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日期:2019-11-29 12:57

Detecting Handwritten Prime Digits

with Neural Networks

November 20, 2019

100 points

1 Introduction

Prime numbers are fantastic. In this assignment we will use Multi Layer Perceptrons

to detect handwritten prime digits. But before doing such a difficult

task I suggest to try and solve an easier problem with MLP. If you succeed,

which I know you will, you can proceed with tackling the challenging problem

of detecting handwritten prime digits.

2 Regression - Toy Data

The first task is to learn the function that generated the following data, using

a simple neural network.

The function that produced this data is actually y = x

2+, where  ~ N(0, σ)

is a random noise from a normal distribution with a small variance. We are going

to use an MLP with one hidden layer, which each has 5 neurons, to learn

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an approximation of this function using the data that we have. This assignment

comes with a starting code, which is incomplete and you are supposed to

complete it.

2.1 Technical Details

2.1.1 Code

The code that comes with this assignment has multiple files including:

assignment

toy example regressor.py

layers.py

neural network.py

utils.py

? toy_example_regressor.py contains most of the codes that are related

to the training procedure, including loading data and iteratively feeding

mini-batches of data to the neural network, plotting the approximated

function and the data, etc. Please read this file and understand it but you

don’t need to modify this.

? layers.py contains definition of the layers that we use in this assignment,

including DenseLayer, SigmoidLayer, and L2LossLayer. Your main responsibility

is to implement forward and backward functions for these

layers.

? neural_network.py contains the definition of a neural network (NeuralNet

class), which is an abstract class. This class basically takes care of running

forward pass and propagating the gradients, backwards, from loss to the

first layer.

? utils.py contains some useful functions.

2.1.2 Data

The training data for this problem, which consists of input data and labels, can

be generated by the function get_data(), which you can find in the main file,

toy_example_regressor.py.

2.1.3 Network Structure

For the regression problem (i.e. the first task) we defined a new class, SimpleNet

, which is inherited from NeuralNet. SimpleNet contains two DenseLayer

s, which one of them has hidden neurons with Sigmoid activation functions.

Network definition can be found in toy_example_regressor.py.

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2.2 Your Task (total: 80 points)

2.2.1 Implementing compute_activations, compute_gradients, and update_weights

functions

There are three type of layers completely implemented in the layers.py file:

DenseLayer, Sigmoid, and L2LossLayer. However, implementation of DenseLayer

is incomplete. You are supposed to implement the following functions

? DenseLayer: This is a simple dense (or linear or fully connected) layer

that has two types of parameters: weights w and biases b.

– compute_activations (15 points): The value of every output neuron

is oi = x.wi + bi

. The number of input and output neurons are

specified in the __init__ function.

– compute_gradients (20 points): Assume that gradient of the loss

with respect to the output neurons, self._output_error_gradient

, are computed by the next layer already. You need to compute the

gradients of the loss with respect to all the parameters of this layer

(i.e. b and w) and store them in self.dw and self.db so that you

can use them the update the parameters later. Needless to say that

shape of dw and w should be equal, and same goes for db and b. In

addition, you should compute the gradient of the loss with respect

to the input, which is the output of the previous layer, and store it

in self._input_error_gradient. This value will be passed on to

the previous layer in the network, which will be used to compute the

gradients recursively (Back Propagation).

– update_weights (10 points): You should perform Stochastic Gradient

Descent and update the weights using the current weights, gradients,

and the given learning rate ( newweights = currentweights ?

learningrate ? gradient )

You can refer to the implementations of Sigmoid and L2LossLayer. Note: It’s

up to you how to implement these functions. However, it would be

computationally less expensive if you use numpy matrix operations to

compute the value of the neurons or gradients.

Your next task is to implement same functions for the NeuralNet class.

? compute_activations (10 points): Iterate over all layers starting from the

first one and compute the activations for each layer. At the end return

the output of the last layer along with the value of loss.

? compute_gradients (10 points): You are supposed to perform back propagation.

