can be high enough to “hide” the typical peak
you would expect to be near the start of the backward pass.

For this reason, we also record and display the memory allocated at the end
of the forward pass as an approximation, and to demonstrate that we indeed
allocate one fewer buffer per fused `conv-bn`

pair.

```
from statistics import mean
torch.backends.cudnn.enabled = True
if use_cuda:
peak_memory_allocated = []
for fused in (True, False):
torch.manual_seed(123456)
model = Net(fused=fused).to(device)
optimizer = optim.Adadelta(model.parameters(), lr=1.0)
scheduler = StepLR(optimizer, step_size=1, gamma=0.7)
for epoch in range(1):
train(model, device, train_loader, optimizer, epoch)
test(model, device, test_loader)
scheduler.step()
peak_memory_allocated.append(torch.cuda.max_memory_allocated())
torch.cuda.reset_peak_memory_stats()
print("cuDNN version:", torch.backends.cudnn.version())
print()
print("Peak memory allocated:")
print(f"fused: {peak_memory_allocated[0]/1024**3:.2f}GB, unfused: {peak_memory_allocated[1]/1024**3:.2f}GB")
print("Memory allocated at end of forward pass:")
print(f"fused: {mean(memory_allocated[0])/1024**3:.2f}GB, unfused: {mean(memory_allocated[1])/1024**3:.2f}GB")
```

```
Train Epoch: 0 [0/60000 (0%)] Loss: 2.348850
Train Epoch: 0 [4096/60000 (7%)] Loss: 7.906003
Train Epoch: 0 [8192/60000 (13%)] Loss: 3.856603
Train Epoch: 0 [12288/60000 (20%)] Loss: 2.177455
Train Epoch: 0 [16384/60000 (27%)] Loss: 1.875102
Train Epoch: 0 [20480/60000 (33%)] Loss: 1.706392
Train Epoch: 0 [24576/60000 (40%)] Loss: 1.608641
Train Epoch: 0 [28672/60000 (47%)] Loss: 1.696115
Train Epoch: 0 [32768/60000 (53%)] Loss: 1.412414
Train Epoch: 0 [36864/60000 (60%)] Loss: 1.309023
Train Epoch: 0 [40960/60000 (67%)] Loss: 1.213390
Train Epoch: 0 [45056/60000 (73%)] Loss: 1.128412
Train Epoch: 0 [49152/60000 (80%)] Loss: 0.853271
Train Epoch: 0 [53248/60000 (87%)] Loss: 0.860089
Train Epoch: 0 [57344/60000 (93%)] Loss: 0.757735
Test set: Average loss: 0.3410, Accuracy: 9148/10000 (91%)
Train Epoch: 0 [0/60000 (0%)] Loss: 2.349130
Train Epoch: 0 [4096/60000 (7%)] Loss: 7.946079
Train Epoch: 0 [8192/60000 (13%)] Loss: 3.232958
Train Epoch: 0 [12288/60000 (20%)] Loss: 2.598448
Train Epoch: 0 [16384/60000 (27%)] Loss: 1.940814
Train Epoch: 0 [20480/60000 (33%)] Loss: 2.433658
Train Epoch: 0 [24576/60000 (40%)] Loss: 2.066980
Train Epoch: 0 [28672/60000 (47%)] Loss: 1.749354
Train Epoch: 0 [32768/60000 (53%)] Loss: 1.305229
Train Epoch: 0 [36864/60000 (60%)] Loss: 1.221748
Train Epoch: 0 [40960/60000 (67%)] Loss: 1.200668
Train Epoch: 0 [45056/60000 (73%)] Loss: 1.162338
Train Epoch: 0 [49152/60000 (80%)] Loss: 1.239552
Train Epoch: 0 [53248/60000 (87%)] Loss: 0.965104
Train Epoch: 0 [57344/60000 (93%)] Loss: 0.828900
Test set: Average loss: 0.5110, Accuracy: 8368/10000 (84%)
cuDNN version: 8902
Peak memory allocated:
fused: 4.44GB, unfused: 1.90GB
Memory allocated at end of forward pass:
fused: 0.99GB, unfused: 1.37GB
```

**Total running time of the script:** ( 0 minutes 21.896 seconds)