def finalTest(size_training, size_test, hidden_layers, lambd, num_iterations):
print "\nBeginning of the finalTest... \n"
images_training, labels_training, images_test, labels_test = read_dataset(size_training, size_test)
# Setup the parameters you will use for this exercise
input_layer_size = 784 # 28x28 Input Images of Digits
num_labels = 10 # 10 labels, from 0 to 9 (one label for each digit)
layers = [input_layer_size] + hidden_layers + [num_labels]
num_of_hidden_layers = len(hidden_layers)
# Fill the randInitializeWeights.py in order to initialize the neural network weights.
Theta = randInitializeWeights(layers)
# Unroll parameters
nn_weights = unroll_params(Theta)
res = fmin_l_bfgs_b(costFunction, nn_weights, fprime=backwards, args=(layers, images_training, labels_training, num_labels, lambd), maxfun = num_iterations, factr = 1., disp = True)
Theta = roll_params(res[0], layers)
print "\nTesting Neural Network... \n"
pred_training = predict(Theta, images_training)
print '\nAccuracy on training set: ' + str(mean(labels_training == pred_training) * 100)
pred = predict(Theta, images_test)
print '\nAccuracy on test set: ' + str(mean(labels_test == pred) * 100)
# Display the images where the algorithm got wrong
temp = (labels_test == pred)
indexes_false = []
for i in range(size_test):
if temp[i] == 0:
indexes_false.append(i)
displayData(images_training[indexes_false, :])
def backwards(nn_weights, layers, X, y, num_labels, lambd):
# Computes the gradient fo the neural network.
# nn_weights: Neural network parameters (vector)
# layers: a list with the number of units per layer.
# X: a matrix where every row is a training example for a handwritten digit image
# y: a vector with the labels of each instance
# num_labels: the number of units in the output layer
# lambd: regularization factor
# Setup some useful variables
m = X.shape[0]
num_layers = len(layers)
# Roll Params
# The parameters for the neural network are "unrolled" into the vector
# nn_params and need to be converted back into the weight matrices.
Theta = roll_params(nn_weights, layers)
# You need to return the following variables correctly
Theta_grad = [zeros(w.shape) for w in Theta]
# ================================ TODO ================================
# The vector y passed into the function is a vector of labels
# containing values from 1..K. You need to map this vector into a
# binary vector of 1's and 0's to be used with the neural network
# cost function.
yv = zeros((num_labels, m))
for i in range(m):
yv[y[i]][i] = 1
# ================================ TODO ================================
# In this point implement the backpropagaition algorithm
a = [[] for i in range(num_layers)]
z = [[] for i in range(num_layers)]
delta=[[] for i in range(num_layers)]
for t in range(m):
a[0] = X[t]
for i in range(0, num_layers - 1):
a[i] = insert(a[i], 0, 1)
z[i] = Theta[i].dot(transpose(a[i]))
a[i + 1] = sigmoid(z[i])
delta[-1] = a[-1] - yv[:,t]
for i in range(num_layers - 1, 0, -1):
if i > 1:
delta[i - 1] = (transpose(Theta[i-1][:, 1:]).dot(delta[i])) * sigmoidGradient(z[i - 2]) #because z[0] corresponds to z2
for i in range(0, num_layers - 1):
Theta_grad[i] += atleast_2d(delta[i+1]).T.dot(atleast_2d(a[i]))
# regularization
for l in range(0, num_layers - 1):
for i in range(Theta[l].shape[0]):
for j in range(1, Theta[l].shape[1]):
Theta_grad[l][i][j] += lambd * Theta[l][i][j]
# Unroll Params
Theta_grad = unroll_params(Theta_grad)
return Theta_grad/m
def costFunction(nn_weights, layers, X, y, num_labels, lambd):
