deeplearning.ai-lecture1-building deep neural network steps

 该实验主要是实现一些“Helper function”,为下一步实现两层神经网络和L层神经网络做准备，实现一个两层网络或深层网络的步骤如下：

Step 1.分别初始化一个两层神经网络和L层神经网络的参数

Step 2: 前向传播的实现：

1.完成一个网络的前向传播的线性部分（linear part）,即计算出 Z [l]     

2.实现relu和 sigmoid激活函数

3.联合前两步，实现网络前向传播的一个【linear->activation】 层 函数

4.实现前向传播的前L-1层【linear->relu】最后一层的【linear->sigmoid】函数

Step 3:计算损失函数

Step 4:反向传播的实现：

1.计算神经网络线性部分（linear part）的反向传播 

2.求出relu和sigmoid函数的梯度函数（relu_backward/relu_backward）

3.联合前两步，实现一个新的【linear->Activation】反向函数

4.整合，实现最后一层的【linear->sigmoid】和前L-1层的【linear->relu】的反向函数

Step 5:更新参数

下面开始实现神经网络的函数

Step 1:

1.   2层神经网络参数初始化

def initialize_parameters(n_x, n_h, n_y):
"""
Argument:
n_x -- size of the input layer
n_h -- size of the hidden layer
n_y -- size of the output layer

Returns:
parameters -- python dictionary containing your parameters:
W1 -- weight matrix of shape (n_h, n_x)
b1 -- bias vector of shape (n_h, 1)
W2 -- weight matrix of shape (n_y, n_h)
b2 -- bias vector of shape (n_y, 1)
"""

np.random.seed(1)

W1 = np.random.randn(n_h, n_x)*0.01
b1 = np.zeros((n_h, 1))
W2 = np.random.randn(n_y, n_h)*0.01
b2 = np.zeros((n_y, 1))

assert(W1.shape == (n_h, n_x))
assert(b1.shape == (n_h, 1))
assert(W2.shape == (n_y, n_h))
assert(b2.shape == (n_y, 1))

parameters = {"W1": W1,
"b1": b1,
"W2": W2,
"b2": b2}

return parameters

2. L层神经网络参数初始化

def initialize_parameters_deep(layer_dims):
"""
Arguments:
layer_dims -- python array (list) containing the dimensions of each layer in our network

Returns:
parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":
Wl -- weight matrix of shape (layer_dims[l], layer_dims[l-1])
bl -- bias vector of shape (layer_dims[l], 1)
"""
np.random.seed(3)
parameters={}
L=len(layer_dims)
for l in range(1,L):
parameters['W'+str(l)]=np.random.randn(layer_dims[l],layer_dims[l-1])*0.01
parameters['b'+str(l)]=np.zeros((layer_dims[l],1))
assert(parameters['W'+str(l)].shape==(layer_dims[l],layer_dims[l-1]))
assert(parameters['b'+str(l)].shape==(layer_dims[l],1))
return parameters

Step 2:

1.网络的前向传播的线性部分

def linear_forward(A,W,b):
"""
Implement the linear part of a layer's forward propagation.
​
Arguments:
A -- activations from previous layer (or input data): (size of previous layer, number of examples)
W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
b -- bias vector, numpy array of shape (size of the current layer, 1)
​
Returns:
Z -- the input of the activation function, also called pre-activation parameter
cache -- a python dictionary containing "A", "W" and "b" ; stored for computing the backward pass efficiently
"""

Z=np.dot(W,A)+b
assert (Z.shape==(W.shape[0],A.shape[1]))
cache=(A,W,b)
return Z,cache

2.实现relu和 sigmoid激活函数

def sigmoid(Z):
"""
Implements the sigmoid activation in numpy

Arguments:
Z -- numpy array of any shape

Returns:
A -- output of sigmoid(z), same shape as Z
cache -- returns Z as well, useful during backpropagation
"""

A = 1/(1+np.exp(-Z))
cache = Z

return A, cache

def relu(Z):
"""
Implement the RELU function.

