C RESCENDO N ET : A S IMPLE D EEP C ONVOLUTIONAL N EURAL N ETWORK WITH E NSEMBLE B EHAVIOR
Anonymous authors
Paper under double-blind review
A BSTRACT
We introduce a new deep convolutional neural network, CrescendoNet, by stacking simple building blocks without residual connections.
Each Crescendo block contains independent convolution paths with increased depths.
The numbers of convolution layers and parameters are only increased linearly in Crescendo blocks.
In experiments, CrescendoNet with only 15 layers outperforms almost all networks without residual connections on benchmark datasets, CIFAR10,
CIFAR100, and SVHN.
Given sufﬁcient amount of data as in SVHN dataset CrescendoNet with 15 layers and 4.1M parameters can match the performance of DenseNet-BC with 250 layers and 15.3M parameters.
CrescendoNet provides a new way to construct high performance deep convolutional neural networks without residual connections.
Moreover, through investigating the behavior and performance of subnetworks in CrescendoNet, we note that the high performance of CrescendoNet may come from its implicit ensemble behavior, which differs from the FractalNet that is also a deep convolutional neural network without residual connections.
Furthermore, the independence between paths in CrescendoNet allows us to introduce a new path-wise training procedure, which can reduce the memory needed for training.
I NTRODUCTION
Deep convolutional neural networks (CNNs) have signiﬁcantly improved the performance of image classiﬁcation (Krizhevsky et al., 2012;
He et al., 2016a; Szegedy et al., 2015).
However, training a CNN also becomes increasingly difﬁcult with the network deepening.
One of important research efforts to overcome this difﬁculty is to develop new neural network architectures (Huang et al., 2016a; Larsson et al., 2017).
Recently, residual network (He et al., 2016a) and its variants (Huang et al., 2017) have used residual connections among layers to train very deep CNN.
The residual connections promote the feature reuse, help the gradient ﬂow, and reduce the need for massive parameters.
The ResNet (He et al., 2016a) and DenseNet (Huang et al., 2017) achieved state-of-the-art accuracy on benchmark datasets.
Alternatively, FractalNet (Larsson et al., 2017) expanded the convolutional layers in a fractal form to generate deep CNNs.
Without residual connections (He et al., 2016a) and manual deep supervision (Lee et al., 2014), FractalNet achieved high performance on image classiﬁcation based on network structural design only.
Many studies tried to understand reasons behind the representation view of deep CNNs.
Veit et al.
(2016) showed that residual network can be seen as an ensemble of relatively shallow effective paths.
However, Greff et al.
(2017) argued that ensembles of shallow networks cannot explain the experimental results of lesioning, layer dropout, and layer reshufﬂing on ResNet.
They proposed that residual connections have led to unrolled iterative estimation in ResNet.
Meanwhile, there is no ensemble behavior in FractalNet and Larsson et al.
(2017) speculated that the high performance of FractalNet was due to the unrolled iterative estimation of features of the longest path using features of shorter paths.
Although unrolled iterative estimation model can explain many experimental results, it is unclear how it helps improve the classiﬁcation performance of ResNet and FractalNet.
On the other hand, the ensemble model can explain the performance improvement easily.
In this work, we propose CrescendoNet, a simple deep convolutional neural network with ensemble behavior.
Same as other deep CNNs, CrescendoNet is created by stacking simple building blocks, called Crescendo blocks (Figure 1).
Each Crescendo block comprises a set of independent feed-forward paths with increased number of convolution and batch-norm layers (Ioffe & Szegedy, 2015a).
We only use the identical size, 3 × 3 , for all convolutional ﬁlters in the entire network.
Despite its simplicity, CrescendoNet shows competitive performance on benchmark CIFAR10, CIFAR100, and SVHN datasets.
Similar to FractalNet, CrescendoNet does not include residual connections.
The high performance of CrescendoNet also comes completely from its network structural design.
Unlike the FractalNet, in which the numbers of convolutional layers and associated parameters are increased exponentially, the numbers of convolutional layers and parameters in Crescendo blocks are increased linearly.
CrescendoNet shows clear ensemble behavior (Section 3.4).
In CrescendoNet, although the longer paths have better performances than those of shorter paths, the combination of different length paths have even better performance.
A set of paths generally outperform its subsets.
This is different from
FractalNet, in which the longest path alone achieves the similar performance as the entire network does, far better than other paths do.