# Computes the cost function of the neural network.
# nn_weights: Neural network parameters (vector)
# layers: a list with the number of units per layer.
# X: a matrix where every row is a training example for a handwritten digit image
# y: a vector with the labels of each instance
# num_labels: the number of units in the output layer
# lambd: regularization factor
# Setup some useful variables
m = X.shape[0]
num_layers = len(layers)
# Unroll Params
Theta = roll_params(nn_weights, layers)
J = 0;
yv = np.zeros((num_labels, m))
for i in range(m):
yv[y[i]][i] = 1
# Cost of the neural network (feedforward)
# Activation of the k-th layer for the i-th example
def scores_layer(i, k):
# k = 0 for the input layer
# k = l for the l-th hidden layer
x_vect = np.append([1], X[i, :]) # insert 1 at the beginning of the input image
if k == 0:
return sigmoid(np.dot(Theta[0], x_vect))
# Insert 1 at the beginning of the activation of the previous layer
res_with_bias = np.append([1], scores_layer(i, k-1))
return sigmoid(np.dot(Theta[k], res_with_bias))
# Cost function for the i-th example
def cost_i(i):
activation_layer = scores_layer(i, num_layers - 2) # output: activation of the outer layer
y_i = yv[:, i]
return (-y_i * np.log(activation_layer) - (1 - y_i) * np.log(1 - activation_layer)).sum()
# Total cost J
for i in range(m):
J += cost_i(i)
J /= m
# Regularization
coeff_reg = lambd / (2 * m)
# Loop on the weight matrixes
for h in range(num_layers - 1):
sub_weights = Theta[h][:, 1:] # the terms corresponding to the bias factors are not regularized
J += coeff_reg * (sub_weights * sub_weights).sum()
return J
def backwards(nn_weights, layers, X, y, num_labels, lambd):
# Computes the gradient fo the neural network.
# nn_weights: Neural network parameters (vector)
# layers: a list with the number of units per layer.
# X: a matrix where every row is a training example for a handwritten digit image
# y: a vector with the labels of each instance
# num_labels: the number of units in the output layer
# lambd: regularization factor
# Setup some useful variables
m = X.shape[0]
num_layers = len(layers)
# Roll Params
# The parameters for the neural network are "unrolled" into the vector
# nn_params and need to be converted back into the weight matrices.
Theta = roll_params(nn_weights, layers)
# You need to return the following variables correctly
Theta_grad = [zeros(w.shape) for w in Theta]
# ================================ TODO ================================
# The vector y passed into the function is a vector of labels
# containing values from 1..K. You need to map this vector into a
# binary vector of 1's and 0's to be used with the neural network
# cost function.
yv = zeros((num_labels, m))
# ================================ TODO ================================
# In this point implement the backpropagaition algorithm
# Unroll Params
Theta_grad = unroll_params(Theta_grad)
return Theta_grad
def cross_validation(lambd_values=[0.1], maxfun_values=[200]):
"""Function that trains the neural network and then tests its accuracy
Parameters : lambd, which measures the coefficient of the regularization
maxfun, which counts the number of iterations of the backpropagation
"""
n_lambd, n_maxfun = len(lambd_values), len(maxfun_values)
# Creation of the DataFrame where the results are to be stored
df_results = pd.DataFrame(index=range(n_lambd * n_maxfun))
df_results['Maxfun'], df_results['Lambd'] = list(
maxfun_values) * n_lambd, list(lambd_values) * n_maxfun
df_results['Hidden layers'] = num_of_hidden_layers
nodes_avg = np.mean(layers[1:-1])
df_results['Nodes per hidden layer (avg)'] = nodes_avg
accuracy_col = []
for lambd in lambd_values:
for maxfun in maxfun_values:
start = time() # start of the timer
res = opt.fmin_l_bfgs_b(costFunction,
nn_weights,
fprime=backwards,
args=(layers, images_validation,
labels_training, num_labels, lambd),
maxfun=maxfun,
factr=1.,
disp=True)
Theta = roll_params(res[0], layers)
