Dive into Deep Learning
Table Of Contents
Dive into Deep Learning
Table Of Contents

6.10. Networks with Parallel Concatenations (GoogLeNet)

During the ImageNet Challenge in 2014, a new architecture emerged that outperformed the rest. Szegedy et al., 2014 proposed a structure that combined the strengths of the NiN and repeated blocks paradigms. At its heart was the rather pragmatic answer to the question as to which size of convolution is ideal for processing. After all, we have a smorgasbord of choices, \(1 \times 1\) or \(3 \times 3\), \(5 \times 5\) or even larger. And it isn’t always clear which one is the best. As it turns out, the answer is that a combination of all the above works best. Over the next few years, researchers made several improvements to GoogLeNet. In this section, we will introduce the first version of this model series in a slightly simplified form - we omit the peculiarities that were added to stabilize training, due to the availability of better training algorithms.

6.10.1. Inception Blocks

The basic convolutional block in GoogLeNet is called an Inception block, named after the movie of the same name. This basic block is more complex in structure than the NiN block described in the previous section.

Structure of the Inception block.

Fig. 6.13 Structure of the Inception block.

As can be seen in the figure above, there are four parallel paths in the Inception block. 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 will perform a \(1\times 1\) convolution on the input to reduce the number of input channels, so as to reduce the model’s complexity. The fourth path uses the \(3\times 3\) maximum pooling layer, followed by the \(1\times 1\) 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, we concatenate the output of each path on the channel dimension and input it to the next layer. The customizable parameters of the Inception block are the number of output channels per layer, which can be used to control the model complexity.

In [1]:
import sys
sys.path.insert(0, '..')

import d2l
from mxnet import gluon, init, nd
from mxnet.gluon import nn

class Inception(nn.Block):
    # c1 - c4 are the number of output channels for each layer in the 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')
        self.p2_2 = nn.Conv2D(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(c3[0], kernel_size=1, activation='relu')
        self.p3_2 = nn.Conv2D(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_1 = nn.MaxPool2D(pool_size=3, strides=1, padding=1)
        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 nd.concat(p1, p2, p3, p4, dim=1)

To understand why this works as well as it does, 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).

6.10.2. GoogLeNet Model

GoogLeNet uses an initial long range feature convolution, a stack of a total of 9 inception blocks and global average pooling to generate its estimates. Maximum pooling between inception blocks reduced the dimensionality. The first part is identical to AlexNet and LeNet, the stack of blocks is inherited from VGG and the global average pooling that avoids a stack of fully connected layers at the end. The architecture is depicted below.

Full GoogLeNet Model

Fig. 6.14 Full GoogLeNet Model

Let’s build the network piece by piece. The first block uses a 64-channel 7×7 convolutional layer.

In [2]:
b1 = nn.Sequential()
b1.add(nn.Conv2D(64, kernel_size=7, strides=2, padding=3, activation='relu'),
       nn.MaxPool2D(pool_size=3, strides=2, padding=1))

The second block 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.

In [3]:
b2 = nn.Sequential()
b2.add(nn.Conv2D(64, kernel_size=1),
       nn.Conv2D(192, kernel_size=3, padding=1),
       nn.MaxPool2D(pool_size=3, strides=2, padding=1))

The third block 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 ratio to the output channels of 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 ratio to the number of output channels per path 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.

In [4]:
b3 = nn.Sequential()
b3.add(Inception(64, (96, 128), (16, 32), 32),
       Inception(128, (128, 192), (32, 96), 64),
       nn.MaxPool2D(pool_size=3, strides=2, padding=1))

The fourth block 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 the ratio. These ratios are slightly different in different Inception blocks.

In [5]:
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),
       nn.MaxPool2D(pool_size=3, strides=2, padding=1))

The fifth block 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.

In [6]:
b5 = nn.Sequential()
b5.add(Inception(256, (160, 320), (32, 128), 128),
       Inception(384, (192, 384), (48, 128), 128),

net = nn.Sequential()
net.add(b1, b2, b3, b4, b5, nn.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 is demonstrated below.

In [7]:
X = nd.random.uniform(shape=(1, 1, 96, 96))
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)

6.10.3. Data Acquisition and 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.

In [8]:
lr, num_epochs, batch_size, ctx = 0.1, 5, 128, d2l.try_gpu()
net.initialize(force_reinit=True, ctx=ctx, init=init.Xavier())
trainer = gluon.Trainer(net.collect_params(), 'sgd', {'learning_rate': lr})
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size, resize=96)
d2l.train_ch5(net, train_iter, test_iter, batch_size, trainer, ctx,
training on gpu(0)
epoch 1, loss 1.7410, train acc 0.360, test acc 0.709, time 84.0 sec
epoch 2, loss 0.6428, train acc 0.760, test acc 0.832, time 74.2 sec
epoch 3, loss 0.4411, train acc 0.833, test acc 0.855, time 74.2 sec
epoch 4, loss 0.3766, train acc 0.857, test acc 0.870, time 74.2 sec
epoch 5, loss 0.3340, train acc 0.874, test acc 0.873, time 74.5 sec

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

6.10.5. Exercises

  1. There are several iterations of GoogLeNet. Try to implement and run them. Some of them include the following:
    • Add a batch normalization layer, as described later in this chapter [2].
    • Make adjustments to the Inception block [3].
    • Use “label smoothing” for model regularization [3].
    • Include it in the residual connection, as described later in this chapter [4].
  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 large range convolution initially?

6.10.6. References

[1] Szegedy, C., Liu, W., Jia, Y., Sermanet, P., Reed, S., & Anguelov, D. & Rabinovich, A. (2015). Going deeper with convolutions. In Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 1-9).

[2] Ioffe, S., & Szegedy, C. (2015). Batch normalization: Accelerating deep network training by reducing internal covariate shift. arXiv preprint arXiv:1502.03167.

[3] Szegedy, C., Vanhoucke, V., Ioffe, S., Shlens, J., & Wojna, Z. (2016). Rethinking the inception architecture for computer vision. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (pp. 2818-2826).

[4] Szegedy, C., Ioffe, S., Vanhoucke, V., & Alemi, A. A. (2017, February). Inception-v4, inception-resnet and the impact of residual connections on learning. In Proceedings of the AAAI Conference on Artificial Intelligence (Vol. 4, p. 12).

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