Approximate Computing for Deep Learning in
TensorFlow
Chiang Chi-An
T
H
E
U
N I V E R
S
I
T
Y
O
F
E
D I N B
U
R
G
H
Master of Science
School of Informatics
University of Edinburgh
2017
Abstract
Nowadays, many machine learning techniques are applied on the smart phone to do
things like image classificatin, audio recognization and object detection to make smart
phone even smarter. Since deep learning has achieved the best result in many fields.
More and more people want to use deep neural netowrk model in the smart phone.
However, deep neural netowrk model can be large, need large amount of computa-
tion that takes too much time and power. There are a few methods of approximate
computing proposed to address this problem in recent years. The method I use in this
paper is mobilenet model using tensorflow which is just published by Google in this
year. I will conduct experiments to show whether mobilenet can decrease model size,
increase speed while at the same time keep decent accuracy. I will compare metrics
of the mobilenet with other traditional models such as VGG model. I will also show
how the parameters of width multiplier and resolution multiplier impact the trade off
between model size, speed and accuracy.
i
Acknowledgements
First of all, I would like to thank my dissertation supervisor, Dr. Pramod Bhatotia,
for teaching me how to conduct rigorous research, organize my thoughts, and produce
a well-structured thesis. From beginning the proposal to finishing the dissertation,
provided me with precious and invaluable intellectual advice. Thanks to him, I did not
have to learn through trial and error, and had the freedom to explore the areas that I
found most interesting and satisfying while I was doing my dissertation. I would also
like to thank Dr. Bob Fisher, my personal tutor. In both our one-on-one meetings and
our group meetings, he demonstrated his investment in me. In addition, the courses
that he advised me to take were very practical and provided me with a comprehensive
course schedule, enabling me to learn all that I could in this MSc program. Finally,
I would like to thank my mother, whose financial and mental support allowed me to
focus on my studies without worry.
ii
Declaration
I declare that this thesis was composed by myself, that the work contained herein is
my own except where explicitly stated otherwise in the text, and that this work has not
been submitted for any other degree or professional qualification except as specified.
(Chiang Chi-An)
iii
Table of Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Achieved results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Dissertation outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Background 4
2.1 Relevant work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Deep Learning . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Approximate Computing . . . . . . . . . . . . . . . . . . . . 4
2.2 TensorFlow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Advantage . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.4 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Methods 10
3.1 Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.1 Activation Function . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.2 Fully Connected Layer . . . . . . . . . . . . . . . . . . . . . 13
3.1.3 Convolutional Layer . . . . . . . . . . . . . . . . . . . . . . 14
3.1.4 Pooling layer . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Loss function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1 Cross Entropy loss . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.2 Hinge Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.3 Loss Functions Comparison . . . . . . . . . . . . . . . . . . 19
3.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
iv
3.3.1 Mini-batch gradient descent . . . . . . . . . . . . . . . . . . 19
3.3.2 Learning Rate Decay . . . . . . . . . . . . . . . . . . . . . . 20
3.3.3 Mini-batch gradient descent extensions . . . . . . . . . . . . 21
3.3.4 Forward Propagation and Backpropagation . . . . . . . . . . 22
3.4 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4.1 L2 regularization . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4.2 L1 regularization . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4.3 Dropout Layer . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.4 Batch Normalization . . . . . . . . . . . . . . . . . . . . . . 27
3.5 Depthwise Separable Convolution . . . . . . . . . . . . . . . . . . . 28
3.6 Transfer Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Results and Evaluation 33
4.1 Resource and tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1.1 Checkpoint File . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1.2 Model File . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2.1 CIFAR 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3.1 Training set and test set . . . . . . . . . . . . . . . . . . . . . 35
4.3.2 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3.3 Mobilenet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3.4 Inception V3 . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3.5 ResNet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4.1 Top-1 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4.2 Top-5 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4.3 Inference Time . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4.4 Model File Size . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.6 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 Conclusion and Discussion 44
5.1 Remarks and observations . . . . . . . . . . . . . . . . . . . . . . . 44
5.2 Limitation and Further work . . . . . . . . . . . . . . . . . . . . . . 44
5.2.1 More approximate computing techniques . . . . . . . . . . . 44
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5.2.2 More extensive Experiment . . . . . . . . . . . . . . . . . . 45
5.2.3 Application into Practice . . . . . . . . . . . . . . . . . . . . 45
5.2.4 Model Architecture Improvement . . . . . . . . . . . . . . . 45
Bibliography 46
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Chapter 1
Introduction
1.1 Motivation
In recent years, machine learning techniques, especially deep learning which uses mul-
tiple layers of artificial neural networks, have achieved remarkable breakthroughs in
many fields. From image classification to Go game AI player AlphaGo [1], deep learn-
ing exhibits the best performance.
At the same time, more and more people use smartphone. Undoubtedly, AI tech-
niques such as deep learning will make smartphones even smarter. Functions such
as face recognition, audio recognition and image classification will be added to many
mobile apps.
The deep learning model training part can be done offline in the server clusters.
For the inference part, although we can send the data to the server, and let the server
do the prediction job and reply with the result. In some cases, if the data is sensitive,
the client may wish not to send out to servers. One example is the bank card number
recognition application. Even without security concerns, network traffic can be slow
and expensive and building reliable servers increases the operation cost.
Thus, if we can do prediction on the smart phone, then there is no data security
concern, no network traffic delay and cost and no need to maintain a potentially ex-
pensive server cluster. However, this approach also has its drawbacks. It requires the
model to be stored in the smart phone’s limited storage and inference computing in the
mobile can be slow and cost battery power.
A deep neural network typically has many layers with a large number of parame-
ters. It requires large storage and a large number of math operations. For example, one
traditional image classification model VGG [2] has about 100 million parameters, need
1
Chapter 1. Introduction 2
more than 1GB to store the model and takes more than 10000 million math operations.
Thus it is not fit in the mobile phone.
To use deep learning models in the mobile phone, we must find a way to signifi-
cantly decrease the model size and the number of computing operations to make the
model file reasonably small and computing fast with less power. In the meantime, per-
formance should be maintained as high as possible. We need to find a suitable trade-off
between them.
1.2 Objective
MobileNet [3] is a new deep neural network model proposed by Google that are spe-
cially designed for mobile and embedded devices using approximate computing tech-
niques. Although the experiments in its paper show that it has strong performance com-
pared to other popular models on ImageNet [4] classification, a useful model should
also have good performance on new datasets using the transfer learning technique.
In this project, we will compare MobileNet with other popular models in accuracy,
model size and inference time in mobile devices to investigate whether approximate
computing used in MobileNet can achieve a better trade off between accuracy and
efficiency to be suitable for mobile device. We will also investigate how the two pa-
rameters width multiplier and resolution multiplier of MobileNet affect the accuracy,
model size and inference time.