Arguments:
Z -- Output of the linear layer, of any shape

Returns:
A -- Post-activation parameter, of the same shape as Z
cache -- a python dictionary containing "A" ; stored for computing the backward pass efficiently
"""

A = np.maximum(0,Z)

assert(A.shape == Z.shape)

cache = Z
return A, cache

3.联合前两步，实现网络前向传播的一个【linear->activation】 层 函数

def linear_activation_forward(A_prev, W, b, activation):
"""
Implement the forward propagation for the LINEAR->ACTIVATION layer

Arguments:
A_prev -- activations from previous layer (or input data): (size of previous layer, number of examples)
W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
b -- bias vector, numpy array of shape (size of the current layer, 1)
activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"

Returns:
A -- the output of the activation function, also called the post-activation value
cache -- a python dictionary containing "linear_cache" and "activation_cache";
stored for computing the backward pass efficiently
"""

if activation == "sigmoid":
# Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
Z, linear_cache = linear_forward(A_prev, W, b)
A, activation_cache = sigmoid(Z)

elif activation == "relu":
# Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
Z, linear_cache = linear_forward(A_prev, W, b)
A, activation_cache = relu(Z)

assert (A.shape == (W.shape[0], A_prev.shape[1]))
cache = (linear_cache, activation_cache)

return A, cache

4.实现前向传播的前L-1层【linear->relu】最后一层的【linear->sigmoid】函数

def L_model_forward(X, parameters):
"""
Implement forward propagation for the [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID computation

Arguments:
X -- data, numpy array of shape (input size, number of examples)
parameters -- output of initialize_parameters_deep()

Returns:
AL -- last post-activation value
caches -- list of caches containing:
every cache of linear_relu_forward() (there are L-1 of them, indexed from 0 to L-2)
the cache of linear_sigmoid_forward() (there is one, indexed L-1)
"""

caches = []
A = X
L = len(parameters) // 2                  # number of layers in the neural network

# Implement [LINEAR -> RELU]*(L-1). Add "cache" to the "caches" list.
for l in range(1, L):
A_prev = A
A, cache = linear_activation_forward(A_prev, parameters['W' + str(l)], parameters['b' + str(l)], activation = "relu")
caches.append(cache)

# Implement LINEAR -> SIGMOID. Add "cache" to the "caches" list.
AL, cache = linear_activation_forward(A, parameters['W' + str(L)], parameters['b' + str(L)], activation = "sigmoid")
caches.append(cache)

assert(AL.shape == (1,X.shape[1]))

return AL, caches

Step 3:计算损失函数

def compute_cost(AL, Y):
"""
Implement the cost function defined by equation (7).

Arguments:
AL -- probability vector corresponding to your label predictions, shape (1, number of examples)
Y -- true "label" vector (for example: containing 0 if non-cat, 1 if cat), shape (1, number of examples)

Returns:
cost -- cross-entropy cost
"""

m = Y.shape[1]

# Compute loss from aL and y.
cost = (1./m) * (-np.dot(Y,np.log(AL).T) - np.dot(1-Y, np.log(1-AL).T))

cost = np.squeeze(cost)      # To make sure your cost's shape is what we expect (e.g. this turns [[17]] into 17).
assert(cost.shape == ())

return cost

Step 4:反向传播的实现：

1.计算神经网络线性部分（linear part）的反向传播 （假设你已经知道dZ[l]，计算dW[l],db[l],dA[l-1]）

def linear_backward(dZ, cache):
"""
Implement the linear portion of backward propagation for a single layer (layer l)

Arguments:
dZ -- Gradient of the cost with respect to the linear output (of current layer l)
cache -- tuple of values (A_prev, W, b) coming from the forward propagation in the current layer

Returns:
dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
dW -- Gradient of the cost with respect to W (current layer l), same shape as W
db -- Gradient of the cost with respect to b (current layer l), same shape as b
"""
A_prev, W, b = cache
m = A_prev.shape[1]

dW = 1./m * np.dot(dZ,A_prev.T)
db = 1./m * np.sum(dZ, axis = 1, keepdims = True)
dA_prev = np.dot(W.T,dZ)

assert (dA_prev.shape == A_prev.shape)
assert (dW.shape == W.shape)
assert (db.shape == b.shape)

return dA_prev, dW, db

2.求出relu和sigmoid函数的梯度函数（relu_backward/relu_backward）

假设dA已经，

def relu_backward(dA, cache):
"""
Implement the backward propagation for a single RELU unit.

Arguments:
dA -- post-activation gradient, of any shape
cache -- 'Z' where we store for computing backward propagation efficiently

Returns:
dZ -- Gradient of the cost with respect to Z
"""

Z = cache
dZ = np.array(dA, copy=True) # just converting dz to a correct object.

# When z <= 0, you should set dz to 0 as well.
dZ[Z <= 0] = 0

assert (dZ.shape == Z.shape)

return dZ

def sigmoid_backward(dA, cache):
"""
Implement the backward propagation for a single SIGMOID unit.