# input('\nProgram paused. Press enter to continue!!!')
# print("\nTesting Neural Network... \n")
pred = predict(Theta, images_test)
end = time() # end of the timer
accuracy = np.mean(labels_test == pred) * 100
print('\nLambda =', lambd)
print('Maxfun =', maxfun)
time_complexity = end - start
print('Time:', time_complexity, 'seconds')
print('Accuracy =', accuracy, '%')
# Modification of the 'Accuracy' column
accuracy_col.append(accuracy)
# Accuracy values stored into the dataframe
df_results['Accuracy'] = accuracy_col
return df_results
def costFunction(nn_weights, layers, X, y, num_labels, lambd):
# Computes the cost function of the neural network.
# nn_weights: Neural network parameters (vector)
# layers: a list with the number of units per layer.
# X: a matrix where every row is a training example for a handwritten digit image
# y: a vector with the labels of each instance
# num_labels: the number of units in the output layer
# lambd: regularization factor
# Setup some useful variables
m = X.shape[0]
num_layers = len(layers)
# Unroll Params
Theta = roll_params(nn_weights, layers)
# The vector y passed into the function is a vector of labels
# containing values from 1..K. You need to map this vector into a
# binary vector of 1's and 0's to be used with the neural network
# cost function.
yv = zeros((num_labels, m))
for i in range(m):
yv[y[i],i] = 1
# In this point calculate the cost of the neural network (feedforward)
# a: the result obtained after each layer
a = ones(X.shape[0])
a = vstack((a,X.transpose()))
for i in range(num_layers-1):
z = dot(Theta[i],a)
a = sigmoid(z)
if i != num_layers-2:
a = vstack((ones(a.shape[1]),a))
#h: final result
h = a.transpose()
#calculate of the cost J
J = 0
for i in range(m):
for k in range(num_labels):
J = J + (-yv[k,i] * log(h[i][k]) - (1-yv[k,i]) * log(1.0 - h[i][k]))
J = J/m;
#regularization
tmp = 0
for i in range(num_layers-1):
for j in range(Theta[i].shape[0]):
for k in range(1,Theta[i].shape[1]):
tmp = tmp + Theta[i][j][k] * Theta[i][j][k]
J = J + tmp * lambd/(2.0*m)
return J
def costFunction(nn_weights, layers, X, y, num_labels, lambd):
# Computes the cost function of the neural network.
# nn_weights: Neural network parameters (vector)
# layers: a list with the number of units per layer.
# X: a matrix where every row is a training example for a handwritten digit image
# y: a vector with the labels of each instance
# num_labels: the number of units in the output layer
# lambd: regularization factor
# Setup some useful variables
m = X.shape[0]
num_layers = len(layers)
# Unroll Params
Theta = roll_params(nn_weights, layers)
# You need to return the following variables correctly
J = 0;
# ================================ TODO ================================
# The vector y passed into the function is a vector of labels
# containing values from 1..K. You need to map this vector into a
# binary vector of 1's and 0's to be used with the neural network
# cost function.
yv = zeros((num_labels, m))
for i in range(m):
yv[y[i]][i] = 1
# ================================ TODO ================================
# In this point calculate the cost of the neural network (feedforward)
for i in range(m):
layer_output = layerOutput(Theta, X, num_layers, i)
for j in range(num_labels):
cost = -yv[j, i] * log(layer_output[j])
cost -= (1 - yv[j, i]) * log(1 - layer_output[j])
J += cost
J /= m
# Regularization
regulation_term = 0
for i in range(len(Theta)):
for j in range(Theta[i].shape[0]):
for k in range(1, Theta[i].shape[1]):
regulation_term += (Theta[i][j][k]) ** 2
J += lambd / m * regulation_term / (num_layers - 1 )
return J
def costFunction(nn_weights, layers, X, y, num_labels, lambd):
