# 7.4. Networks with Parallel Concatenations (GoogLeNet)¶ Open the notebook in Colab Open the notebook in Colab Open the notebook in Colab

In 2014, GoogLeNet won the ImageNet Challenge, proposing a structure that combined the strengths of NiN and paradigms of repeated blocks [Szegedy et al., 2015]. One focus of the paper was to address the question of which sized convolution kernels are best. After all, previous popular networks employed choices as small as $$1 \times 1$$ and as large as $$11 \times 11$$. One insight in this paper was that sometimes it can be advantageous to employ a combination of variously-sized kernels. In this section, we will introduce GoogLeNet, presenting a slightly simplified version of the original model: we omit a few ad-hoc features that were added to stabilize training but are unnecessary now with better training algorithms available.

## 7.4.1. Inception Blocks¶

The basic convolutional block in GoogLeNet is called an Inception block, likely named due to a quote from the movie Inception (“We Need To Go Deeper”), which launched a viral meme.

Fig. 7.4.1 Structure of the Inception block.

As depicted in Fig. 7.4.1, the inception block consists of four parallel paths. The first three paths use convolutional layers with window sizes of $$1\times 1$$, $$3\times 3$$, and $$5\times 5$$ to extract information from different spatial sizes. The middle two paths perform a $$1\times 1$$ convolution on the input to reduce the number of channels, reducing the model’s complexity. The fourth path uses a $$3\times 3$$ maximum pooling layer, followed by a $$1\times 1$$ convolutional layer to change the number of channels. The four paths all use appropriate padding to give the input and output the same height and width. Finally, the outputs along each path are concatenated along the channel dimension and comprise the block’s output. The commonly-tuned hyperparameters of the Inception block are the number of output channels per layer.

from d2l import mxnet as d2l
from mxnet import np, npx
from mxnet.gluon import nn
npx.set_np()

class Inception(nn.Block):
# c1--c4 are the number of output channels for each path
def __init__(self, c1, c2, c3, c4, **kwargs):
super(Inception, self).__init__(**kwargs)
# Path 1 is a single 1 x 1 convolutional layer
self.p1_1 = nn.Conv2D(c1, kernel_size=1, activation='relu')
# Path 2 is a 1 x 1 convolutional layer followed by a 3 x 3
# convolutional layer
self.p2_1 = nn.Conv2D(c2[0], kernel_size=1, activation='relu')
activation='relu')
# Path 3 is a 1 x 1 convolutional layer followed by a 5 x 5
# convolutional layer
self.p3_1 = nn.Conv2D(c3[0], kernel_size=1, activation='relu')
activation='relu')
# Path 4 is a 3 x 3 maximum pooling layer followed by a 1 x 1
# convolutional layer
self.p4_2 = nn.Conv2D(c4, kernel_size=1, activation='relu')

def forward(self, x):
p1 = self.p1_1(x)
p2 = self.p2_2(self.p2_1(x))
p3 = self.p3_2(self.p3_1(x))
p4 = self.p4_2(self.p4_1(x))
# Concatenate the outputs on the channel dimension
return np.concatenate((p1, p2, p3, p4), axis=1)

from d2l import torch as d2l
import torch
from torch import nn
from torch.nn import functional as F

class Inception(nn.Module):
# c1--c4 are the number of output channels for each path
def __init__(self, in_channels, c1, c2, c3, c4, **kwargs):
super(Inception, self).__init__(**kwargs)
# Path 1 is a single 1 x 1 convolutional layer
self.p1_1 = nn.Conv2d(in_channels, c1, kernel_size=1)
# Path 2 is a 1 x 1 convolutional layer followed by a 3 x 3
# convolutional layer
self.p2_1 = nn.Conv2d(in_channels, c2[0], kernel_size=1)
self.p2_2 = nn.Conv2d(c2[0], c2[1], kernel_size=3, padding=1)
# Path 3 is a 1 x 1 convolutional layer followed by a 5 x 5
# convolutional layer
self.p3_1 = nn.Conv2d(in_channels, c3[0], kernel_size=1)
self.p3_2 = nn.Conv2d(c3[0], c3[1], kernel_size=5, padding=2)
# Path 4 is a 3 x 3 maximum pooling layer followed by a 1 x 1
# convolutional layer
self.p4_2 = nn.Conv2d(in_channels, c4, kernel_size=1)

def forward(self, x):
p1 = F.relu(self.p1_1(x))
p2 = F.relu(self.p2_2(F.relu(self.p2_1(x))))
p3 = F.relu(self.p3_2(F.relu(self.p3_1(x))))
p4 = F.relu(self.p4_2(self.p4_1(x)))
# Concatenate the outputs on the channel dimension

