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Assignment 1 Backpropagation

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ELEC 576 / COMP 576
Assignment 1
Due: https://elec576.rice.edu/assignments via Canvas
Submission Instructions
Every student must submit their work in a zip file in the following format: netid-assignment1.zip.
You should also provide intermediate and final results as well as any necessary code. Submit
your zip file on Canvas.
GPU Resource from AWS
To accelerate the training using GPU, you can optionally use Amazon Web Services(AWS)
GPU instance using AWS Education credits. You can also get additional AWS credits from
Github Student Developer Pack.
After having an AWS account, You can either create a fresh ubuntu instance and install
software dependencies by yourself or use off-the-shelf TensorFlow ready image from AWS
Marketplace.
1 Backpropagation in a Simple Neural Network
In this problem, you will learn how to implement the backpropagation algorithm for a
simple neural network. To make your job easier, we provide you with starter code in
three layer neural network.py. You will fill in this starter code to build a 3-layer neural
network (see Fig. 1) and train it using backpropagation.
a) Dataset
We will use the Make-Moons dataset available in Scikit-learn. Data points in this dataset
form two interleaving half circles corresponding to two classes (e.g. “female” and “male”).
In the main() function of three layer neural network.py, uncomment the “generate and
visualize Make-Moons dataset” section (see below) and run the code. Include the generated
figure in your report.
# generate and visualize Make-Moons dataset
X, y = generate_data()
plt.scatter(X[:, 0], X[:, 1], s=40, c=y, cmap=plt.cm.Spectral)
b) Activation Function
Tanh, Sigmoid and ReLU are popular activation functions used in neural networks.
You will implement them and their derivatives.
1. Implement function actFun(self, z, type) in three layer neural network.py.
This function computes the activation function where z is the net input and type ∈
{‘Tanh’, ‘Sigmoid’, ‘ReLU’}.
2. Derive the derivatives of Tanh, Sigmoid and ReLU
3. Implement function diff actFun(self, z, type) in three layer neural network.py.
This function computes the derivatives of Tanh, Sigmoid and ReLU.
c) Build the Neural Network
Lets now build a 3-layer neural network of one input layer, one hidden layer, and one
output layer. The number of nodes in the input layer is determined by the dimensionality
of our data, 2. The number of nodes in the output layer is determined by the number of
classes we have, also 2. The input to the network will be x- and y- coordinates and its
output will be two probabilities, one for class 0 (“female”) and one for class 1 (“male”).
The network looks like the following.
Mathematically, the network is defined as follows.
z1 = W1x + b1 (1)
a1 = actFun(z1) (2)
z2 = W2a1 + b2 (3)
a2 = ˆy = softmax(z2) (4)
where zi
is the input of layer i and ai
is the output of layer i after applying the activation
function. θ ≡ {W1, b1, W2, b2} are the parameters of this network, which we need to learn
from the training data.
If we have N training examples and C classes then the loss for the prediction ˆy with
respect to the true labels y is given by:
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Figure 1: A three-layer neural network
L(y, yˆ) = −
1
N
X
n∈N
X
i∈C
yn,i log ˆyn,i (5)
Note that y are one-hot-encoding vectors and ˆy are vectors of probabilities.
1. In three layer neural network.py, implement the function feedforward(self,
X, actFun). This function builds a 3-layer neural network and computes the two
probabilities (self.probs in the code or a2 in Eq. 4), one for class 0 and one for
class 1. X is the input data, and actFun is the activation function. You will pass the
function actFun you implemented in part b into feedforward(self, X, actFun).
2. In three layer neural network.py, fill in the function calculate loss(self, X,
y). This function computes the loss for prediction of the network. Here X is the input
data, and y is the given labels.
d) Backward Pass – Backpropagation
It’s time to implement backpropagation, finally!
1. Derive the following gradients: ∂L
∂W2
,
∂L
∂b2
,
∂L
∂W1
,
∂L
∂b1
mathematically
2. In three layer neural network.py, implement the function backprop(self, X,
y). Again, X is the input data, and y is the given labels. This function implements
backpropagation (i.e., computing the gradients above).