In other words, first compute the gradient of the lass layer and

pass it to the last layer. Then starting from the last layer iterate over

all layers backwards, first compute its gradient and then pass it to the

previous layer.

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? update_weights (5 points): You should update the parameters of all the

layers (i.e. those who have parameters). You can use the update_weight

function of each layer

2.2.2 Training the model (10 points)

Once you are done with implementing and testing the correctness of the implemented

functions, you are ready to build a multi layer Perceptron and train

it.

There is already an existing starter code for you at toy_example_regressor

.py. It’s a script that contains definition of a simple two layer MLP with scripts

that loads the data and trains the MLP. At the end of the training the code

plots data and the approximated function. Also the network weights will be

saved to a file with this name simple_net_weights-{timestamp}.pkl. You

should change its name to simple_net_weights.pkl and include it in

your submission.. The timestamp is added to make sure you don’t overwrite

some previously well trained model.

In addition, you should check the loss and the saved image. Check if the

predicted function is similar to f(x) = x

2 and matches the validation data.

You should include the saved image, data_function.png, both in your

report and in your submission.

Note that this is a regression task, so the last layer of the MLP only has one

neuron without any activation functions.

3 Detecting Prime Digits

Now we can use the same layers to distinguish one digit prime numbers (i.e.

2,3,5, and 7) from one digit composite numbers (i.e. 1, 4, 6, 8, and 9). This is

a binary classification problem. So the MLP that we are going to use will have

only one neuron in the last layer with a sigmoid activation function.

3.1 Data

The dataset that you will be using for this task is the MNIST dataset, which

contains gray scale images of hand written digits. Sizes of images are 28 by

28 pixels. We have already preprocessed it for you. We have set the labels for

prime digits to 1, and 0 otherwise. Also we have normalized the values of pixels

(i.e. pixel values are in the interval of [0, 1]).

3.2 Network Structure

The input of the network is a vector with 784 neurons (28 ? 28) The network

has one hidden layer with 20 neurons with sigmoid activation functions. The

output of the network is one neuron with sigmoid activation function.

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3.3 Your Task (20 points)

For this task, all you need to do is to read and understand the prime_classifier

.py code and then run it. Over the course of training, loss values and accuracies

on the validation set is printed. At the end of the training, network parameters

will be saved in a file with the following format prime_net_weights-{

timestamp}.pkl. You should change its name to prime_net_weights.pkl

and include it in your submission.. The timestamp is added to make sure

you don’t overwrite some previously well trained model.

4 Testing Your Code

To help you with testing your code, a number of tester files have been included.

You can use them to test your implementations. For example if you run:

$ python test_layers.py

You can see if the three functions (DenseLayer.compute_activations(),

DenseLayer.compute_gradients(), and DenseLayer.update_weights()) that

you implemented in layers.py file work properly or not. We recommend to

use all the four test files that are included:

testers

test layers.py

test neural network.py

test toy example regressor.py

test prime classifier.py

Also, you can estimate the total points that you might get for this assignment

by running the following code:

$ python evaluate_assignment.py

Note: We will use stronger test cases to test your code and grade

your assignment. So passing these tests does not guarantee anything.

These tests are only meant to help you with this assignment.

5 What to submit

You should include all the following files in a tar.gz or zip file with your

student id (either YOUR_STUDENT_ID.tar.gz or YOUR_STUDENT_ID.zip).

1. Your code. Do not change the signature of the functions that you were

supposed to implement. Do NOT include the dataset (assignment4-

mnist.pkl)

2. You should include the following files that are automatically saved in your

submission:

? prime_net_weights.pkl

? simple_net_weights.pkl

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? data_function.png

3. A report.pdf file concisely explaining what you did in this assignment.

Also in your report include your model’s loss (for both problems) and

accuracy (only for prime digit detection).

Note: your code will be evaluated with an automated script. So if you

don’t follow the above steps, you will lose all or a portion of your points for this

assignment.

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