# Computes the cost function of the neural network.
# nn_weights: Neural network parameters (vector)
# layers: a list with the number of units per layer.
# X: a matrix where every row is a training example for a handwritten digit image
# y: a vector with the labels of each instance
# num_labels: the number of units in the output layer
# lambd: regularization factor
# Setup some useful variables
m = X.shape[0]
num_layers = len(layers)
# Unroll Params
Theta = roll_params(nn_weights, layers)
# ================================ TODO ================================
# The vector y passed into the function is a vector of labels
# containing values from 1..K. You need to map this vector into a
# binary vector of 1's and 0's to be used with the neural network
# cost function.
yv = np.zeros((num_labels, m))
for i in range(len(y)):
yv[int(y[i]), i] = 1
yv = np.transpose(yv)
# ================================ TODO ================================
# In this point calculate the cost of the neural network (feedforward)
x = np.copy(X)
for i in range(num_layers - 1):
s = np.shape(Theta[i])
theta = Theta[i][:, 1:s[1]]
x = np.dot(x, np.transpose(theta))
x = x + Theta[i][:, 0]
x = sigmoid(x)
cost = (yv * np.log(x) + (1 - yv) * np.log(1 - x)) / m
cost = -np.sum(cost)
somme = 0
for i in range(num_layers - 1):
somme += lambd * np.sum(Theta[i] ** 2) / (2 * m)
cost += somme
return cost
def costFunction(nn_weights, layers, X, y, num_labels, lambd):
# Computes the cost function of the neural network.
# nn_weights: Neural network parameters (vector)
# layers: a list with the number of units per layer.
# X: a matrix where every row is a training example for a handwritten digit image
# y: a vector with the labels of each instance
# num_labels: the number of units in the output layer
# lambd: regularization factor
# Setup some useful variables
m = X.shape[0]
num_layers = len(layers)
# Unroll Params
Theta = roll_params(nn_weights, layers)
# You need to return the following variables correctly
J = 0
# ================================ TODO ================================
# The vector y passed into the function is a vector of labels
# containing values from 1..K. You need to map this vector into a
# binary vector of 1's and 0's to be used with the neural network
# cost function.
yv = zeros((num_labels, m))
for i in range(m):
yv[y[i],i] = 1.0
# ================================ TODO ================================
# In this point calculate the cost of the neural network (feedforward)
activation = transpose(concatenate(ones(m,1),X),axis=1)
activations = [activation]
for i in range(num_layers-1):
z = dot(Theta[i],activation)
zs.append(z)
if i == (num_layers-1):
activation = sigmoid(z)
else:
activation = concatenate((ones(1,m),sigmoid(z)), axis = 0)
activations.append(activation)
J = (1.0/m)*(sum(-1*yv*log(activations[-1]) -
(1 - yv) * log(1 - activations[-1])))
return J
def backwards(nn_weights, layers, X, y, num_labels, lambd):
# Computes the gradient fo the neural network.
# nn_weights: Neural network parameters (vector)
# layers: a list with the number of units per layer.
# X: a matrix where every row is a training example for a handwritten digit image
# y: a vector with the labels of each instance
# num_labels: the number of units in the output layer
# lambd: regularization factor
# Setup some useful variables
m = X.shape[0]
num_layers = len(layers)
# Roll Params
# The parameters for the neural network are "unrolled" into the vector
# nn_params and need to be converted back into the weight matrices.
Theta = roll_params(nn_weights, layers)
# You need to return the following variables correctly
Theta_grad = [zeros(w.shape) for w in Theta]
# The vector y passed into the function is a vector of labels
# containing values from 1..K. You need to map this vector into a
# binary vector of 1's and 0's to be used with the neural network
# cost function.
yv = zeros((num_labels, m))
for i in range(m):
yv[y[i],i] = 1
# In this point implement the backpropagaition algorithm
A = []
a = ones(X.shape[0])
a = vstack((a,X.transpose()))
Z = []
Z.append(a)
for i in range(num_layers-1):
A.append(a.transpose())
z = dot(Theta[i],a)
Z.append(z)
a = sigmoid(z)
if i != num_layers-2:
a = vstack((ones(a.shape[1]),a))
# A: list of result after each layer
A.append(a.transpose())
h = a.transpose()
# delta for the last layer
delta = h - yv.transpose()
# calculate of gradients
for j in range(num_layers-2,0,-1):
Theta_grad[j] = Theta_grad[j] + dot(delta.transpose(),A[j])
# calculate of delta for current layer(have to remove the first column of Theta)
tmp = dot(Theta[j][:,1:].transpose(),delta.transpose())
tmp = tmp.transpose()
tmp_matrix = zeros(tmp.shape)
for i in range(m):
tmp_matrix[i] = sigmoidGradient(Z[j].transpose()[i])
delta = tmp_matrix * tmp
Theta_grad[0] = Theta_grad[0] + dot(delta.transpose(),A[0])
# regularization
for i in range(num_layers-1):
for j in range((Theta_grad[i].shape)[0]):
for k in range((Theta_grad[i].shape)[1]):
Theta_grad[i][j,k] = Theta_grad[i][j,k]/m
if k >=1:
Theta_grad[i][j,k] = Theta_grad[i][j,k] + lambd/m*Theta[i][j,k]
# Unroll Params
Theta_grad = unroll_params(Theta_grad)
return Theta_grad
def backwards(nn_weights, layers, X, y, num_labels, lambd):