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e) Time to Have Fun – Training!
You already have all components needed to run the training. In three layer neural network.py,
we also provide you function visualize decision boundary(self, X, y) to visualize the
decision boundary. Let’s have fun with your network now.
1. Train the network using different activation functions (Tanh, Sigmoid and ReLU).
Describe and explain the differences that you observe. Include the figures generated
in your report. In order to train the network, uncomment the main() function in
three layer neural network.py, take out the following lines, and
run three layer neural network.py.
plt.scatter(X[:, 0], X[:, 1], s=40, c=y, cmap=plt.cm.Spectral)
plt.show()
2. Increase the number of hidden units (nn hidden dim) and retrain the network using Tanh as the activation function. Describe and explain the differences that you
observe. Include the figures generated in your report.
f) Even More Fun – Training a Deeper Network!!!:
Let’s have some more fun and be more creative now. Write your own n layer neural network.py
that builds and trains a neural network of n layers. Your code must be able to accept
as parameters (1) the number of layers and (2) layer size. We provide you hints below
to help you organize and implement the code, but if you have better ideas, please
feel free to implement them and ignore our hints. In your report, please tell us why
you made the choice(s) you did.
Hints:
1. Create a new class, e.g DeepNeuralNetwork, that inherits NeuralNetwork in
three layer neural network.py
2. In DeepNeuralNetwork, change function feedforward, backprop, calculate loss
and fit model
3. Create a new class, e.g. Layer(), that implements the feedforward and backprop steps for a single layer in the network
4. Use Layer.feedforward to implement DeepNeuralNetwork.feedforward
5. Use Layer.backprop to implement DeepNeuralNetwork.backprop
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6. Notice that we have L2 weight regularizations in the final loss function in addition to the cross entropy. Make sure you add those regularization terms in
DeepNeuralNetwork.calculate loss and their derivatives in
DeepNeuralNetwork.fit model.
Train your network on the Make Moons dataset using different number of layers,
different layer sizes, different activation functions and, in general, different network
configurations. In your report, include generated images and describe what you observe and what you find interesting (e.g. decision boundary of deep vs shallow neural
networks).
Next, train your network on another dataset different from Make Moons. You
can choose datasets provided by Scikit-learn (more details here) or any dataset of
your interest. Make sure that you have the correct number of input and output
nodes. Again, play with different network configurations. In your report, describe
the dataset you choose and tell us what you find interesting.
Be curious and creative!!! You are exploring Deep Learning. 🙂
2 Training a Simple Deep Convolutional Network on MNIST
Deep Convolutional Networks (DCN) have been state-of-the-art in many perceptual
tasks including object recognition, image segmentation, and speech recognition. In
this problem, you will build and train a simple 4-layer DCN on MNIST Dataset.
We provide you with starter code in dcn mnist.py. You will fill in this starter code
to complete task (a), (b), and (c) below. Also, since one of the purposes of this
assignment is to get you familiar with Tensorflow, dcn mnist.py is purposely written
in a very simple form, a.k.a no object-oriented structure, no fancy tool in both Tensorflow and Python. That says, you are encouraged (but not required) to re-organize
dcn mnist.py as you like, but be sure to explain your code in the report. For acknowledgement, dcn mnist.py inherits from this tutorial Deep MNIST for Expert.
MNIST is a dataset of handwritten digits (from 0 to 9). This dataset is one of
the most popular benchmarks in machine learning and deep learning. If you develop an algorithm to learn from static images for tasks such as object recognition,
most likely, you will want to debug your algorithm on MNIST first before testing it
on more complicated datasets such as CIFAR10 and SVHN. There are also modified
versions of MNIST, such as permutation invariant MNIST, which will come in handy
for benchmarking at times.
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More details, the MNIST data is split into three parts: 55,000 data points of
training data (mnist.train), 10,000 points of test data (mnist.test), and 5,000 points of
validation data (mnist.validation). The digits have been size-normalized and centered
in a fixed-size image. MNIST images are of size 28 x 28. When loaded in Tensorflow,
each image is flattened into a vector of 28×28=784 numbers. Each MNIST image
will have a corresponding label which is a number between 0 and 9 corresponding to
the digit that is drawn in that image. In Tensorflow, the following code will load the
MNIST dataset.