1.3 Achieved results
MobileNets with different width multipliers and resolution multipliers are successfully
trained on the CIFAR-100 using transfer learning with pre-trained model on ImageNet.
GoogLeNet Inception V3 [5] and ResNet [6] models are also trained on the CIFAR-
100 using transfer learning. Top-1 and top-5 accuracy on test set are computed for each
model. The size of model files to be deployed in mobile app is recorded. The inference
time of each model in Android device is computed. Comparison of the results shows
that MobileNet with width multiplier 1 and resolution multiplier 1 have speedup more
than 17× and shrink the model file more than 6× both compared with GoogLeNet
Inception V3 and ResNet models. It has 18.3% loss in top-1 accuracy and 8.5% loss
in top-5 accuracy compared with GoogLeNet Inception V3 and with almost no loss
in both top-1 and top-5 accuracy compared with ResNet. The results also show that as
Chapter 1. Introduction 3
we decrease width multiplier, model size becomes smaller and inference time quicker
but with more accuracy loss. The resolution multiplier has the similar effect except
that it doesn’t affect model size.
1.4 Dissertation outline
Chapter 2 will introduce various approximate computing techniques for deep learning
which can be divided into 3 general categories such as low rank approximation to
which techniques used in this project belong, network pruning and quantization. The
introduction of Tensorflow [7] which is the deep learning framework used in this
project is also included in Chapter 1.
Chapter 3 will elaborate both the theory and implementation of the deep learning
models in detail. They include loss function, optimization algorithm, regularization
method, various kinds of layers used, transfer learning and the particular approxi-
mate computing technique used in this project: approximating traditional convolu-
tional layer with depth-wise separable convolution layer.
Chapter 4 describes experiment results and analysis.
Chapter 5 gives the project conclusion and discussion, then future work.
Chapter 2
Background
2.1 Relevant work
2.1.1 Deep Learning
Deep learning techniques have achieved state-of-art results in many areas of
machine learning. The achievements are remarkable, especially for the success of deep
convolutional neural network (CNNs) in image classification. CNNs have the best re-
sults in all the standard image datasets such as MNIST [8], CIFAR-10 [9], CIFAR-100
[9] and ImageNet [4]. Many different CNNs models have been developed such as
ResNet, VGG and Inception. Because convolutional layers can make better use of
image spatial information, these models typically have a sequence of many convolu-
tional layers.
2.1.2 Approximate Computing
Until recently, deep learning researchers have been primarily focused on improving
model’s accuracy. However, the use of multiple convolutional layers also results in
large number of parameters requiring large memory for model storage and increases
the computational cost.
With the widespread use of mobile devices and the application of deep learning
in mobile apps, more and more researchers are aware that to have a good mobile user
experience, accuracy is not enough – the model must also be efficient: less memory,
quicker inference and less energy consumption. Because mobile consumers do not
want a single app to take too much space of limited memory and want the app to
respond instantly.
4
Chapter 2. Background 5
They resort to approximate computing techniques to make a better trade-off be-
tween accuracy and efficiency. The goal is to make model size smaller and inference
time quicker to be suitable for mobile device while at the same keep as much accuracy
as possible.
[10] shows that significant redundancy often exists in deep learning models. Through
approximate computing, we can remove the redundancy, saving both memory and
computation cost. The approximate computing for deep learning can be divided into
roughly three general approaches: pruning, quantization and low rank approximation.
2.1.2.1 Low Rank Approximation of Filters
This approach decomposes the filters in convolutional layers into a series of separable
smaller filters which are a low-rank approximation of original filters and reduce time
complexity. The optimal decomposition can be found by minimizing the reconstruc-
tion error of the filters or the layer output. Since convolutional layers are the most
time-consuming parts in CNNs, this low-rank decomposition will generate significant
speed up.
[11] uses SVD decomposition to make convolutional layers 1.6 times faster while
sacrificing 1% accuracy. [12] uses rank 1 filter basis approximation that achieves
speedup by factor 2.5 without sacrifice of accuracy and by factor 4.5 with less than
1% accuracy decrease for a text character recognition network. [11] and [12] can only
decompose linear filters in a single layer. [13] further develops this method to take
into account the nonlinearity, such as Rectified Linear Units (ReLU), which makes the
approximation more accurate. It also invents new algorithms to optimize the whole
network to reduce the accumulated errors when approximating multiple convolutional
layers. It achieves speed up of factor 4 on a large pre-trained model on ImageNet with
only 0.9% e top-5 error rate increase.
Instead of finding low-rank approximation of convolutional layers of pre-trained
networks, some papers replace traditional convolutional layers with layers that has sim-
ilar function but with smaller computation cost. Flattened networks [14] replaces 3D
filters in conventional convolutional networks with consecutive sequence of 1-D filters
in all 3 dimensions which reduces the parameters significantly and make the feedfor-
ward computation about 2 times faster. Factorized networks [15] factor the convolu-
tion operation by unravelling the convolutional layer with a sequence of combination
of single channel convolution and linear channel projection. It achieves similar ac-
curacy but with much less computation compared with traditional deep convolutional
Chapter 2. Background 6
neural networks models. MobileNets [3] uses a similar approach to those of flattened
networks [14] and factorized networks [15]. Its model is based on depthwise separable
convolutions which separate traditional convolutions into depthwise convolutions that
apply a single filter for each input channel and pointwise convolutions that combine
the results linearly. The MobileNet model has smaller size and comparable accuracy
with models such as GoogleNet [5] and VGG 16 [2]. It provides two hyperparameters
width multiplier and resolution multiplier to adjust the trade-off between latency and
accuracy.
2.1.2.2 Network Pruning
This approach tries to remove parts of the models that are not important to reduce
number of parameters and computation.
[16] first learns the importance of network connections and remove those that are
not important, then retrains the connections left. Its experiments show that the number
of parameters in VGG-16 model can be reduced by 13×and AlexNet [17] model by 9×
with no loss of accuracy using this method.
[18] and [19] aim to prune whole filters together instead of weights, which can
induce more speedup in the convolutional layers. [18] reports inference time
decreases by 34% for VGG-16 and 38% for ResNet-110 on CIFAR-10 almost without
loss of accuracy. [19] reports 3.31× FLOPs reduction and 16.63× compression on
VGG-16 with 0.52% top-5 accuracy drop.
[20]’s pruning algorithm aims specifically at reducing energy consumption of CNNs
instead of computation and memory cost. It reports energy consumption for AlexNet
decreases by 3.7× and GoogLeNet decreases 1.6× both with top-5 accuracy loss less
than 1%.
2.1.2.3 Network Quantization
Network Quantization quantitizes the parameters of neural network models and en-
codes them with fewer bits to reduce their memory storage. For example, using 8 bits
instead of 32 bits will require only about 25% of previously needed storage. Another
benefit of quantization is to make the inference computation faster and use less power.