# Computes the gradient fo the neural network.
# nn_weights: Neural network parameters (vector)
# layers: a list with the number of units per layer.
# X: a matrix where every row is a training example for a handwritten digit image
# y: a vector with the labels of each instance
# num_labels: the number of units in the output layer
# lambd: regularization factor
# Setup some useful variables
m = X.shape[0]
num_layers = len(layers)
# Roll Params
# The parameters for the neural network are "unrolled" into the vector
# nn_params and need to be converted back into the weight matrices.
Theta = roll_params(nn_weights, layers)
# You need to return the following variables correctly
Theta_grad = [zeros(w.shape) for w in Theta]
# ================================ DONE ================================
# The vector y passed into the function is a vector of labels
# containing values from 1..K. You need to map this vector into a
# binary vector of 1's and 0's to be used with the neural network
# cost function.
yv = zeros((m, num_labels))
for i in range(m):
yv[i][y[i]] += 1
# ================================ DONE ================================
# In this point implement the backpropagation algorithm
# In this point calculate the cost of the neural network (feedforward)
# Step 1: Initialization of useful variables
# Z and A will store the hidden states of the network, as lists of matrices, of size num_layers
A = [addColumnOne(X)]
Z = [addColumnOne(X)]
# delta will store the delta for each layer from the last to the second layer (in reverse order)
delta = []
# Step 2: Feedforward
for i in range(num_layers - 1):
h = A[i].dot(Theta[i].T)
Z.append(h)
h = addColumnOne(sigmoid(h))
A.append(h)
# Step 3: Backpropagation
d = removeFirstColumn(A[-1]) - yv
delta.append(d)
for i in range(num_layers - 2, 0, -1):
d = removeFirstColumn(d.dot(Theta[i])) * sigmoidGradient(Z[i])
delta.append(d)
delta.reverse()
# delta is of size num_layers-1 (no delta for the input layer)
for i in range(num_layers - 1):
Theta_grad[i] += delta[i].T.dot(A[i])
# DONE: no regularization on the bias weights !!
Theta_grad[i] += lambd * Theta[i]
for j in range(Theta[i].shape[0]):
Theta_grad[i][j, 0] -= lambd * Theta[i][j, 0]
Theta_grad[i] /= m
# Unroll Params
Theta_grad = unroll_params(Theta_grad)
return Theta_grad
def backwards(nn_weights, layers, X, y, num_labels, lambd):
# Computes the gradient fo the neural network.
# nn_weights: Neural network parameters (vector)
# layers: a list with the number of units per layer.
# X: a matrix where every row is a training example for a handwritten digit image
# y: a vector with the labels of each instance
# num_labels: the number of units in the output layer
# lambd: regularization factor
# Setup some useful variables
m = X.shape[0]
num_layers = len(layers)
