Logistic Regression as a Neural Network

Logistic Regression Recap

Computation Graph

• $u = bc$
• $v = a+u$
• $J = 3v$

• $\frac{dJ}{dv} = 3$
• $\frac{dJ}{du} = \frac{dJ}{dv} \times \frac{dv}{du} = 3 \times 1 = 3$
• $\frac{dJ}{da} = \frac{dJ}{dv} \times \frac{dv}{da} = 3 \times 1 = 3$
• $\frac{dJ}{db} = \frac{dJ}{dv} \times \frac{dv}{du} \times \frac{du}{db} = 3 \times 1 \times c = 3c$
• $\frac{dJ}{dc} = \frac{dJ}{dv} \times \frac{dv}{du} \times \frac{du}{dc} = 3 \times 1 \times b = 3b$

Loss Function

• $da = \frac {dL(a,y)} {da} = - \frac{y}{a} + \frac{1-y}{1-a}$
• $dz = \frac {dL(a,y)} {dz} = \frac {dL(a,y)}{da} \times \frac {da}{dz} = a-y$
• $dw_1 = \frac {dL(a,y)} {dw_1} = x_1 \times dz$
• $dw_2 = \frac {dL(a,y)} {dw_2} = x_2 \times dz$

Cost Function

J=0, dw1=0, dw2=0, db=0
for i=1 to m
z[i] = w.tx[i] + b
a[i] = sigmoid(z[i])
J += -y[i]*log(a[i]) + (1-y[i])log(1-a[i])
dz[i] = a[i] - y[i]
for j=1 to n
dw[j] += x[i][j] * dz[i] #第i组样本的第j个feature
db += dz[i]
# if n = 2
#dw1 += x[i][1] * dz[i]
#dw2 += x[i][2] * dz[i]
dw1 = dw1 / m
dw2 = dw2 / m
db  = db / m

w1 = w1 - alpha*dw1
w2 = w2 - alpha*dw2
b  = b - alpha*db

Vectorization

#for loop
z = 0
for i in range(1,n):
z += w[i] * x[i]
z+=b

# use numpy
# vectorized version of doing
# matrix multiplications
z = np.dot(w.T,x)+b

numpy数组的另一个特点是可以做element-wise的矩阵运算，这样让我们避开了for循环的使用

a = np.ones([1,2])  #[1,1]
a = a*2 #[2,2]

J=0,db=0
dw = np.zeros([n,1])
for i=1 to m
#z是1xm的
#x是nxm的
z[i] = w.tx[i] + b
a[i] = sigmoid(z[i])
J += -y[i]*log(a[i]) + (1-y[i])log(1-a[i])
dz[i] = a[i] - y[i]
# for j=1 to n
# dw[j] += x[i][j] * dz[i] #第i组样本的第j个feature
dw += x[i]*dz[i]
db += dz[i]
dw = dw/m

Compute Forward Propagation

Z = np.dot(w.T,X) + b #b is a 1x1 number

Compute Backwward Propagation

def initialize_with_zeros(dim):
"""
This function creates a vector of zeros of shape (dim, 1) for w and initializes b to 0.

Argument:
dim -- number of features in a given example

Returns:
w -- initialized vector of shape (dim, 1)
b -- initialized scalar (corresponds to the bias)
"""
w = np.zeros([dim,1]) # dimx1
b = 0
assert(w.shape == (dim, 1))
assert(isinstance(b, float) or isinstance(b, int))

return w, b

def sigmoid(z):
"""
Compute the sigmoid of z
Arguments:
z -- A scalar or numpy array of any size.
Return:
s -- sigmoid(z)
"""
s = 1/(1+np.exp(-z))
return s

def propagate(w, b, X, Y):
"""
Arguments:
w -- weights, a numpy array of size (num_px * num_px * 3, 1)
b -- bias, a scalar
X -- data of size (features, #examples)
Y -- true "label" vector of size (1, #examples)

Return:
cost -- negative log-likelihood cost for logistic regression
dw -- gradient of the loss with respect to w, thus same shape as w
db -- gradient of the loss with respect to b, thus same shape as b
"""

m = X.shape[1]
A = sigmoid(np.dot(w.T, X)+b) # compute activation
cost = - 1/m * np.sum(Y*np.log(A) + (1-Y)*np.log(1-A))  #compute cost
# BACKWARD PROPAGATION (TO FIND GRAD)
dw = 1/m * np.dot(X,(A-Y).T)
db = 1/m * np.sum(A-Y)

assert(dw.shape == w.shape)
assert(db.dtype == float)
cost = np.squeeze(cost)
assert(cost.shape == ())

