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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
Step1:  Implementing a denoising autoencoder
Complete and hand in this completed worksheet (including its outputs and any supporting code outside of the worksheet) with your assignment submission. For more details see the HW page on the course website.
In this exercise we will develop a denoising autoencoder, and test it out on the MNIST dataset.
Step2:  We will use the class DenoisingAutoencoder in the file METU/denoising_autoencoder.py to represent instances of our network. The network parameters are stored in the instance variable self.params where keys are string parameter names and values are numpy arrays. Below, we initialize toy data and a toy model that we will use to develop your implementation.
Step3:  Forward pass
Step4:  Backward pass
Implement the rest of the function. This will compute the gradient of the loss with respect to the variables W1, b1, W2, and b2. Now that you (hopefully!) have a correctly implemented forward pass, you can debug your backward pass using a numeric gradient check
Step5:  Train the network
To train the network we will use stochastic gradient descent (SGD). Look at the function DenoisingAutoencoder.train_with_SGD and fill in the missing sections to implement the training procedure. This should be very similar to the training procedures you used in the first HW. 
Once you have implemented the method, run the code below to train the network on toy data. You should achieve a training loss less than 2.0.
Step6:  Load the data
Now that you have implemented a DAE network that passes gradient checks and works on toy data, it's time to load up the MNIST dataset so we can use it to train DAE on a real dataset. Make sure that you have run "cs231n/datasets/get_datasets.sh" script before you continue with this step.
Step7:  Train a network
To train our network we will use SGD with momentum. In addition, we will adjust the learning rate with an exponential learning rate schedule as optimization proceeds; after each epoch, we will reduce the learning rate by multiplying it by a decay rate.
Step8:  Debug the training
With the default parameters we provided above, you should get a validation accuracy of about 0.29 on the validation set. This isn't very good.
One strategy for getting insight into what's wrong is to plot the loss function and the accuracies on the training and validation sets during optimization.
Another strategy is to visualize the weights that were learned in the first layer of the network. In most neural networks trained on visual data, the first layer weights typically show some visible structure when visualized.
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Python Code:
# A bit of setup
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
plt.rcParams['figure.figsize'] = (10.0, 8.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'
# for auto-reloading external modules
# see http://stackoverflow.com/questions/1907993/autoreload-of-modules-in-ipython
%load_ext autoreload
%autoreload 2
def rel_error(x, y):
   returns relative error 
  return np.max(np.abs(x - y) / (np.maximum(1e-8, np.abs(x) + np.abs(y))))
Explanation: Implementing a denoising autoencoder
Complete and hand in this completed worksheet (including its outputs and any supporting code outside of the worksheet) with your assignment submission. For more details see the HW page on the course website.
In this exercise we will develop a denoising autoencoder, and test it out on the MNIST dataset.
End of explanation
from METU.denoising_autoencoder import DenoisingAutoencoder
from METU.Noise import Noise, GaussianNoise
# Create a small net and some toy data to check your implementations.
# Note that we set the random seed for repeatable experiments.
input_size = 4
hidden_size = 2
num_inputs = 100
# Outputs are equal to the inputs
network_size = (input_size, hidden_size, input_size)
def init_toy_model(num_inputs, input_size):
  np.random.seed(0)
  net = DenoisingAutoencoder((input_size, hidden_size, input_size))
  net.init_weights()
  return net
def init_toy_data(num_inputs, input_size):
  np.random.seed(1)
  X = np.random.randn(num_inputs, input_size)
  return X
net = init_toy_model(num_inputs, input_size)
X = init_toy_data(num_inputs, input_size)
print "Ok, now we have a toy network"
Explanation: We will use the class DenoisingAutoencoder in the file METU/denoising_autoencoder.py to represent instances of our network. The network parameters are stored in the instance variable self.params where keys are string parameter names and values are numpy arrays. Below, we initialize toy data and a toy model that we will use to develop your implementation.
End of explanation
loss,_ = net.loss(GaussianNoise(0.5)(X), X, reg=3e-3, activation_function='sigmoid')
correct_loss = 2.42210627243
print 'Your loss value:' + str(loss)
print 'Difference between your loss and correct loss:'
print np.sum(np.abs(loss - correct_loss))
Explanation: Forward pass: compute loss
Open the file METU/denoising_autoencoder.py and look at the method DenoisingAutoencoder.loss. This function is very similar to the loss functions you have written in the first HW: It takes the data and weights and computes the class scores, the loss, and the gradients on the parameters. 
Implement the first part of the forward pass which uses the weights and biases to compute the scores for the corrupted input. In the same function, implement the second part that computes the data and the regularization losses.