Because using fewe bits saves memory bandwidth and RAM access time and allows
more operations done in one cycle for SIMD instructions.
During the training phase, in each step, the parameters of neural networks adjusts
Chapter 2. Background 7
a little using back propagation and gradient descent algorithm which requires high-
precision number format such as 32 bits floating number. Thus, instead of training a
quantized model from scratch, we usually quantize a pre-trained model.
Quantization for deep networks typically doesn’t decrease the accuracy of infer-
ence. Because deep networks are often very robust and good at ignoring the noise
including the precision error noise introduced by quantization.
One simple way to quantize is to store the minimum and maximum values of the
floating numbers set, then using an integer to represent the floating number. For ex-
ample, if we use 8 bits to represent floating numbers in the range [-20.0, 50], then 0
represents -20.0, 255 represents 50.0, 128 represents 35.0 and so on.
[21] uses k-means clustering algorithm and product quantization method to quan-
tize the network parameters layer by layer. It achieves 16-24 times compression of
CNN on ImageNet with 1% loss of accuracy.
[22] uses Hessian-weighted k-means clustering and fixed-length binary encoding
to do the quantization. Hessian-weighting also takes into account the across layers im-
pact of quantization errors aside from within impact and thus can quantize the whole
network at once. This paper also employs Huffman coding to further compress the net-
work. It reports that the quantize models are 1.95%, 4.51% and 2.46% respectively of
the original model sizes for LeNet, ResNet and AlexNet at no or marginal performance
loss.
Network quantization can also be combined with other approximate computing
techniques. Deep compression [23] combines network pruning and quantization. It
first prunes the model connections and only keeps most important connection to reduce
parameters by 9-13 times. Then it quantizes the weights so that we can use only 5 bits
to represent a weight instead of 32 bits. Finally it uses Huffman coding to reduce the
model further. This method compresses AlexNet model by 35 times, from 240MB to
6.9MB, increases the speed by 3-4 times and costs 3-7 times less power.
2.2 TensorFlow
2.2.1 Introduction
TensorFlow is the second generation machine learning system published by Google.
It is a successor for Google’s previous DistBelief system. Its computation is based on
data flow graph with takes math operation as node and multidimensional data arrays
Chapter 2. Background 8
(tensors) flows through edges.
It is open-sourced and can be used in either single machine or multi-server clusters.
It can be run in CPU or GPU and even specialized computation device such as TPU
(Tensor Processing Units) which are used in Google. It enables the researchers to
easily implement various deep learning algorithms and has attracted much attention
from research communities.
The main components of TensorFlow consist of client, master and working pro-
cesses. Client sends request to the master and the master schedules the working pro-
cesses to do computation in available devices. TensorFlow can be used both in single-
machine and distributed clusters where client, master and working processes run in
different machines .
2.2.2 Advantage
One of the many useful features is that TensorFlow can differentiate symbolic expres-
sion and derive the backpropagation automatically for neural network training which
greatly reduce the work on programmer and the chance to make mistakes.
The TensorFlow is designed based on dataflow graph model. It provides python and
C++ interface for programmers to easily construct the graph which makes architecture,
algorithm and parameters experimentation very easy.
After the user constructs the dataflow graph, the TensorFlow system will optimize
the graph and actually execute the operations in machines. Through this first con-
structing graph then actually executing approach, it enables the TensorFlow to know
the whole information before executing and thus can do optimization as much as pos-
sible.
All computations are encoded as nodes in a data graph. The dependency of the data
between different operations are explicitly encoded in the graph, so the TensorFlow can
partition the graph according to the dependencies and run the subgraph computations
in parallel in different devices.
The TensorFlow allows the user to specify the subgraphs that need to be computed.
The user can feed tensors to none or some of the input place holders. The TensorFlow
system only runs the computation that is necessary and prune the irrelevant graph away.
The tensor flow’s data graph model not only makes it easy to run concurrently and
also easy to distribute computation to multiple devices.
In TensorFlow, the data flowing through graph are called tensors. A tensor is a
Chapter 2. Background 9
multi-dimensional array of primitive types such as int32. It represents the input and
the output of the operations, which are represented in the vertices. Every operation has
a type and none or more attributes. An operation that contains mutable state is called a
stateful operation. Variable is one of such kind of operation. Another special operation
is queue operation
User can use TensorFlow’s checkpoint files to periodically save training models
and reload the model later. This facility not only improves the fault tolerance, it also
can be used for transfer learning.
2.2.3 Architecture
The TensorFlow adopts a layered architecture. On the top level are training and infer-
ence libraries. The next level is python and C++ API which are built on the C API.
Below C API level are distributed master and dataflow executor.
The distributed master accepts a data flow graph as input, it will prune the unnec-
essary part of the graph and divide the graph into subgraphs to distribute computation
to different devices. Many optimizations, such as constant folding and subexpression
elimination, are done by it.
The dataflow executor’s task is to execute the computation of the subgraph dis-
tributed by the distributed master.
The next level is kernel implementations which has more than 200 operations im-
plemented including often used operation such as Const, Var, MatMul, Conv2D and
ReLU.
Apart from above core components, the TensorFlow system also includes several
useful tools such as a dashboard to visualize the data flow graph and training progress
and a profiler that shows the running time of different tasks in different devices.
2.2.4 Performance
In Chintala’s benchmark of convolutional models testing, the results show that Ten-
sorFlow has shorter training step time than Caffe and with a similar one to Torch.
Experiments have shown that TensorFlow can scale well in problems such as image
classification and language modelling.
Chapter 3
Methods
3.1 Network Architecture
The neural networks are organized as layers of neurons with input layer as the first
layer, output layer as the last and hidden layers between them. A simple neural network
is shown in figure 3.1.
Figure 3.1: A simple neural network.
10
Chapter 3. Methods 11
3.1.1 Activation Function
Each neuron is a computing unit that applies linear transformation to its inputs, fol-
lowed by an activation function to generate the output (Figure 3.2).
Figure 3.2: Computation in a single neuron. wi is the weight for input xi, b is the bias
and f is the activation function.
We typically use a non-linear function as the activation function, because if the
activation function is linear, it can be incorporated into the previous linear transfor-
mation. There are many different activation functions. The most commonly used are
sigmoid, tanh and rectified linear unit (ReLU). In deep neural networks, ReLU is found
to have better results than sigmoid and tanh.
• Sigmoid
f (x) =
1
1+ e−x
(3.1)
Chapter 3. Methods 12
Figure 3.3: sigmoid plot
• Tanh
f (x) = tanh(x) =
ex− e−x
ex + e−x
(3.2)
Figure 3.4: Tanh plot
• ReLU
Chapter 3. Methods 13
f (x) = max(0,x) (3.3)
Figure 3.5: ReLU plot
3.1.2 Fully Connected Layer
The neurons in the neighbouring layers are fully connected. Every pair of neurons has
a connection. If the two layers have M neurons and N neurons respectively, then there
are M×N connections between them each with different weight parameters. This is
the traditional layer type often used in regular neural network. An example is given in
figure 3.6.