from d2l import tensorflow as d2l
import tensorflow as tf

class Inception(tf.keras.Model):
# c1--c4 are the number of output channels for each path
def __init__(self, c1, c2, c3, c4):
super().__init__()
# Path 1 is a single 1 x 1 convolutional layer
self.p1_1 = tf.keras.layers.Conv2D(c1, 1, activation='relu')
# Path 2 is a 1 x 1 convolutional layer followed by a 3 x 3
# convolutional layer
self.p2_1 = tf.keras.layers.Conv2D(c2[0], 1, activation='relu')
activation='relu')
# Path 3 is a 1 x 1 convolutional layer followed by a 5 x 5
# convolutional layer
self.p3_1 = tf.keras.layers.Conv2D(c3[0], 1, activation='relu')
activation='relu')
# Path 4 is a 3 x 3 maximum pooling layer followed by a 1 x 1
# convolutional layer
self.p4_2 = tf.keras.layers.Conv2D(c4, 1, activation='relu')

def call(self, x):
p1 = self.p1_1(x)
p2 = self.p2_2(self.p2_1(x))
p3 = self.p3_2(self.p3_1(x))
p4 = self.p4_2(self.p4_1(x))
# Concatenate the outputs on the channel dimension
return tf.keras.layers.Concatenate()([p1, p2, p3, p4])


To gain some intuition for why this network works so well, consider the combination of the filters. They explore the image in varying ranges. This means that details at different extents can be recognized efficiently by different filters. At the same time, we can allocate different amounts of parameters for different ranges (e.g., more for short range but not ignore the long range entirely).

As shown in Fig. 7.4.2, GoogLeNet uses a stack of a total of 9 inception blocks and global average pooling to generate its estimates. Maximum pooling between inception blocks reduces the dimensionality. The first module is similar to AlexNet and LeNet. The stack of blocks is inherited from VGG and the global average pooling avoids a stack of fully-connected layers at the end.

We can now implement GoogLeNet piece by piece. The first module uses a 64-channel $$7\times 7$$ convolutional layer.

b1 = nn.Sequential()

b1 = nn.Sequential(nn.Conv2d(1, 64, kernel_size=7, stride=2, padding=3),
nn.ReLU(),

def b1():
return tf.keras.models.Sequential([
activation='relu'),


The second module uses two convolutional layers: first, a 64-channel $$1\times 1$$ convolutional layer, then a $$3\times 3$$ convolutional layer that triples the number of channels. This corresponds to the second path in the Inception block.

b2 = nn.Sequential()

b2 = nn.Sequential(nn.Conv2d(64, 64, kernel_size=1),
nn.ReLU(),

def b2():
return tf.keras.Sequential([
tf.keras.layers.Conv2D(64, 1, activation='relu'),


The third module connects two complete Inception blocks in series. The number of output channels of the first Inception block is $$64+128+32+32=256$$, and the number-of-output-channel ratio among the four paths is $$64:128:32:32=2:4:1:1$$. The second and third paths first reduce the number of input channels to $$96/192=1/2$$ and $$16/192=1/12$$, respectively, and then connect the second convolutional layer. The number of output channels of the second Inception block is increased to $$128+192+96+64=480$$, and the number-of-output-channel ratio among the four paths is $$128:192:96:64 = 4:6:3:2$$. The second and third paths first reduce the number of input channels to $$128/256=1/2$$ and $$32/256=1/8$$, respectively.

b3 = nn.Sequential()
b3.add(Inception(64, (96, 128), (16, 32), 32),
Inception(128, (128, 192), (32, 96), 64),

b3 = nn.Sequential(Inception(192, 64, (96, 128), (16, 32), 32),
Inception(256, 128, (128, 192), (32, 96), 64),

def b3():
return tf.keras.models.Sequential([
Inception(64, (96, 128), (16, 32), 32),
Inception(128, (128, 192), (32, 96), 64),


The fourth module is more complicated. It connects five Inception blocks in series, and they have $$192+208+48+64=512$$, $$160+224+64+64=512$$, $$128+256+64+64=512$$, $$112+288+64+64=528$$, and $$256+320+128+128=832$$ output channels, respectively. The number of channels assigned to these paths is similar to that in the third module: the second path with the $$3\times 3$$ convolutional layer outputs the largest number of channels, followed by the first path with only the $$1\times 1$$ convolutional layer, the third path with the $$5\times 5$$ convolutional layer, and the fourth path with the $$3\times 3$$ maximum pooling layer. The second and third paths will first reduce the number of channels according to the ratio. These ratios are slightly different in different Inception blocks.

b4 = nn.Sequential()
b4.add(Inception(192, (96, 208), (16, 48), 64),
Inception(160, (112, 224), (24, 64), 64),
Inception(128, (128, 256), (24, 64), 64),
Inception(112, (144, 288), (32, 64), 64),
Inception(256, (160, 320), (32, 128), 128),

b4 = nn.Sequential(Inception(480, 192, (96, 208), (16, 48), 64),
Inception(512, 160, (112, 224), (24, 64), 64),
Inception(512, 128, (128, 256), (24, 64), 64),
Inception(512, 112, (144, 288), (32, 64), 64),
Inception(528, 256, (160, 320), (32, 128), 128),

def b4():
return tf.keras.Sequential([
Inception(192, (96, 208), (16, 48), 64),
Inception(160, (112, 224), (24, 64), 64),
Inception(128, (128, 256), (24, 64), 64),
Inception(112, (144, 288), (32, 64), 64),
Inception(256, (160, 320), (32, 128), 128),