Furthermore, the independence between paths in CrescendoNet allows us to introduce a new pathwise training procedure, in which paths in each building block are trained independently and sequentially.
The path-wise procedure can reduce the memory needed for training.
Especially, we can reduce the amortized memory used for training CrescendoNet to about one fourth.
We summarize our contribution as follows:
• We propose the Crescendo block with linearly increased convolutional and batch-norm layers.
The CrescendoNet generated by stacking Crescendo blocks further demonstrates that the high performance of deep CNNs can be achieved without explicit residual learning.
• Through our analysis and experiments, we note that the implicit ensemble behavior of CrescendoNet leads to high performance.
• We introduce a path-wise training approach for CrescendoNet, which can lower the memory requirements without signiﬁcant loss of accuracy given sufﬁcient data.
C RESCENDO N ET 2. A RCHITECTURE D ESIGN
Crescendo Block The Crescendo block is built by two layers, the convolution layer and the following batch normalization layer (Ioffe & Szegedy, 2015b).
The convolutional layers have the identical size, 3 × 3 .
The Conv-BatchNorm unit f 1 , deﬁned in the Eq.1 is the base branch of the Crescendo block.
[Equation]
The variable z denotes the input feature maps.
We use two hyper-parameters, the scale S and the interval I to deﬁne the structure of the Crescendo block H S .
The interval I speciﬁes the depth difference between every two adjacent branches and the scale S sets the number of branches per block.
The structure of the n th branch is deﬁned by the following equation:
[Equation]
where the superscript nI is the number of recursion times of the function f 1 .
The structure of
Crescendo block H S can be obtained below:
[Equation]
where ⊕ denotes an element-wise averaging operation.
Note that the feature maps from each path are averaged element-wise, leaving the width of the channel unchanged.
A Crescendo block with
S =
and I = 1 is shown in Figure 1.
The structure of Crescendo block is designed for exploiting more feature expressiveness.
The different depths of parallel paths lead to different receptive ﬁelds and therefore generate features in different abstract levels.
In addition, such an incremental and parallel form explicitly supports the ensemble effects, which shows excellent characteristics for efﬁcient training and anytime classiﬁcation.
We will explain and demonstrate this in the following sections.
CrescendoNet Architecture The main body of CrescendoNet is composed of stacked Crescendo blocks with max-pooling layers between adjacent blocks (Figure 1).
Following the main body, like most deep CNNs, we use fully connected layers and a soft-max layer as the classiﬁer.
The overall structure of CrescendoNet is simple and we only need to tune the Crescendo block to modify the entire network.
2. P ATH - WISE TRAINING
To reduce the memory consumption during training CrescendoNet, we propose a path-wise training procedure, leveraging the independent multi-path structure of our model.
We denote stacked ConvBatchNorm layers in one Crescendo block as one path.
We train each path individually, from the shortest to the longest repetitively.
When we are training one path, we freeze the parameters of other paths.
In other words, these frozen layers only provide learned features to support the training.
Figure 2 illustrates the procedure of path-wise training within a CrescendoNet block containing four paths.
There are two advantages of path-wise training.
First, path-wise training procedure signiﬁcantly reduces the memory requirements for convolutional layers, which constitutes the major memory cost for training CNNs.
For example, the higher bound of the memory required can be reduced to about 40% for a Crescendo block with 4 paths where interval = 1 .
Second, path-wise training works well with various optimizers and regularizations.
Even dropout and drop-path can be applied to the model during the training.
2. R EGULARIZATION
Dropout (Hinton et al., 2012) and drop-connect (Wan et al., 2013), which randomly set a selected subset of activations or weights to zero respectively, are effective regularization techniques for deep neural networks.
Their variant, drop-path (Larsson et al., 2017), shows further performance improvement by dropping paths when training FractalNet.
We use both dropout and drop-path for regularizing the Crescendo block.
We drop the branches in each block with a predeﬁned probability.
For example, given drop-path rate, p = 0 .
3 , the expectation of the number of dropped branches is 1 . 2 for a Crescendo block with four branches.
For the fully connected layers, we use L2 norm of their weights as an additional term to the loss.
E XPERIMENTS 3. D ATASETS
We evaluate our models with three benchmark datasets: CIFAR10, CIFAR100 (Krizhevsky et al., 2014), and Street View House Numbers (SVHN) (Netzer et al., 2011).