"db": db}

def optimize(w, b, X, Y, num_iterations, learning_rate, print_cost = False):
"""
This function optimizes w and b by running a gradient descent algorithm

Arguments:
w -- weights, a numpy array of size (num_px * num_px * 3, 1)
b -- bias, a scalar
X -- data of shape (features, #examples)
Y -- true "label" vector of shape (1, #examples)
num_iterations -- number of iterations of the optimization loop
learning_rate -- learning rate of the gradient descent update rule
print_cost -- True to print the loss every 100 steps

Returns:
params -- dictionary containing the weights w and bias b
grads -- dictionary containing the gradients of the weights and bias with respect to the cost function
costs -- list of all the costs computed during the optimization, this will be used to plot the learning curve.
"""

costs = []
for i in range(num_iterations):
# Run propagation
# update rule
w = w - learning_rate * dw
b = b - learning_rate * db
# Record the costs
if i % 100 == 0:
costs.append(cost)
# Print the cost every 100 training iterations
if print_cost and i % 100 == 0:
print ("Cost after iteration %i: %f" %(i, cost))

params = {"w": w,
"b": b}

"db": db}

def predict(w, b, X):
'''
Predict whether the label is 0 or 1 using learned logistic regression parameters (w, b)

Arguments:
w -- weights, a numpy array of size (num_px * num_px * 3, 1)
b -- bias, a scalar
X -- data of size (features, #examples)

Returns:
Y_prediction -- a numpy array (vector) containing all predictions (0/1) for the examples in X
'''

m = X.shape[1]
Y_prediction = np.zeros((1,m))
w = w.reshape(X.shape[0], 1)
# Compute vector "A" predicting the probabilities of a cat being present in the picture
A = sigmoid(np.dot(w.T,X)+b)
for i in range(A.shape[1]):
# Convert probabilities A[0,i] to actual predictions p[0,i]
if (A[0][i] > 0.5):
Y_prediction[0][i] = 1
else:
Y_prediction[0][i] = 0
assert(Y_prediction.shape == (1, m))
return Y_prediction

def model(X_train, Y_train, X_test, Y_test, num_iterations = 2000, learning_rate = 0.5, print_cost = False):
"""
Builds the logistic regression model by calling the function you've implemented previously

Arguments:
X_train -- training set represented by a numpy array of shape (num_px * num_px * 3, m_train)
Y_train -- training labels represented by a numpy array (vector) of shape (1, m_train)
X_test -- test set represented by a numpy array of shape (num_px * num_px * 3, m_test)
Y_test -- test labels represented by a numpy array (vector) of shape (1, m_test)
num_iterations -- hyperparameter representing the number of iterations to optimize the parameters
learning_rate -- hyperparameter representing the learning rate used in the update rule of optimize()
print_cost -- Set to true to print the cost every 100 iterations

Returns:
d -- dictionary containing information about the model.
"""
#1. initialize parameters with zeros
w, b = initialize_with_zeros(X_train.shape[0])

parameters, grads, costs = optimize(w,b,X_train,Y_train,num_iterations, learning_rate, print_cost)

#3. Retrieve parameters w and b from dictionary "parameters"
w = parameters["w"]
b = parameters["b"]

#4. Predict test/train set examples
Y_prediction_test = predict(w,b,X_test)
Y_prediction_train = predict(w,b,X_train)

#5. Print train/test Errors
print("train accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_train - Y_train)) * 100))
print("test accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_test - Y_test)) * 100))

d = {"costs": costs,
"Y_prediction_test": Y_prediction_test,
"Y_prediction_train" : Y_prediction_train,
"w" : w,
"b" : b,
"learning_rate" : learning_rate,
"num_iterations": num_iterations}

return d

#Run the following cell to train your model.
d = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 2000, learning_rate = 0.005, print_cost = True)

# result
# Cost after iteration 0: 0.693147
# Cost after iteration 100: 0.584508
# Cost after iteration 200: 0.466949
# Cost after iteration 300: 0.376007
# Cost after iteration 400: 0.331463
# Cost after iteration 500: 0.303273
# Cost after iteration 600: 0.279880
# Cost after iteration 700: 0.260042
# Cost after iteration 800: 0.242941
# Cost after iteration 900: 0.228004
# Cost after iteration 1000: 0.214820
# Cost after iteration 1100: 0.203078
# Cost after iteration 1200: 0.192544
# Cost after iteration 1300: 0.183033
# Cost after iteration 1400: 0.174399
# Cost after iteration 1500: 0.166521
# Cost after iteration 1600: 0.159305
# Cost after iteration 1700: 0.152667
# Cost after iteration 1800: 0.146542
# Cost after iteration 1900: 0.140872
# train accuracy: 99.04306220095694 %
# test accuracy: 70.0 %