End of explanation
from METU.gradient_check import eval_numerical_gradient
reg = 3e-3
# Use numeric gradient checking to check your implementation of the backward pass.
# If your implementation is correct, the difference between the numeric and
# analytic gradients should be less than 1e-8 for each of W1, W2, b1, and b2.
net.init_weights()
noisy_X = GaussianNoise(0.5)(X)
loss, grads = net.loss(noisy_X, X, reg, activation_function='tanh')
# these should all be less than 1e-5 or so
f = lambda W: net.loss(noisy_X, X, reg, activation_function='tanh')[0]
W1_grad = eval_numerical_gradient(f, net.weights[1]['W'], verbose=False)
print '%s max relative error: %e' % ("W1", rel_error(W1_grad, grads[1]['W']))
W0_grad = eval_numerical_gradient(f, net.weights[0]['W'], verbose=False)
print '%s max relative error: %e' % ("W0", rel_error(W0_grad, grads[0]['W']))
b1_grad = eval_numerical_gradient(f, net.weights[1]['b'], verbose=False)
print '%s max relative error: %e' % ("b1", rel_error(b1_grad, grads[1]['b']))
b0_grad = eval_numerical_gradient(f, net.weights[0]['b'], verbose=False)
print '%s max relative error: %e' % ("b0", rel_error(b0_grad, grads[0]['b']))
Explanation: Backward pass
Implement the rest of the function. This will compute the gradient of the loss with respect to the variables W1, b1, W2, and b2. Now that you (hopefully!) have a correctly implemented forward pass, you can debug your backward pass using a numeric gradient check:
End of explanation
net = init_toy_model(num_inputs, input_size)
reg = 3e-3
stats = net.train_with_SGD(X, noise=GaussianNoise(sd=0.5),
            learning_rate=0.02, learning_rate_decay=0.95, 
            reg=reg, batchsize=100, num_iters=500, verbose=False, activation_function='sigmoid')
print 'Final training loss: ', stats['loss_history'][-1]
# plot the loss history
plt.plot(stats['loss_history'])
plt.xlabel('iteration')
plt.ylabel('training loss')
plt.title('Training Loss history')
plt.show()
Explanation: Train the network
To train the network we will use stochastic gradient descent (SGD). Look at the function DenoisingAutoencoder.train_with_SGD and fill in the missing sections to implement the training procedure. This should be very similar to the training procedures you used in the first HW. 
Once you have implemented the method, run the code below to train the network on toy data. You should achieve a training loss less than 2.0.
End of explanation
from cs231n.data_utils import load_mnist
X_train, y_train, X_val, y_val, X_test, y_test = load_mnist()
X_train = X_train.reshape(X_train.shape[0], -1)
X_val = X_val.reshape(X_val.shape[0], -1)
X_test = X_test.reshape(X_test.shape[0], -1)
    
print 'Train data shape: ', X_train.shape
print 'Train labels shape: ', y_train.shape
print 'Test data shape: ', X_test.shape
print 'Test labels shape: ', y_test.shape
#Visualize some samples
x = np.reshape(X_train[100], (28,28))
plt.imshow(x)
plt.title(y_train[0])
plt.show()
plt.imshow(GaussianNoise(rate=0.5,sd=0.5)(x))
plt.show()
# Yes, DAE will learn to reconstruct from such corrupted data
Explanation: Load the data
Now that you have implemented a DAE network that passes gradient checks and works on toy data, it's time to load up the MNIST dataset so we can use it to train DAE on a real dataset. Make sure that you have run "cs231n/datasets/get_datasets.sh" script before you continue with this step.
End of explanation
import time
input_size = 28 * 28
hidden_size = 300 # Try also sizes bigger than 28*28
reg = 0.003 # 3e-3
net = DenoisingAutoencoder((input_size, hidden_size, input_size))
net.init_weights()
# Train with SGD
tic = time.time()
stats = net.train_with_SGD(X_train, noise=GaussianNoise(rate=0.5,sd=0.5),
            learning_rate=0.4, learning_rate_decay=0.99, 
            reg=reg, num_iters=1000, batchsize=128, momentum='classic', mu=0.9, verbose=True, 
            activation_function='sigmoid')
toc = time.time()
print toc-tic, 'sec elapsed'
Explanation: Train a network
To train our network we will use SGD with momentum. In addition, we will adjust the learning rate with an exponential learning rate schedule as optimization proceeds; after each epoch, we will reduce the learning rate by multiplying it by a decay rate.
End of explanation
# Plot the loss function and train / validation accuracies
plt.subplot(2, 1, 1)
plt.plot(stats['loss_history'])
plt.title('Loss history')
plt.xlabel('Iteration')
plt.ylabel('Loss')
plt.show()
#from cs231n.vis_utils import visualize_grid
#from cs231n.vis_utils import visualize_grid_2D
# SHOW SOME WEIGHTS
W0 = net.weights[0]['W']
W0 = W0.T
num_of_samples=100
for i in range(0,10):
    for j in range(0,10):
        plt.subplot(10, 10, i*10+j+1)
        rand_index = np.random.randint(0,W0.shape[0]-1,1)
        plt.imshow(W0[rand_index].reshape(28,28))
        plt.axis('off')
plt.show()
# SHOW SOME RECONSTRUCTIONS
plt_index=1
for i in range(0,10):
    rand_index = np.random.randint(0,X_train.shape[0]-1,1)
    x = X_train[rand_index]
    x_noisy = GaussianNoise(rate=0.5,sd=0.5)(x)
    x_recon = net.predict(x_noisy)
    #x_loss,_ = net.loss(x_noisy, x, reg=0.0, activation_function='sigmoid')
    
    plt.subplot(10,3,plt_index)
    plt.imshow(x.reshape(28,28))
    plt.axis('off')
    if i == 0: plt.title('input')
    plt_index+=1
    plt.subplot(10,3,plt_index)
    plt.imshow(x_noisy.reshape(28,28))
    plt.axis('off')
    if i == 0: plt.title('corrupted input')
    plt_index+=1
    plt.subplot(10,3,plt_index)
    plt.imshow(x_recon.reshape(28,28))
    plt.axis('off')
    if i == 0: plt.title('reconstruction')
    plt_index+=1
Explanation: Debug the training
With the default parameters we provided above, you should get a validation accuracy of about 0.29 on the validation set. This isn't very good.
One strategy for getting insight into what's wrong is to plot the loss function and the accuracies on the training and validation sets during optimization.
Another strategy is to visualize the weights that were learned in the first layer of the network. In most neural networks trained on visual data, the first layer weights typically show some visible structure when visualized.
End of explanation
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