CIFAR10 and CIFAR each have 50,000 training images and 10,000 test images, belonging to 10 and 100 classes respectively.
All the images are in RGB format with the size of 32 × 32 -pixel.
SVHN are color images, with the same size of 32 × 32 -pixel, containing 604,388 and 26,032 images for training and testing respectively.
Note that these digits are cropped from a series of numbers.
Thus, there may be more than one digit in an image, but only the one in the center is used as the label.
For data augmentation, we use a widely adopted scheme (Lin et al., 2013; Larsson et al., 2017; Huang et al., 2016a;b; Srivastava et al., 2015b; Springenberg et al., 2014; He et al., 2016a).
We ﬁrst pad images with 4 zero pixels on each side, then crop padded images to 32 × 32 -pixel randomly and horizontally ﬂipping with a 50% probability.
We preprocess each image in all three datasets by subtracting off the mean and dividing the variance of the pixels.
3. T RAINING
We use Mini-batch gradient descent to train all our models.
We implement our models using TensorFlow distributed computation framework (Abadi et al., 2016) and run them on NVidia P100 GPU.
We also optimize our models by adaptive momentum estimation (Adam) optimization (Kingma &
Ba, 2014) and Nesterov Momentum optimization (Nesterov, 1983) respectively.
For Adam optimization, we set the learning rate hyper-parameter to 0 .
001 and let Adam adaptively tune the learning rate during the training.
We choose the momentum decay hyper-parameter β 1 = 0 .
9 and β 2 = 0 .
And we set the smoothing term (cid:15) =
10 − 8 .
This conﬁguration is the default setting for the AdamOptimizer class in TensorFlow.
For Nesterov Momentum optimization, we set the hyper-parameter momentum = 0 .
We decay the learning rate from 0 .
1 to 0 .
01 after 512 epochs for CIFAR and from 0 .
05 to 0 .
005 , then to 0 .
0005 , after 42 epochs and 63 epochs respectively for
SVHN.
We use truncated normal distribution for parameter initialization.
The standard deviation of hyper-parameters is 0 .
05 for convolutional weights and 0 .
04 for fully connected layer weights.
For all datasets, we use the batch size of 128 on each training replica.
For the whole net training, we run 700 epochs on CIFAR and 70 epochs on SVHN.
For the path-wise training, we run 1400 epochs on CIFAR and 100 epochs on SVHN.
Using a CrescendoNet model with three blocks each contains four branches as illustrated in Figure 1, we investigate the following preliminary aspects: the model performance under different block widths, the ensemble effect, and the path-wise training performance.
We study the Crescendo block with three different width conﬁgurations: equal width globally, equal width within the block, and increasing width.
All the three conﬁgurations have the same fully connected layers.
For the ﬁrst one, we set the number of feature maps to 128 for all the convolutional layers.
For the second, the numbers of feature maps are (128, 256, 512) for convolutional layers in each block.
For the last, we gradually increase the feature maps for each branch in three blocks to (128, 256, 512) correspondingly.
For example, the number of feature maps for the second and fourth branches in the second block is (192, 256) and (160, 192, 224, 256).
The exact number of maps for each layer is deﬁned by the following equation:
[Equation]
where n maps denotes the number of feature maps for a layer, n inmaps and n outmaps are number of input and output maps respectively, n layers is the number of layers in the block, and i layer is the index of the layer in the branch, starting from 1 .
To inspect the ensemble behavior of CrescendoNet, we compare the performance of models with and without drop-path technique and subnets composed by different combinations of branches in each block.
For the simplicity, we denote the branch combination as a set P containing the index of the branch.
For example, P = { 1 , 3 } means the blocks in the subnet only contains the ﬁrst and third branches.
The same notation is used in Table 2 and Figure 3.
3. R ESULTS OF THE WHOLE NET
Table 1 gives a comparison among CrescendoNet and other representative models on CIFAR and
SVHN benchmark datasets.
For ﬁve datasets, CrescendoNet with only 15 layers outperforms almost all networks without residual connections, plus original ResNet and ResNet with Stochastic
Depth.
For CIFAR10 and CIFAR100 without data augmentation, CrescendoNet also performs better than all the given models except DenseNet with bottleneck layers and compression (DenseNet-BC) with 250 layers.
However, CrescendoNet’s error rate 1.76% matches the 1.74% error rate of given
DenseNet-BC, on SVHN dataset which has plentiful data for each class.