Arguments:
dA -- post-activation gradient, of any shape
cache -- 'Z' where we store for computing backward propagation efficiently

Returns:
dZ -- Gradient of the cost with respect to Z
"""

Z = cache

s = 1/(1+np.exp(-Z))
dZ = dA * s * (1-s)

assert (dZ.shape == Z.shape)

return dZ

3.联合前两步，实现一个新的【linear->Activation】反向函数

def linear_activation_backward(dA, cache, activation):
"""
Implement the backward propagation for the LINEAR->ACTIVATION layer.

Arguments:
dA -- post-activation gradient for current layer l
cache -- tuple of values (linear_cache, activation_cache) we store for computing backward propagation efficiently
activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"

Returns:
dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
dW -- Gradient of the cost with respect to W (current layer l), same shape as W
db -- Gradient of the cost with respect to b (current layer l), same shape as b
"""
linear_cache, activation_cache = cache

if activation == "relu":
dZ = relu_backward(dA, activation_cache)
dA_prev, dW, db = linear_backward(dZ, linear_cache)

elif activation == "sigmoid":
dZ = sigmoid_backward(dA, activation_cache)
dA_prev, dW, db = linear_backward(dZ, linear_cache)

return dA_prev, dW, db

4.整合，实现最后一层的【linear->sigmoid】和前L-1层的【linear->relu】的反向函数

def L_model_backward(AL, Y, caches):
"""
Implement the backward propagation for the [LINEAR->RELU] * (L-1) -> LINEAR -> SIGMOID group

Arguments:
AL -- probability vector, output of the forward propagation (L_model_forward())
Y -- true "label" vector (containing 0 if non-cat, 1 if cat)
caches -- list of caches containing:
every cache of linear_activation_forward() with "relu" (there are (L-1) or them, indexes from 0 to L-2)
the cache of linear_activation_forward() with "sigmoid" (there is one, index L-1)

Returns:
grads -- A dictionary with the gradients
grads["dA" + str(l)] = ...
grads["dW" + str(l)] = ...
grads["db" + str(l)] = ...
"""
grads = {}
L = len(caches) # the number of layers
m = AL.shape[1]
Y = Y.reshape(AL.shape) # after this line, Y is the same shape as AL

# Initializing the backpropagation
dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))

# Lth layer (SIGMOID -> LINEAR) gradients. Inputs: "AL, Y, caches". Outputs: "grads["dAL"], grads["dWL"], grads["dbL"]
current_cache = caches[L-1]
grads["dA" + str(L)], grads["dW" + str(L)], grads["db" + str(L)] = linear_activation_backward(dAL, current_cache, activation = "sigmoid")

for l in reversed(range(L-1)):
# lth layer: (RELU -> LINEAR) gradients.
current_cache = caches[l]
dA_prev_temp, dW_temp, db_temp = linear_activation_backward(grads["dA" + str(l + 2)], current_cache, activation = "relu")
grads["dA" + str(l + 1)] = dA_prev_temp
grads["dW" + str(l + 1)] = dW_temp
grads["db" + str(l + 1)] = db_temp

return grads

Step 5:更新参数

def update_parameters(parameters, grads, learning_rate):
"""
Update parameters using gradient descent

Arguments:
parameters -- python dictionary containing your parameters
grads -- python dictionary containing your gradients, output of L_model_backward

Returns:
parameters -- python dictionary containing your updated parameters
parameters["W" + str(l)] = ...
parameters["b" + str(l)] = ...
"""

L = len(parameters) // 2 # number of layers in the neural network

# Update rule for each parameter. Use a for loop.
for l in range(L):
parameters["W" + str(l+1)] = parameters["W" + str(l+1)] - learning_rate * grads["dW" + str(l+1)]
parameters["b" + str(l+1)] = parameters["b" + str(l+1)] - learning_rate * grads["db" + str(l+1)]

return parameters

6.实现一个预测函数，来预测测试集的正确率

def predict(X, y, parameters):
"""
This function is used to predict the results of a  L-layer neural network.

Arguments:
X -- data set of examples you would like to label
parameters -- parameters of the trained model

Returns:
p -- predictions for the given dataset X
"""

m = X.shape[1]
n = len(parameters) // 2 # number of layers in the neural network
p = np.zeros((1,m))

# Forward propagation
probas, caches = L_model_forward(X, parameters)

# convert probas to 0/1 predictions
for i in range(0, probas.shape[1]):
if probas[0,i] > 0.5:
p[0,i] = 1
else:
p[0,i] = 0
print("p="+str(p))
#print results
#print ("predictions: " + str(p))
#print ("true labels: " + str(y))
print("Accuracy: "  + str(np.sum((p == y)/m)))

return p