# Roll Params
# The parameters for the neural network are "unrolled" into the vector
# nn_params and need to be converted back into the weight matrices.
Theta = roll_params(nn_weights, layers)
Theta_grad = [np.zeros(w.shape) for w in Theta]
yv = np.zeros((num_labels, m))
for i in range(m):
yv[y[i]][i] = 1
# Implementation of the backpropagation algorithm
for i in range(m):
a_values, z_values = [], [
] # arrays where the values of the activations are to be stored
a = np.append([1], X[i, :])
a_values.append(a)
# Loop of the feedforward algorithm
for k in range(num_layers - 1):
z = np.dot(Theta[k], a)
z_values.append(z)
a = np.append([1], sigmoid(z))
a_values.append(a)
delta_layer = a[1:] - yv[:, i] # error array of the outer layer
# np.outer to calculate the matrix product of delta_layer.T and a_values[-2]
Theta_grad[-1] += np.outer(delta_layer, a_values[-2]) / m
# Descending loop
for h in range(num_layers - 2):
# Error of the (num_layers - 2 - h)-th hidden layer
# The error that corresponds to the bias factors is not taken into account
delta_layer = np.dot(Theta[-1 - h].T,
delta_layer)[1:] * sigmoidGradient(
z_values[-2 - h])
# Calculation of the gradient
Theta_grad[-2 - h] += np.outer(delta_layer, a_values[-3 - h]) / m
#Regularization
for h in range(num_layers - 1):
# The terms corresponding to the bias factors are not regularized
Theta_grad[h][:, 1:] += lambd * Theta[h][:, 1:] / m
# Unroll Params
Theta_grad = unroll_params(Theta_grad)
return Theta_grad
def costFunction(nn_weights, layers, X, y, num_labels, lambd):
# Computes the cost function of the neural network.
# nn_weights: Neural network parameters (vector)
# layers: a list with the number of units per layer.
# X: a matrix where every row is a training example for a handwritten digit image
# y: a vector with the labels of each instance
# num_labels: the number of units in the output layer
# lambd: regularization factor
# Setup some useful variables
m = X.shape[0]
num_layers = len(layers)
# Unroll Params
Theta = roll_params(nn_weights, layers)
# You need to return the following variables correctly
J = 0
# ================================ DONE ================================
# The vector y passed into the function is a vector of labels
# containing values from 1..K. You need to map this vector into a
# binary vector of 1's and 0's to be used with the neural network
# cost function.
yv = np.zeros((m, num_labels))
for i in range(m):
yv[i][y[i]] += 1
# ================================ DONE ================================
# In this point calculate the cost of the neural network (feedforward)
# Step 1: Initialization of useful variables
# H will store the hidden states of the network, H is a list of matrices, of size num_layers
H = [X]
# Step 2: Feedforward
for i in range(num_layers-1):
h = sigmoid(addColumnOne(H[i]).dot(Theta[i].T))
H.append(h)
# The end layer is H[num_layers]
yv_pred = H[num_layers-1]
# Step 3: Compute cost
# We create the variable S, a matrix of size (m, K) which we will sum afterwards
S = np.zeros((m, num_labels))
temp = np.log(yv_pred)
temp = yv*temp
temp2 = np.log(1.0-yv_pred)
temp2 = (1.0-yv)*temp2
S += - temp - temp2
J += np.sum(S)
J = J/m
reg = 0
for i in range(num_layers-1):
# No regularization on the bias weights
reg += np.sum(removeFirstColumn(Theta[i])**2)
J += lambd * reg / (2.0 * m)
return J
def costFunction(nn_weights, layers, X, y, num_labels, lambd):
# Computes the cost function of the neural network.
# nn_weights: Neural network parameters (vector)
# layers: a list with the number of units per layer.
# X: a matrix where every row is a training example for a handwritten digit image
# y: a vector with the labels of each instance
# num_labels: the number of units in the output layer
# lambd: regularization factor
# Setup some useful variables
m = X.shape[0]
num_layers = len(layers)
# Unroll Params
Theta = roll_params(nn_weights, layers)