Comparing with FractalNet, another outstanding model without residual connection, CrescendoNet has a simpler structure, fewer parameters, but higher accuracies.
The lower rows in Table 1 compare the performance of our model given different conﬁguration.
In three different widths, the performance simultaneously grows with the number of feature maps.
In other words, there is no over-ﬁtting when we increase the capacity of CrescendoNet in an appropriate scope.
Thus, CrescendoNet demonstrates a potential to further improve its performance by scaling up.
In addition, the drop-path technique shows its beneﬁts to our models on all the datasets, just as it does to FractalNet.
Another interesting result from Table 1 is the performance comparison between Adam and Nesterov
Momentum optimization methods.
Comparing with Nesterov Momentum method, Adam performs similarly on CIFAR10 and SVHN, but worse on CIFAR100.
Note that there are roughly 60000 , 5000 , and 500 training images for each class in SVHN, CIFAR10, and CIFAR100 respectively.
This implies that Adam may be a better option for training CrescendoNet when the training data is abundant, due to the convenience of its adaptive learning rate scheduling.
The last row of Table 1 gives the result from path-wise training.
Training the model with less memory requirement can be achieved at the cost of some performance degradation.
However, Pathwise trained CrescendoNet still outperform many of networks without residual connections on given datasets.
3. R ESULTS OF S UBNETS
Table 2 provides a performance comparison among different path combinations of CrescendoNet, trained by Adam optimization, with block-wise width (128 , 256 , 512) .
, the results show the ensemble behavior of our model.
Speciﬁcally, the more paths contained in the network, the better the performance.
And the whole net outperforms any single path network with a large margin.
For example, the whole net and the net based on the longest path show the inference error rate of 6 . 90% and 10 .
69% respectively, for CIFAR10 without data augmentation.
This implicit ensemble behavior [Table] - with Dropout 2.7M - - - - 1. ResNet (pre-activation) 1.7M5.
- 10.2M4.
DenseNet (k = 12) 1.0M 7.
DenseNet-BC (k = 24) 15.3M 5.
CrescendoNet Nesterov (128, 128, 128) 4.1M 7.
(128, 256, 512)-W 18.3M 7.
(128, 256, 512) 27.7M 6.
without drop-path 27.7M 8.
CrescendoNet Adam (128, 128, 128) 4.1M 7.
(128, 256, 512) 27.7M 6.
without drop-path 27.7M 9.
path-wise training 27.7M 8.
Branches
[Table]
+
SVHN
P= { 1, 2, 3, 4 } 6.
P= { 2, 3, 4 } 6.
P= { 1, 2, 4 } 7.
P= { 1, 2, 3 } 7.
P= { 3, 4 } 7.
P= { 2, 4 } 7.
P= { 2, 3 } 8.
P= { 1, 4 } 8.
P= { 1, 2 } 10.
P= { 4 } 10.
P= { 3 } 11.
P= { 2 } 12.
P= { 1 } 28.
differentiates CrescendoNet from FractalNet, which shows a student-teacher effect.
Speciﬁcally, the longest path in FractalNet can achieve a similar or even lower error rate compared to the whole net.
To investigate the dynamic behavior of subnets, we test the error rate changes of subnets during the training.
We use Adam to train the CrescendoNet with the structure shown in Figure 1 on CIFAR for 450 epochs.
Figure 3 illustrates the behavior of different path combinations during the training.
It shows that the inference accuracy of the whole net grows simultaneously with all the subnets, which demonstrates the ensemble effect.
Second, for any single path network, the performance grows with the depth.
This behavior of the anytime classiﬁer is also shown by FractalNet.
In other words, we could use the short path network to give a rough but quick inference, then use more paths to gradually increase the accuracy.
This may be useful for time-critical applications, like integrated recognition system for autonomous driving tasks.
R ELATED W ORK
Conventional deep CNNs, such as AlexNet (Krizhevsky et al., 2012) and VGG-19 (Simonyan &
Zisserman, 2014), directly stacked the convolutional layers.
However, the vanishing gradient problem makes it difﬁcult to train and tune very deep CNN of conventional structures.
Recently, stacking small convolutional blocks has become an important method to build deep CNNs.
Introducing new building blocks becomes the key to improve the performance of deep CNN.
Lin et al.