Chapter 3. Methods 14
Figure 3.6: An example of fully connected layer
3.1.3 Convolutional Layer
For image and other high-dimensional data, convolutional layer is often preferable to
fully connected layer. Because fully connected layer will create too many connec-
tions, and thus has much more parameters which can be slow to train and easy to
overfit. For example, if the input image is 30x30x3 and fully connected layer is used
as the first hidden layer, then every neuron in the fully connected layer will connect to
30x30x3=2700 neurons in the input layer. For such small image, it may not be a prob-
lem. But for larger image such as 300x300x3, there will be 270000 connections for a
single neuron which is difficult to handle. Another problem is that high-dimensional
data such as image often has inherent spatial structure. But for fully connected layer,
the input is just a vector of pixel values. The relative position of the pixels has no effect
and the spatial structure information is lost.
To address these problems, convolutional layer is invented. To be suitable for image
data, the layout of neurons in convolutional layer is three-dimensional instead of one-
dimensional in the fully connected layer. The 3 dimensions are width, height and
depth respectively. Each neuron in the convolutional layer now only connects to a
small regions of neurons of previous layer. The small region is small in width and
height but includes all depth. The width and height of the region is called receptive
Chapter 3. Methods 15
field or filter size. So the receptive field controls how large the connection region will
be. In this way, we reduce the connections dramatically. For example, If the receptive
field is 3×3, the input volume is 300x300x3, then one neuron will connect to 3x3x3=27
neurons of the previous layer instead of 270000 in a fully connected layer. Apart from
the benefit of reducing the number of connections, it is also helpful for learning the
local feature of the image.
To reduce the number of parameters further, the convolutional layer let neurons
in the same depth slice share the same weights. Thus, for different positions in the
image, the filter uses the same weights to extract the features, which makes the feature
extracting translation invariant.
During forward propagation phase, we slide a window of size defined by receptive
field over all the input volume and compute the dot product of filter weights and the
pixel values in the window to get a single number in the output volume. The dot
products of all positions constitute the activation map. And the activation maps for all
filters stacked in the depth dimension to constitute the total output volume.
In summary, by arranging layer of neurons in 3D space, constraining the connec-
tions to local area and sharing the weights, convolutional layer can make better use
of spatial information with much less parameters. The local connection and weight
sharing are illustrated in figure 3.7. An exmaple of a 3D convolutional layer is given
in figure 3.8.
Chapter 3. Methods 16
Figure 3.7: An example of 1D convolutional layer. For illustration purpose, the graph
shows connections of 1 dimensional convolutional layer instead of the usual 3 dimen-
sional convolutional layer used for image data. The filter size is 1. The connections with
the same color share the same weight parameters.
Figure 3.8: An example of 3D convolutional layer. The input size is 32×32×3. There
are 5 filters. The connection is local in width and hight dimension but across all depth
dimension.
Chapter 3. Methods 17
3.1.3.1 Convert Fully connected layer to Convolutional Layer
Fully connected layer can be converted to convolutional layer. For example, if the fully
connected layer accepts 5×5×128 input volume and outputs volume 1×1×10, then
a convolutional layer with 10 filters of size 5×5 will give the same effect. Replacing
the fully connected layer with a convolutional layer has the advantage that when the
input image has a larger size than the trained image, we can process multiple areas
of the input image in a single forward pass instead of multiple forward passes to get
multiple class score vectors and the final prediction can be done using their average,
which can improve the prediction accuracy.
3.1.4 Pooling layer
Pooling layer is often used in the convolutional neural networks and can decrease the
size of features significantly. It works by sliding a small window over input volume,
using a non-linear function to computing a number with the values in the small window
as input. The computation is conducted for each input depth independently. The most
commonly used non-linear function is max function. Other functions, such as average
(figure 3.9) and L2-norm, are also used. By reducing multiple values in a local region
to only one number, the pooling layer has the effect of extracting more abstract fea-
tures which helps the model to generalize and reduce overfitting. The pooling layer
introduces no additional parameters and it will reduce the width and height by factor
2 or more with depth unchanged. Thus, the number of parameters of the later layers is
reduced. The most commonly used filter size is 2×2, which results in output volume
of 1/4 input volume size. Larger filter size is rarely used, because it will discard too
much information and often result in bad performance.
Chapter 3. Methods 18
Figure 3.9: An example of average pooling operation for a single depth slice with a 2×2
filter and stride of 2.
3.2 Loss function
Suppose we have n classes, for sample x, we have computed a score vector f of n
elements. f j is the class score of sample xi for class j. Larger score indicates it is more
likely for xi to belong that class. The loss function takes the score vector as input and
output a single number to indicate how well the score outcome matches with the true
class label. Intuitively, if the score for the true class is relatively higher than others’,
then the loss function value should be smaller.
3.2.1 Cross Entropy loss
We can use the softmax function to convert class score vector to class probability
vector with each value in range [0,1] and the total sum as 1.
The probability of data sample xi belonging to class k given the class score vector
f is:
P(y = k|xi) =
e fk
∑ j e
f j
(3.4)
That is, for each score, take its exponentiation and then divided by sum of exponen-
tiations to normalize the value to 0-1. We want the loss to be small when the predicted
probability for the correct class is larger. We can take negative log of P(yi|xi) where yi
is the correct class for xi to get the loss. The loss for sample xi is as follows:
Li =− log(P(yi|xi)) =− log(
e fyi
∑ j e
f j
) =− fyi + log∑
j
e f j (3.5)
Chapter 3. Methods 19
3.2.2 Hinge Loss
Another commonly used loss function is hinge loss. The loss for sample (xi,yi) given
class score vector f is:
Li = ∑
j 6=yi
max(0, f j− fi +1) (3.6)
Intuitively, this loss function wants the score for the true class to be larger than
others at least by 1. Otherwise, the loss will increase for each violation.
3.2.3 Loss Functions Comparison
The cross-entropy unlike hinge loss provides probability for each class which is easier
for human to interpret than raw class score. Another difference is that, once the mar-
gins between true class score and other class scores are large enough, the hinge loss
becomes zero and cannot decrease further, whereas the cross-entropy loss can always
decrease. The hinge loss and cross-entropy loss often have similar performance.
3.3 Optimization
3.3.1 Mini-batch gradient descent
The training process is to use optimization algorithm to update the parameters so that
the loss is minimized. Most common used optimization algorithm for neural network.
θn+1 = θn−η∇L(θn) (3.7)
θ is the parameter vector, L(θ) is the loss, ∇L(θ) is its gradient and η is the learning
rate. The gradient descent is an iterative algorithm that updates the parameters though
the negative direction of gradient at each iteration and the step size is controlled by the
learning rate.