#input_data file is provided to load the mnist
import input_data
mnist = input_data.read_data_sets(‘MNIST_data’, one_hot=True)
The tutorial on how to install Tensorflow can be found at TensorFlow Download
and Setup.
a) Build and Train a 4-layer DCN
The architecture of the DCN that you will implement is as follows.
conv1(5-5-1-32) – ReLU – maxpool(2-2) – conv2(5-5-32-64) – ReLU – maxpool(2-2)
– fc(1024) – ReLU – DropOut(0.5) – Softmax(10)
More details on the architecture can be found in this tutorial Deep MNIST for
Expert.
Follow the tutorial Deep MNIST for Expert to fill in dcn mnist.py. Particularly,
1. Read the tutorial Deep MNIST for Expert to learn how to use Tensorflow.
2. Complete functions weight variable(shape), bias variable(shape), conv2d(x,
W), max pool 2×2(x) in dcn mnist.py. The first two functions initialize the
weights and biases in the network, and the last two functions will implement
convolution and max-pooling operators, respectively.
3. Build your network: In dcn mnist.py, you will see ”FILL IN THE CODE
BELOW TO BUILD YOUR NETWORK”. Complete the following sections
in dcn mnist.py: placeholders for input data and input labeles, first
convolutional layer, convolutional layer, densely connected layer, dropout,
softmax.
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4. Set up Training: In dcn mnist.py, you will see ”FILL IN THE FOLLOWING
CODE TO SET UP THE TRAINING”. Complete section setup training in
dcn mnist.py.
5. Run Training: Study the rest of dcn mnist.py. Notice that, different from the
tutorial Deep MNIST for Expert, I use the summary operation (e.g. summary op,
summary writer, …) to monitor the training. Here, I only monitor the training
loss value. Now, run dcn mnist.py. What is the final test accuracy of your network? Note that I set the batch size to 50, and to save time, I set the max step
to only 5500. Batch size is the number of MNIST images that are sent to the
DCN at each iteration, and max step is the maximum number of training iterations. max step = 5500 means the training will stop after 5500 iterations no
matter what. When batch size is 50, 5500 iterations is equivalent to 5 epochs.
Remind that, in each epoch, the DCN will see the whole training set once. In
this case, since there are 55K training images, each epoch is consisted of 55K/50
= 1100 iterations.
6. Visualize Training: In your terminal, type tensorboard –logdir=path/to/results
where path/to/results is result dir in dcn mnist.py. Follow the instruction in
your terminal to visualize the training loss in the training. You will be asked to
navigate to a website to see the results, e.g. http://172.28.29.81:6006. Include
the figures generated by TensorBoard in your report.
b) More on Visualizing Your Training
In part (a) of this problem, you only monitor the training loss during the training. Now, let’s visualize your training more! Study dcn mnist.py and this tutorial
TensorBoard: Visualizing Learning to learn how to monitor a set of variables during the
training. Then, modify dcn mnist.py so that you can monitor the statistics (min, max,
mean, standard deviation, histogram) of the following terms after each 100 iterations:
weights, biases, net inputs at each layer , activations after ReLU at each layer, activations
after Max-Pooling at each layer. Also monitor the test and validation error after each
1100 iterations (equivalently, after each epoch). Run the training again and visualize the
monitored terms in TensorBoard. Include the resultant figures in your report.
c) Time for More Fun!!!
As you have noticed, I use ReLU non-linearity, random initialization, and Adam training algorithm in dcn mnist.py. In this section, run the network training with different nonlinearities (tanh, sigmoid, leaky-ReLU, MaxOut,…), initialization techniques (Xavier…)
and training algorithms (SGD, Momentum-based Methods, Adagrad..). Make sure you
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still monitor the terms specified in part (b). Include the figures generated by TensorBoard
and describe what you observe. Again, be curious and creative! You are encouraged to
work in groups, but you need to submit separate reports.
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Assignment 1 Backpropagation
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