(2013) ﬁrst introduced the NetworkInNetwork module which is a micro neural network using a multiple layer perceptron (MLP) for local modeling.
Then, they piled the micro neural networks into a deep macro neural network.
Szegedy et al.
(2015) introduced a new building block called Inception, based on which they built
GoogLeNet.
Each Inception block has four branches of shallow CNNs, building by convolutional kernels with size 1 × 1 , 3 × 3 , 5 × 5 , and max-pooling with kernel size 3 × 3 .
Such a multiple-branch scheme is used to extract diversiﬁed features while reducing the need for tuning the convolutional sizes.
The main body of GoogLeNet has 9 Inception blocks stacked each other.
Stacking multiplebranch blocks can create an exponential combination of feed-forward paths.
Such a structure combined with the dropout technique can show an implicit ensemble effect (Veit et al., 2016; Srivastava et al., 2014).
GoogLeNet was further improved with new blocks to more powerful models, such as Xception (Chollet, 2016) and Inception-v4 (Szegedy et al., 2016).
To improve the scalability of
GoogLeNet, Szegedy et al.
(2016) used convolution factorization and label-smoothing regularization in Inception-v4.
In addition, Chollet (2016) explicitly deﬁned a depth-wise separable convolution module replacing Inception module.
Recently, Larsson et al.
(2017) introduced FractalNet built by stacked Fractal blocks, which are the combination of identical convolutional layers in a fractal expansion fashion.
FractalNet showed that it is possible to train very deep neural network through the network architecture design.
FractalNet implicitly also achieved deep supervision and student-teacher learning by the fractal architecture.
However, the fractal expansion form increases the number of convolution layers and associated parameters exponentially.
For example, the original FractalNet model with 21 layers has 38.6 million parameters, while a ResNet of depth 1001 with similar accuracy has only 10.2 million parameters (Huang et al., 2016a).
Thus, the exponential expansion reduced the scalability of FractalNet.
Another successful idea in network architecture design is the use of skip-connections (He et al., 2016a;b; Huang et al., 2016a; Zagoruyko & Komodakis, 2016;
Xie et al., 2016).
ResNet (He et al., 2016a) used the identity mapping to short connect stacked convolutional layers, which allows the data to pass from a layer to its subsequent layers.
With the identity mapping, it is possible to train a 1000-layer convolutional neural network.
Huang et al.
(2016a) recently proposed DenseNet with extremely residual connections.
They connected each layer in the Dense block to every subsequent layer.
DenseNet achieved the best performance on benchmark datasets so far.
On the other hand,
Highway networks (Srivastava et al., 2015a) used skip-connections to adaptively infuse the input and output of traditional stacked neural network layers.
Highway networks have helped to achieve high performance in language modeling and translation.
D ISCUSSION AND C ONCLUSION
CNNs have shown excellent performance on image recognition tasks.
However, it is still challenging to tune, modify, and redesign the existing architectures.
We propose CrescendoNet, which has a simple convolutional neural network architecture without residual connections (He et al., 2016a).
Crescendo block uses convolutional layers with same size 3 × 3 and joins feature maps from each branch by the averaging operation.
The number of convolutional layers grows linearly in CrescendoNet while exponentially in FractalNet (Larsson et al., 2017).
This leads to a signiﬁcant reduction of computational complexity.
Even with much fewer layers and simpler structure, CrescendoNet matches the performance of the original and most of the variants of ResNet on CIFAR10 and CIFAR100 classiﬁcation tasks.
Like FractalNet (Larsson et al., 2017), we use dropout and drop-path as regularization mechanisms, which can train CrescendoNet to be an anytime classiﬁer, namely, CrescendoNet can perform inference with any combination of the branches according to the latency requirements.
Our experiments also demonstrated that CrescendoNet synergized well with Adam optimization, especially when the training data is sufﬁcient.
In other words, we can avoid scheduling the learning rate which is usually performed empirically for training existing CNN architectures.
CrescendoNet shows a different behavior from FractalNet.
In FractalNet (Larsson et al., 2017), the longest path alone achieves the similar performance as the entire network, far better than other paths, which shows the student-teacher effect.
The whole FractalNet except the longest path acts as a scaffold for the training and becomes dispensable later.
On the other hand, CrescendoNet shows the ensemble behavior as the whole network signiﬁcantly outperforms any path within it.
This fact sheds the light on exploring the implicit ensemble behavior and design strategy for deep CNNs.