When the training data is huge, for example ImageNet has over 10 million images,
computing the gradient using the entire data set is costly. In this situation, we need to
use mini-batch gradient descent. In this method, we take a small subset of samples (a
mini-batch) from the data set at each step and then use this mini-batch instead of the
whole data set in normal gradient descent algorithm to compute the gradient and do the
parameter updating. Due to the correlation between samples in the training data set,
the gradient of the loss function over the mini-batch is often very approximate to the
Chapter 3. Methods 20
gradient of the loss function over the whole training data set. Since the computation
cost is much cheaper in the mini-batch gradient descent algorithm than the normal
gradient descent algorithm at each parameter updating step, much more updates can
be performed and thus, the loss function can converge much more quickly in mini-
batch gradient descent algorithm
The learning rate in the mini-batch gradient descent algorithm is very important.
When the learning rate is very small, although the loss is guaranteed to decrease, the
converging speed may be too slow. We can increase the learning rate to speed up the
learning, but this may lead to overstep that makes the loss increase. It is very difficult
to set a suitable learning rate. Different datasets or different network architectures may
require different learning rate. We may need to set different learning rate for different
parameters and in different training phases. Learning rate decay and extensions of
mini-batch gradient descent algorithms can be used to solve this problem.
3.3.2 Learning Rate Decay
At the start of training, we may want a relatively larger learning rate so that the loss
function value can decrease quicker. In the later stage, with the improvement getting
smaller in each step, we may want to decay the learning rate so that it can avoid over-
stepping and fine-tune the parameters. We can set the learning rate decay according to
some rule – for example, multiply 0.9 every 1 epoch. Or set the decay manually, for
example, when we see the training loss doesn’t decrease any more, we can try to half
the learning rate.
Let η0 is the initial learning rate, k is decay rate and t is the number of training
steps. Three commonly used rules can be expressed as follows.
3.3.2.1 Natural Exponential decay
η = η0e
−kt (3.8)
3.3.2.2 Exponential decay
η = η0k
t (3.9)
3.3.2.3 Inverse Time Decay
η =
η0
1+ kt
(3.10)
Chapter 3. Methods 21
3.3.3 Mini-batch gradient descent extensions
Many extensions are proposed to improve over the basic mini-batch gradient descent
algorithm. Algorithms such as Adagrad and RMSProp try to setting the learning rate
adaptively during training. Algorithms such as Momentum and Nesterov Momentum
try to adjust the parameter updating direction to reduce oscillations.
3.3.3.1 Adagrad
Adagrad algorithm can adapt the learning rate for each parameter automatically.
C =C+d2
θ
(3.11)
θ = θ−
η
√
C+ ε
dθ (3.12)
ε is used to avoid dividing 0 and it is set to a very small value such as 1e−6.
The above formulae operations are element-wise for each parameter. So each pa-
rameter has its own effective learning rate. AdaGrad keeps track of the sum of gradients
and uses it to adjust the learning rate.
3.3.3.2 RMSProp
One problem of Adagrad is that the effective learning rate η√
C+ε
is always decreasing,
when it is approximate to 0, then the algorithm stops learning.
Another algorithm called RMSProp tries to solve this problem.
C = γC+(1− γ)d2
θ
(3.13)
θ = θ−
η
√
C+ ε
dθ (3.14)
γ is the decay rate. RMSProp makes a simple change which makes C as the mov-
ing average of gradient square instead of accumulated sum in the Adagrad. Now the
effective learning rate is no longer always decreasing.
3.3.3.3 Momentum
v = γv−ηdθ (3.15)
Chapter 3. Methods 22
θ = θ− v (3.16)
γ is another hyperparameter called momentum. v is the velocity. We integrate
previous velocity with gradient to get the current velocity and then using the velocity
to update the θ. This is different from basic gradient descent where we directly update
the parameters using gradient. This algorithm is helpful to reduce oscillating and speed
up convergence.
3.3.3.4 Nesterov Momentum
The Nesterov momentum uses the gradient of the next position instead of current po-
sition and achieves better results over momentum.
θ
′ = θ+ γv (3.17)
v = γv−ηdθ′ (3.18)
θ = θ− v (3.19)
3.3.4 Forward Propagation and Backpropagation
Let ai represents the activation values of layer i. For the input layer, the values are
directly from input x, so we have a1 = x . We can compute all neurons’ value layer by
layer from input layer until output layer.
ai+1 = fi(Wiai +bi) (3.20)
From the output layer’s values, we can compute the loss that measures the error
between the model predicted value and the actual target value.
In the training process, we need to use gradient descent algorithm to update the
parameters to reduce the loss. Backpropagation makes use of chain rule to compute
gradients of all parameters with respect to the output efficiently. The backpropagation
is applied on the computation graph from the last output node backward to all other
nodes. During backpropagation, in a node, for each input, multiply the input gradient
with respect to the local output and the node outputs the gradient with respect to the
final output that is received from later node, and then the process continues for each
input node.
Chapter 3. Methods 23
3.3.4.1 Chain Rule
The derivative of composite functions can be computed using chain rule method. For
example, if variable x is a function of y which in turn is a function of z, then according
to the chain rule:
dx
dz
=
dx
dy
.
dy
dz
(3.21)
3.3.4.2 Example
The following illustrates the forward propagation and backpropagation process of feed-
ing one sample data to a neural network that has one hidden layer with ReLU activation
and uses cross-entropy loss.
W,b,W ′,b′ are the weights and biases for hidden layer and output layer respec-
tively. X ,y are the sample data and class label.
Forward propagation
Compute the affine transform for hidden layer.
Z =W T X +b (3.22)
Compute the ReLU activation for hidden layer.
H = max(Z,0) (3.23)
Compute the affine transform for output layer which is the class score.
S =W ′T H +b′ (3.24)
Convert class score to probability using softmax function.
Pk =
eSy
∑ j e
S j
(3.25)
Compute the loss
L =− logPy (3.26)
Backpropagation
Compute gradient of class score.
∂L
∂Sk
= pk−1(y = k) (3.27)
Compute gradient of weight w′.
∂L
∂W ′
= H
∂L
∂S
T
(3.28)
Chapter 3. Methods 24
Compute gradient of bias b′.
∂L
∂b′k
=
∂L
∂Sk
(3.29)
Backpropagate to hidden layer.
∂L
∂H
=W
∂L
∂S
(3.30)
Set non-positive elements to 0 in ∂L
∂H . Because
∂max(x,0)
∂x = 1 if x > 0 and 0 if x≤ 0.
∂L
∂Z
=
∂L
∂H
�1(H > 0) (3.31)
Compute gradient of weight w.
∂L
∂W
= X
∂L
∂Z
T
(3.32)
Compute gradient of bias b.