# You need to return the following variables correctly
J = 0
# ================================ DONE ================================
# The vector y passed into the function is a vector of labels
# containing values from 1..K. You need to map this vector into a
# binary vector of 1's and 0's to be used with the neural network
# cost function.
yv = np.zeros((m, num_labels))
for i in range(m):
yv[i][y[i]] += 1
# ================================ DONE ================================
# In this point calculate the cost of the neural network (feedforward)
# Step 1: Initialization of useful variables
# H will store the hidden states of the network, H is a list of matrices, of size num_layers
H = [X]
# Step 2: Feedforward
for i in range(num_layers - 1):
h = sigmoid(addColumnOne(H[i]).dot(Theta[i].T))
H.append(h)
# The end layer is H[num_layers]
yv_pred = H[num_layers - 1]
# Step 3: Compute cost
# We create the variable S, a matrix of size (m, K) which we will sum afterwards
S = np.zeros((m, num_labels))
temp = np.log(yv_pred)
temp = yv * temp
temp2 = np.log(1.0 - yv_pred)
temp2 = (1.0 - yv) * temp2
S += -temp - temp2
J += np.sum(S)
J = J / m
reg = 0
for i in range(num_layers - 1):
# No regularization on the bias weights
reg += np.sum(removeFirstColumn(Theta[i])**2)
J += lambd * reg / (2.0 * m)
return J
def backwards(nn_weights, layers, X, y, num_labels, lambd):
"""
:param nn_weights: Neural network parameters (vector)
:param layers: a list with the number of units per layer.
:param X: a matrix where every row is a training example for a handwritten digit image
:param y: a vector with the labels of each instance
:param num_labels: the number of units in the output layer
:param lambd: regularization factor
:return: Computes the gradient fo the neural network.
"""
# Setup some useful variables
m = X.shape[0]
num_layers = len(layers)
# Roll Params
# The parameters for the neural network are "unrolled" into the vector
# nn_params and need to be converted back into the weight matrices.
Theta = roll_params(nn_weights, layers)
# The vector y passed into the function is a vector of labels
# containing values from 1..K. You need to map this vector into a
# binary vector of 1's and 0's to be used with the neural network
# cost function.
yv = np.zeros((num_labels, m))
for i in range(len(y)):
yv[int(y[i]), i] = 1
yv = np.transpose(yv)
a = []
z = []
x = np.copy(X)
a.append(insertOne(x))
z.append(x)
# if you want to be able to follow the training accuracy:
# pred = predict(Theta, X)
# accuracy = np.mean(y == pred) * 100
# print(accuracy)
for i in range(num_layers - 1):
s = np.shape(Theta[i])
theta = Theta[i][:, 1:s[1]]
x = np.dot(x, np.transpose(theta))
x = x + Theta[i][:, 0]
z.append(x)
x = sigmoid(x)
a.append(insertOne(x))
delta = [np.zeros(w.shape) for w in z]
delta[num_layers - 1] = (x - yv)
for i in range(num_layers - 2, 0, -1):
s = np.shape(Theta[i])
theta = np.copy(Theta[i][:, 1:s[1]])
temp = np.dot(np.transpose(theta), np.transpose(delta[i + 1]))
delta[i] = np.transpose(temp) * sigmoidGradient(z[i])
Delta = []
for i in range(num_layers - 1):
temp = np.dot(np.transpose(delta[i + 1]), a[i])
Delta.append(temp)
# if you want to follow the cost during the training:
# cost = (yv * np.log(x) + (1 - yv) * np.log(1 - x)) / m
# cost = -np.sum(cost)
#
# somme = 0
#
# for i in range(num_layers - 1):
# somme += lambd * np.sum(Theta[i] ** 2) / (2 * m)
#
# cost += somme
Theta_grad = [(d / m) for d in Delta]
i = 0
for t in Theta:
current = lambd * t / m
# d'après le poly il faudrait qu'il y ait cette ligne
# mais après quand on son checkNNGradient il vaut mieux enlever
# cette ligne donc je ne sais pas ...:
# current[:, 0] = current[:, 0]*0
Theta_grad[i] += current
i += 1
# Unroll Params
Theta_grad = unroll_params(Theta_grad)
return Theta_grad
checkNNGradients(lambd)
input('\nProgram paused. Press enter to continue!!!')
# ================================ Step 8: Implement Backpropagation with Regularization ================================
print("\nChecking Backpropagation with Regularization ...\n")
lambd = 3.0
checkNNGradients(lambd)
input('\nProgram paused. Press enter to continue!!!')
