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import numpy as np
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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
初始化两层的网络和L层的神经网络的参数。 实现正向传播模块（在下图中以紫色显示）。 - 完成模型正向传播步骤的LINEAR部分（ ）。 - 提供使用的ACTIVATION函数（relu / Sigmoid）。 - 将前两个步骤合并为新的[LINEAR-> ACTIVATION]前向函数。 - 堆叠[LINEAR-> RELU]正向函数L-1次（第1到L-1层），并在末尾添加[LINEAR-> SIGMOID]（最后的 层）。这合成了一个新的L_model_forward函数。 计算损失。 实现反向传播模块（在下图中以红色表示）。 - 完成模型反向传播步骤的LINEAR部分。 - 提供的ACTIVATE函数的梯度（relu_backward / sigmoid_backward） - 将前两个步骤组合成新的[LINEAR-> ACTIVATION]反向函数。 - 将[LINEAR-> RELU]向后堆叠L-1次，并在新的L_model_backward函数中后向添加[LINEAR-> SIGMOID] 最后更新参数。
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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) # number of layers in the network
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))
#校验
# 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
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# 测试下
parameters = initialize_parameters_deep([5,4,3])
parameters
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# 正向传播模块, 其中A[0]=X
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 = (Z,W,b)
return Z,cache
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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".
### START CODE HERE ### (≈ 2 lines of code)
Z, linear_cache = linear_forward(A_prev,W,b)
A, activation_cache = sigmoid(Z)
### END CODE HERE ###
elif activation == "relu":
# Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
### START CODE HERE ### (≈ 2 lines of code)
Z, linear_cache = linear_forward(A_prev,W,b)
A, activation_cache = relu(Z)
### END CODE HERE ###
assert (A.shape == (W.shape[0], A_prev.shape[1]))
cache = (linear_cache, activation_cache)
return A, cache
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# 使用你先前编写的函数
# 使用for循环复制[LINEAR-> RELU]（L-1）次
# 不要忘记在“cache”列表中更新缓存。 要将新值 c添加到list中，可以使用list.append(c)。
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
### START CODE HERE ### (≈ 2 lines of code)
A, cache = linear_activation_forward(A_prev,parameters['W' + str(l)],parameters['b' + str(l)],activation = "relu")
caches.append(cache)
### END CODE HERE ###
# Implement LINEAR -> SIGMOID. Add "cache" to the "caches" list.
### START CODE HERE ### (≈ 2 lines of code)
AL, cache = linear_activation_forward(A,parameters['W' + str(L)],parameters['b' + str(L)],activation = "sigmoid")
caches.append(cache)
### END CODE HERE ###
assert(AL.shape == (1,X.shape[1]))
return AL, caches
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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.
### START CODE HERE ### (≈ 1 lines of code)
cost = -1 / m * np.sum(Y * np.log(AL) + (1-Y) * np.log(1-AL),axis=1,keepdims=True)
### END CODE HERE ###
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
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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]
### START CODE HERE ### (≈ 3 lines of code
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)
### END CODE HERE ###
assert (dA_prev.shape == A_prev.shape)
assert (dW.shape == W.shape)
assert (db.shape == b.shape)
return dA_prev, dW, db
sigmoid_backward：实现SIGMOID单元的反向传播。 你可以这样使用：
dZ = sigmoid_backward(dA, activation_cache)
relu_backward：实现RELU单元的反向传播。 你可以这样使用：
dZ = relu_backward(dA, activation_cache)
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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
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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":
### START CODE HERE ### (≈ 2 lines of code)
dZ = relu_backward(dA, activation_cache)
dA_prev, dW, db = linear_backward(dZ, linear_cache)
### END CODE HERE ###
elif activation == "sigmoid":
### START CODE HERE ### (≈ 2 lines of code)
dZ = sigmoid_backward(dA, activation_cache)
dA_prev, dW, db = linear_backward(dZ, linear_cache)
### END CODE HERE ###
return dA_prev, dW, db
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# dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL)) # derivative of cost with respect to AL
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" (it's caches[l], for l in range(L-1) i.e l = 0...L-2)
the cache of linear_activation_forward() with "sigmoid" (it's caches[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
### START CODE HERE ### (1 line of code)
dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))
### END CODE HERE ###
# Lth layer (SIGMOID -> LINEAR) gradients. Inputs: "AL, Y, caches". Outputs: "grads["dAL"], grads["dWL"], grads["dbL"]
### START CODE HERE ### (approx. 2 lines)
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")
### END CODE HERE ###
for l in reversed(range(L - 1)):
# lth layer: (RELU -> LINEAR) gradients.
# Inputs: "grads["dA" + str(l + 2)], caches". Outputs: "grads["dA" + str(l + 1)] , grads["dW" + str(l + 1)] , grads["db" + str(l + 1)]
### START CODE HERE ### (approx. 5 lines)
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
### END CODE HERE ###
return grads