∂L
∂bk
=
∂L
∂Zk
(3.33)
From above, we can see that during backpropagation, we used many intermediate
results computed in forward propagation. Thus we often save the needed intermediate
values in forward propagation to save computation time by avoiding duplicate compu-
tation in backpropagation.
Although above example is just for a simple neural network, it can be easily ex-
tended to a more complex network. During the forward propagation and backpropaga-
tion process, the computation is local to each layer. Each layer only needs to know the
value propagated to it, compute the values and propagate the values to other layers. It
doesn’t need to care about how other layers do the computation. Thus, different layers
and operations can be used as components to construct deep and very complex neural
networks in many different ways of combination.
3.4 Regularization
We often use the regularization method to reduce overfitting. One way of regularization
is to add weight penalty to the loss. The new loss is the sum of original data loss
and the added regularization loss. The regularization parameter lambda controls the
regularization strength. A large lambda will put more weight to regularization loss
and thus stronger regularization. Small lambda will put more weight to data loss and
thus weaker regularization. Different dataset or network architectures may require
Chapter 3. Methods 25
very different value of lambda. There is no simple way to decide suitable lambda. It
is usually set through cross-validation. By adding regularization loss which penalizes
large weights, it helps to result in networks with smaller weights.
Small weights means a few change of the inputs won’t change the output of the
network too much. Few outliers won’t matter too much for the regularized networks
which make the network less sensitive to the noise in the data. On the other hand, a
little change on some of the inputs may cause the output of network with large weights
change a lot. So large weights will make the model easily adapt to all the training data
including noise.
In summary, regularized networks with small weights tend to be simpler, robust to
noise, less likely to overfit and better to generalize. Unregularized networks with large
weights tend to be more complex, easy to learn the noise and more likely to overfit.
3.4.1 L2 regularization
L =
1
N ∑i
Li︸ ︷︷ ︸
data loss
+
1
2
λ∑
k
∑
l
W 2k,l︸ ︷︷ ︸
regularization loss
(3.34)
3.4.2 L1 regularization
L =
1
N ∑i
Li︸ ︷︷ ︸
data loss
+ λ∑
k
∑
l
|Wk,l|︸ ︷︷ ︸
regularization loss
(3.35)
The L2 regularization and L1 regularization are similar. Both penalize large weights.
But they have different form of weight updating in gradient descent algorithm. For L2
regularization, the additional update of w because of added regularization loss is
w = w−ηλw (3.36)
For L1 regularization, it is
w = w−ηλ sign(w) (3.37)
From above we can see that the updating amount is constant for L1 regularization
and proportional to w for L2 regularization. Thus, the penalty is much larger for L2
regularization when |w| is large and much larger for L1 regularization when |w| is
small. The effect is that weights in L1 are sparse with a small number of relatively
large weights and others driven to 0. On the other hand, L2 regularization weights are
more diffuse. The sparsity feature of L1 regularization makes L1 a better choice for
Chapter 3. Methods 26
feature selection purpose. In other situations, L2 regularization is found usually better
than L1 regularization.
We can also combine these two regularizations which is called Elastic net regular-
ization.
L =
1
N ∑i
Li︸ ︷︷ ︸
data loss
+∑
k
∑
l
λ1|Wk,l|+
1
2
λ2W
2
k,l︸ ︷︷ ︸
regularization loss
(3.38)
Apart from adding regularization loss, another way to avoid weights with too large
magnitude is called Max norm regularization. This method does the weights updating
as normal using gradient descent algorithm and then clips the weights if needed to
ensure each weight vector norm is below a preset maximum value.
3.4.3 Dropout Layer
Dropout is a method to reduce overfitting. In the training stage, we randomly drop
out the neurons and the associated connections according to probability 1− p (Figure
3.10). This has the effect of sampling from a large number of sub-networks. In the
testing stage, we do not drop out neurons. Instead, we use the full networks but with
the neuron’s output weighted with p. In this way, we compute the average output of
all the sub-networks approximately.
By randomly dropping out neurons, the dropout techniques trains over exponen-
tially large number of sub-networks, and using the average prediction of them which is
like a kind of ensemble learning, it reduces the overfitting and also increases the speed
of training.
Chapter 3. Methods 27
Figure 3.10: An example of dropout operation. The first and third neurons and their
associated connections are dropped out.
3.4.4 Batch Normalization
During neural network training, the parameters change of one layer will change the
distribution of inputs of the layers after it. This phenomenon called internal covariate
shiftis is especially true for deep neural network, the impact will be amplified by mul-
tiple layers. To adapt to the input distribution change, it usually requires a low learning
rate, which makes the training slow.
To solve this problem, we can transform inputs to the layer to have mean 0 and
variance 1. This transformation is called whitening. To make the computation fast and
also differentiable required by the backpropagation, we can whiten each dimension of
the input independently.
x =
x−E[x]√
Var[x]
(3.39)
The x is one dimension of the input which is scalar.
To avoid changing the layer’s representation, we add a linear transformation after
the whitening transformation.
y = γx+β (3.40)
Chapter 3. Methods 28
The two transformations together are called batch normalization.
During training, the mean and variance of x are estimated from mini-batch samples.
The population means and variances are also estimated by taking moving averages of
mini-batch statistics during training. During inference, the fixed population means and
variances are used so that the output is only determined by the input.
For a layer in the original network.
z = g(Wu+b) (3.41)
We can apply batch normalization in this way.
z = g(BN(Wu)) (3.42)
The reason to remove b is that it can be cancelled by β parameter in the batch
normalizaton.
In the convolutional layer, the activation map is got by using the same filter applied
on different locations of previous layer. When we use batch normalization for the
convolutional layer, we will normalize all the activations in the activation map together
in the mini-batch. Thus, if the activation map has size p× q and the batch size is m,
then the normalization is applied over the p×q×m values. Just like the activation map
shares the same weights, we use the same parameter γ and β for a activation map.
The batch normalization can reduce layer input distribution change and make the
gradients less sensitive to parameter scales, thus higher learning rate can be used to
speed up the training.
During training, the batch normalization depends on the whole mini-batch samples,
the output of one training sample is not deterministic any more. In this way, batch
normalization has the effect of regulization and can remove other regulization methods
such as dropout.
3.5 Depthwise Separable Convolution
The depthwise separable convolutions factorize the conventional convolution (Figure
3.11) with a depthwise convolution (Figure 3.12) followed by a pointwise convolution
(Figure 3.13).
Chapter 3. Methods 29
Figure 3.11: Conventional convolution example
Figure 3.12: Depthwise convolution example
Chapter 3. Methods 30
Figure 3.13: Pointwise convolution example
The depthwise convolution is done independently for each channel of the input
where a single filter is applied. The pointwise convolution is the same with con-
ventional convolution operation but with kernel size 1×1, which is why it is called
pointwise. It combines the features from depthwise convolution linearly to create
new features.