# ================================ Step 9: Training Neural Networks & Prediction ================================
print("\nTraining Neural Network... \n")
# You should also try different values of the regularization factor
lambd = 3.0
res = fmin_l_bfgs_b(costFunction, nn_weights, fprime = backwards, args = (layers, images_training, labels_training, num_labels, 1.0), maxfun = 50, factr = 1., disp = True)
Theta = roll_params(res[0], layers)
input('\nrogram paused. Press enter to continue!!!')
print("\nTesting Neural Network... \n")
pred = predict(Theta, images_test)
print('\nAccuracy: ' + str(mean(labels_test==pred) * 100))
checkNNGradients(lambd)
raw_input('\nProgram paused. Press enter to continue!!!')
# ================================ Step 8: Implement Backpropagation with Regularization ================================
print "\nChecking Backpropagation with Regularization ...\n"
lambd = 3.0
checkNNGradients(lambd)
raw_input('\nProgram paused. Press enter to continue!!!')
# ================================ Step 9: Training Neural Networks & Prediction ================================
print "\nTraining Neural Network... \n"
# You should also try different values of the regularization factor
lambd = 3.0
res = fmin_l_bfgs_b(costFunction, nn_weights, fprime=backwards, args=(layers, images_training, labels_training, num_labels, 1.0), maxfun=50, factr=1., disp=True)
Theta = roll_params(res[0], layers)
raw_input('\nrogram paused. Press enter to continue!!!')
print "\nTesting Neural Network... \n"
pred = predict(Theta, images_test)
print '\nAccuracy: ' + str(mean(labels_test==pred) * 100)
def backwards(nn_weights, layers, X, y, num_labels, lambd):
# Computes the gradient fo the neural network.
# nn_weights: Neural network parameters (vector)
# layers: a list with the number of units per layer.
# X: a matrix where every row is a training example for a handwritten digit image
# y: a vector with the labels of each instance
# num_labels: the number of units in the output layer
# lambd: regularization factor
# Setup some useful variables
m = X.shape[0]
num_layers = len(layers)
# Roll Params
# The parameters for the neural network are "unrolled" into the vector
# nn_params and need to be converted back into the weight matrices.
Theta = roll_params(nn_weights, layers)
# You need to return the following variables correctly
Theta_grad = [zeros(w.shape) for w in Theta]
# ================================ DONE ================================
# The vector y passed into the function is a vector of labels
# containing values from 1..K. You need to map this vector into a
# binary vector of 1's and 0's to be used with the neural network
# cost function.
yv = zeros((m, num_labels))
for i in range(m):
yv[i][y[i]] += 1
# ================================ DONE ================================
# In this point implement the backpropagation algorithm
# In this point calculate the cost of the neural network (feedforward)
# Step 1: Initialization of useful variables
# Z and A will store the hidden states of the network, as lists of matrices, of size num_layers
A = [addColumnOne(X)]
Z = [addColumnOne(X)]
# delta will store the delta for each layer from the last to the second layer (in reverse order)
delta = []
# Step 2: Feedforward
for i in range(num_layers-1):
h = A[i].dot(Theta[i].T)
Z.append(h)
h = addColumnOne(sigmoid(h))
A.append(h)
# Step 3: Backpropagation
d = removeFirstColumn(A[-1]) - yv
delta.append(d)
for i in range(num_layers-2, 0, -1):
d = removeFirstColumn(d.dot(Theta[i])) * sigmoidGradient(Z[i])
delta.append(d)
delta.reverse()
# delta is of size num_layers-1 (no delta for the input layer)
for i in range(num_layers-1):
Theta_grad[i] += delta[i].T.dot(A[i])
# DONE: no regularization on the bias weights !!
Theta_grad[i] += lambd * Theta[i]
for j in range(Theta[i].shape[0]):
Theta_grad[i][j, 0] -= lambd * Theta[i][j, 0]
Theta_grad[i] /= m
# Unroll Params
Theta_grad = unroll_params(Theta_grad)
return Theta_grad