The fifth module has two Inception blocks with $$256+320+128+128=832$$ and $$384+384+128+128=1024$$ output channels. The number of channels assigned to each path is the same as that in the third and fourth modules, but differs in specific values. It should be noted that the fifth block is followed by the output layer. This block uses the global average pooling layer to change the height and width of each channel to 1, just as in NiN. Finally, we turn the output into a two-dimensional array followed by a fully-connected layer whose number of outputs is the number of label classes.

b5 = nn.Sequential()
b5.add(Inception(256, (160, 320), (32, 128), 128),
Inception(384, (192, 384), (48, 128), 128),
nn.GlobalAvgPool2D())

net = nn.Sequential()
net.add(b1, b2, b3, b4, b5, nn.Dense(10))

b5 = nn.Sequential(Inception(832, 256, (160, 320), (32, 128), 128),
Inception(832, 384, (192, 384), (48, 128), 128),
nn.Flatten())

net = nn.Sequential(b1, b2, b3, b4, b5, nn.Linear(1024, 10))

def b5():
return tf.keras.Sequential([
Inception(256, (160, 320), (32, 128), 128),
Inception(384, (192, 384), (48, 128), 128),
tf.keras.layers.GlobalAvgPool2D(),
tf.keras.layers.Flatten()
])
# Recall that this has to be a function that will be passed to
# d2l.train_ch6() so that model building/compiling need to be within
# strategy.scope() in order to utilize the CPU/GPU devices that we have
def net():
return tf.keras.Sequential([b1(), b2(), b3(), b4(), b5(),
tf.keras.layers.Dense(10)])


The GoogLeNet model is computationally complex, so it is not as easy to modify the number of channels as in VGG. To have a reasonable training time on Fashion-MNIST, we reduce the input height and width from 224 to 96. This simplifies the computation. The changes in the shape of the output between the various modules are demonstrated below.

X = np.random.uniform(size=(1, 1, 96, 96))
net.initialize()
for layer in net:
X = layer(X)
print(layer.name, 'output shape:\t', X.shape)

sequential0 output shape:    (1, 64, 24, 24)
sequential1 output shape:    (1, 192, 12, 12)
sequential2 output shape:    (1, 480, 6, 6)
sequential3 output shape:    (1, 832, 3, 3)
sequential4 output shape:    (1, 1024, 1, 1)
dense0 output shape:         (1, 10)

X = torch.rand(size=(1, 1, 96, 96))
for layer in net:
X = layer(X)
print(layer.__class__.__name__,'output shape:\t', X.shape)

Sequential output shape:     torch.Size([1, 64, 24, 24])
Sequential output shape:     torch.Size([1, 192, 12, 12])
Sequential output shape:     torch.Size([1, 480, 6, 6])
Sequential output shape:     torch.Size([1, 832, 3, 3])
Sequential output shape:     torch.Size([1, 1024])
Linear output shape:         torch.Size([1, 10])

X = tf.random.uniform(shape=(1, 96, 96, 1))
for layer in net().layers:
X = layer(X)
print(layer.__class__.__name__, 'output shape:\t', X.shape)

Sequential output shape:     (1, 24, 24, 64)
Sequential output shape:     (1, 12, 12, 192)
Sequential output shape:     (1, 6, 6, 480)
Sequential output shape:     (1, 3, 3, 832)
Sequential output shape:     (1, 1024)
Dense output shape:  (1, 10)


## 7.4.3. Training¶

As before, we train our model using the Fashion-MNIST dataset. We transform it to $$96 \times 96$$ pixel resolution before invoking the training procedure.

lr, num_epochs, batch_size = 0.1, 10, 128
d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr)

loss 0.239, train acc 0.909, test acc 0.901
2237.2 examples/sec on gpu(0)

lr, num_epochs, batch_size = 0.1, 10, 128
d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr)

loss 0.247, train acc 0.905, test acc 0.879
2773.2 examples/sec on cuda:0

lr, num_epochs, batch_size = 0.1, 10, 128
d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr)

loss 0.232, train acc 0.913, test acc 0.901
3472.7 examples/sec on /GPU:0

<tensorflow.python.keras.engine.sequential.Sequential at 0x7fd5f53067d0>


## 7.4.4. Summary¶

• The Inception block is equivalent to a subnetwork with four paths. It extracts information in parallel through convolutional layers of different window shapes and maximum pooling layers. $$1 \times 1$$ convolutions reduce channel dimensionality on a per-pixel level. Maximum pooling reduces the resolution.

• GoogLeNet connects multiple well-designed Inception blocks with other layers in series. The ratio of the number of channels assigned in the Inception block is obtained through a large number of experiments on the ImageNet dataset.

• GoogLeNet, as well as its succeeding versions, was one of the most efficient models on ImageNet, providing similar test accuracy with lower computational complexity.

## 7.4.5. Exercises¶

1. There are several iterations of GoogLeNet. Try to implement and run them. Some of them include the following:

2. What is the minimum image size for GoogLeNet to work?

3. Compare the model parameter sizes of AlexNet, VGG, and NiN with GoogLeNet. How do the latter two network architectures significantly reduce the model parameter size?

4. Why do we need a long range convolution initially?