Thus the depth separable convolution has the effect of filtering input channel through
depthwise convolution and then combining features to create new ones through point-
wise convolution. The effects are exactly the same with conventional convolution. The
difference is that conventional convolution achieves this using a single step, whereas
depth separable convolution uses two separate steps.
Through the separation of feature filtering and feature combining, depthwise sepa-
rable convolution reduces the amount of computation tremendously.
Assume the input I has size W ×H×M where W is the input width, H is the height
and M is the number of input channels. The filer F has size w× h and the number of
filters is N. With stride as 1 and zero padding, the output of conventional convolution
O will have size W ×H×N. The elements of O are computed as follows:
Oi, j,n = ∑
u,v,m
Ii+u, j+v,m ·Fu,v,m,n (3.43)
It takes O(W ·H ·M ·N ·w ·h)
Chapter 3. Methods 31
For depthwise convolution, we use one filter for each input channel. The filter has
size w×h×M. The output of the depthwise convolution has size W ×H×M.
It is computed as follows:
Oi, j,m = ∑
u,v
Ii+u, j+v,m ·Fu,v,m (3.44)
It takes O(W ·H ·M ·w ·h)
Then for the 1× 1 pointwise convolution, it uses N filters, takes the output of
depthwise convolution and generates output of size W×H×N. It takes O(W ·H ·M ·N)
In total, depthwise separable convolution takes O(W ·H ·M ·w ·h+W ·H ·M ·N) =
O(W ·H ·M · (w ·h+N))
The time ratio between depthwise separable convolution and conventional convo-
lution is 1/N + 1/wh. For a typical convolution, where w = 3,h = 3,N > 100, we
achieve about a 9-fold increase in speed.
3.6 Transfer Learning
Training a good deep convolutional neural network model usually requires large com-
putation resource and long time. For example, training a deep convolutional neural
network model on ImageNet may takes weeks even with GPU clusters. If we cannot
afford the computation resource or time, we can use transfer learning method. We
can use a pre-trained model (there are already many state of the art trained models
available free from internet), replace the last fully-connected layer and retrain it. The
previous layers of neural network model can be seen as a feature extractor. The last
fully connected layer is used to compute class scores using extracted features. We can
use the same features as the pre-trained model, but the classes are often different from
pre-trained model, so we need to replace and retrain the last layer. If retraining only
the last layer doesn’t have a satisfactory performance, we may also need to fine-tune
previous layers: initializing weights with pre-trained model and updating them during
training with smaller learning rate. The reason to use smaller learning rate is that we
expect the weights of pre-trained model are not far from the final optimized weights
and we want to update them little by little and not to overstep. Whether fine-tuning is
needed often depends on the similarity between the new dataset and the dataset used
by the pre-trained model in terms of both image data and class labels. If they are very
similar, the kind of features extracted by the layers before last layer in the pre-trained
Chapter 3. Methods 32
model are likely to also suit the new model, and retraining only the last layer may be
enough.
Apart from saving much training time and computation resources using transfer
learning, it often has better results.
Chapter 4
Results and Evaluation
4.1 Resource and tools
The model training and evaluation is implemented using python with TensorFlow
framework 1.0 on Ubuntu Linux system. I use Amazon Elastic Compute Cloud (EC2)
G2 instance which uses NVIDIA GRID K520 GPUs for my model training.
The image classification app on the mobile is implemented using Android java with
TensorFlow mobile library. Currently the TensorFlow mobile library support 3 plat-
forms: Android, IOS and Raspberry Pi. The library provides APIs that let mobile app
easily load pre-trained model and do inference with it. The Android image classifica-
tion app is developed with Android Studio which is the official IDE for Android.
4.1.1 Checkpoint File
During training, we can use TensorFlow API to save the learned model parameters
periodically to binary checkpoint files. Thereby, the model parameters are backed up.
Next time, the model parameters can be restored by loading data from checkpoint file.
4.1.2 Model File
The model file is in Protocol Buffers format which can be saved and loaded using many
different languages. Thus, we can save the model file using python and load the model
using java in Android app.
The Graph object contains all the information about the model graph. The graph
consists of nodes. Each node stores various information including node name, opera-
tion such as “Add” and “Conv2D”, input nodes and other attributes such as filter size
33
Chapter 4. Results and Evaluation 34
for “Conv2D”.
To make it suitable for deployment, we can use tool from TensorFlow freeze graph.py
to combine the graph definition file and checkpoint file, which contains learned param-
eters into a single model file. The tool achieves this by replacing Variable node with
Const node that contains the parameters and it also removes nodes unnecessary for
inference to simplify graph and decreases file size.
The resulting model file can then be shipped with Android app. In the Android
app, upon starting, we will first load the model file using TensorFlow Mobile java API.
Then we can do inference using the loaded model.
4.2 Dataset
4.2.1 CIFAR 100
Figure 4.1: A sample of 100 images from CIFAR-100
Chapter 4. Results and Evaluation 35
The CIFAR-100 dataset contains 60000 small images of size 32×32. They belong to
100 different classes, with each class containing 600 images. A sample of 100 images
of this dataset is shown in figure 4.1.
4.3 Experimental Setup
4.3.1 Training set and test set
This CIFAR-100 dataset is divided into training set which contains 50000 images and
test set which contains 10000 images.
4.3.2 Preprocessing
During the training, an image is randomly transformed before feeding to the neural
networks. In this way, the neural networks will train on multiple versions of the same
image and the actual training data set size is much larger than original data set size.
This will make the model better generalize and reduce overfitting.
4.3.2.1 Randomly Shift the Image
First pad the image, and then randomly crop the image. In this way, the image will
randomly shift in the four directions.
4.3.2.2 Randomly Flip the Image
The image is flipped left to right with 0.5 probability.
4.3.2.3 Randomly adjust the image brightness
This randomly add a value between -63 and 63 to all RGB components of every pixel.
4.3.2.4 Randomly change the image contrast
Randomly choose a contrast factor 0.2 ≤ f ≤ 1.8. For each RBG channel, compute
the mean m and update the corresponding component of each pixel with:
(x−m)× f +m
After above randomly changing steps of the image, lastly we normalize the image
data to make it have zero mean and unit norm.
Chapter 4. Results and Evaluation 36
4.3.3 MobileNet
Hyperparameters
• Batch Size: 128
• Momentum: 0.9
• Initial learning rate: 0.01
• Learning rate decay: decay with factor 0.94 every 2 epochs
• Weight decay parameter: 0.00004
• Optimizer: RMSProp optimization algorithm with decay rate of 0.9
The initial weights are loaded from mobilenet pre-trained model on ImageNet. In
the first stage, train only on the last fully connected layer and keeping the parameters
of previous layers unchanged. It trains 25000 steps in this phase. Then train all layers
to fine-tune the model. It trains 55000 steps in this phase. During training, random
minor changes are applied on the images to augment the data set.
After training finishes, we use the test set to evaluate the performance. Note that
the prediction on each image is just done once. If using average prediction of multiple
changes on an image is used, the performance is likely to improve.
The models are exported to TensorFlow model file. In the Android mobile image
classification app, the model file is loaded and the inference time is computed by di-
viding the time it takes to classify 100 images one by one with 100. The inference time
on mobile is done on Nexus 6 Android phone.
The experiments are done for width multipliers 1.0, 0.75, 0.5 and 0.25, image sizes
32, 24 and 16. Thus, the above steps are done for a total of 12 models.
The change of losses with training steps for model with width multiplier 1.0 and
image size 32 are as follows. Others are similar. The red line is for first stage and the
green line for the second stage.
Figure 4.2, 4.3 and 4.4 shows the change of total loss, cross entropy loss and regu-
larization loss with the training steps in both stages.
Chapter 4. Results and Evaluation 37
Figure 4.2: Total Loss
Figure 4.3: Cross Entropy Loss
Chapter 4. Results and Evaluation 38
Figure 4.4: Regularization Loss
4.3.4 Inception V3
Google Inception V3 model is proposed in [5]. It adds an auxiliary logits layer in
addition to usual logits layer to speedup convergence during training. For this model
in the experiment, scale the image from 32×32 to 128×128. The first stage trains on
the auxiliary logits layer and logits layer 15000 steps with a fixed learning rate of 0.01.
The second stage trains 30000 steps on all layers with a smaller fixed learning rate of
0.0001. Both stages use weight decay of 0.00004.
Figures 4.5, 4.6 and 4.7 show the change of total loss, cross entropy loss and reg-
ularization loss with the training steps in both training stages for the Inception V3
model.
Chapter 4. Results and Evaluation 39
Figure 4.5: Total Loss
Figure 4.6: Cross Entropy Loss
Figure 4.7: Regularization Loss
Chapter 4. Results and Evaluation 40
4.3.5 ResNet
The ReNet model is proposed in [6]. For this model in the experiment, it undergoes
the same process with the Inception V3 model during training.
Figures 4.8, 4.9 and 4.10 shows the change of total loss, cross entropy loss and
regularization loss with the training steps in both training stages for the ResNet model.
Figure 4.8: Total Loss
Figure 4.9: Cross Entropy Loss
Chapter 4. Results and Evaluation 41
Figure 4.10: Regularization Loss
4.4 Metrics
4.4.1 Top-1 Accuracy
The ratio between the number of images that are predicted correctly and the total num-
ber of images in the test set.
4.4.2 Top-5 Accuracy
Same with top-1 accuracy, it is the ratio between the number of correct predictions
and the total number of images. The difference is the meaning of correct prediction.
For top-5 accuracy, classifier gives five candidate guesses instead of one guess. If the
correct label is one of the five guesses, then the prediction is considered correct.
4.4.3 Inference Time
The average time model takes to classify a single image.
4.4.4 Model File Size
The size of the model file in TensorFlow for deployment. The model file size is mainly
determined by the number of parameters and the number of bits used to encode each
parameter.
Chapter 4. Results and Evaluation 42
4.5 Results
Table 4.1 shows the performance for MobileNets with various width multipliers and
resolution multipliers. Table 4.2 shows performance for full MobileNet, Inception V3
and ResNet.
Table 4.1: Performance for different width multipliers and resolution multipliers.
Table 4.2: Performance of Different Models
Chapter 4. Results and Evaluation 43
4.6 Analysis
Table 4.3: Relative Performance
For comparison purposes, the accuracy loss, inference time speedup and model size
compression ratio of MobileNet model over Inception 3 and ResNet are computed in
table 4.3.
We can see that the MobileNet have significant inference speed up and model size
compression over Inception and ResNet. Its accuracy is similar with ResNet and
have a relatively big loss compared with Inception.
We can also see that smaller width multiplier will decrease inference time, model
size and accuracy. Smaller resolution multiplier will not affect model size and will
decrease inference time and accuracy. Because smaller width multiplier will decrease
the number of channels used in the filters which will decrease the number of parame-
ters, so the model file decreases. Smaller resolution multiplier will decrease the input
image size; thus, the amount of computation decrease, but the number of parameters is
the same. Thus it will speed up inference but not shrink model file size.
The results also show that it is better to decrease width multiplier than resolution
multiplier to speed up inference and shrink model file. For example, using width mul-
tiplier 0.75 and resolution multiplier 1.0 have higher accuracy, quicker inference and
smaller model size than using width multiplier 1.0 and resolution multiplier 0.75.
Chapter 5
Conclusion and Discussion
5.1 Remarks and observations
This project implements the MobileNet model using the Tensorflow framework. The
approximate computing techniques, approximating traditional convolutional layer with
depth-wise separable convolution layer, are used. An Android mobile image classifi-
cation app is built to test the real inference time of each model. In the experiment,
MobileNets with various width multipliers and resolution multipliers are successfully
trained on the CIFAR-100 dataset to compare these two hyperparameters’ effect on
the performance, which show that by adjusting them we can get different trade-offs
between accuracy and efficiency. The decrease of width multiplizer and resolution
multiplier lead to smaller model size and quicker image classification on mobiles
with greater accuracy loss. Thus, mobile developers can adjust them to find the best
trade-off for their applications. Comparison with other models such as Inception
and ResNet are also done in the experiment, which shows that MobileNet has much
speedup in inference time and smaller mobile size with reasonable accuracy sacrifice.
The resulting model is more suitable for mobile deployment which takes much less
memory space and inference time.
5.2 Limitation and Further work
5.2.1 More approximate computing techniques
Currently, the approximate computing technique used is depth-wise separable convolu-
tion which is approximation to traditional convolution. We would like to apply network
44
Chapter 5. Conclusion and Discussion 45
pruning and quantization techniques on the resulting models to further decrease model
size and inference time in future work.
5.2.2 More extensive Experiment
In this project, due to computing resource and time constraint, we use one dataset
CIFAR-100 and two traditional popular models Inception and ResNet in compari-
son. In future work, we will use more datasets and more models to do more extensive
evaluation.
5.2.3 Application into Practice
In future work, we would like to put the approximate computing techniques used in
this project into real practice. Many mobile applications would benefit from approxi-
mate computing techniques used in this project. Two examples are bank card number
recognition and handwritten Chinese character recognition. The first one can be used
in a payment app that let users avoid the hassle of entering card number manually. The
second one can be used in Chinese input app. The computing techniques used in this
project would make the recognition in the two applications much faster and the apps
less memory-consuming.
5.2.4 Model Architecture Improvement
Although the MobileNet achieves significant inference speedup and model size shrink-
ing, it has a relatively big accuracy loss compared with the Inception model. Thus,
we would like to adjust the model architecture to improve its accuracy in future work.
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