1 导入所需的包
import numpy as np import h5py import matplotlib.pyplot as plt from testCases_v2 import * from dnn_utils_v2 import sigmoid, sigmoid_backward, relu, relu_backward %matplotlib inline plt.rcParams['figure.figsize'] = (5.0, 4.0) # set default size of plots plt.rcParams['image.interpolation'] = 'nearest' plt.rcParams['image.cmap'] = 'gray' np.random.seed(1)
包介绍
- numpy 用Python进行科学计算的基本软件包。
- matplotlib 是一个用于在Python中绘制图表的库。
- dnn_utils 提供一些必要的函数。
- testCases 提供一些测试用例来评估函数的正确性。
2 概要
主要步骤如下：
- 为一个2层神经网络和一个L层神经网络初始化参数
- 建立前向传播模型
- 完成一层中前向传播的线性部分（得到\(Z^{[l]}\)）
- 激活函数
relu/sigmoid - 将前两步结合到一个新的
LINEAR->ACTIVATION前向函数中 - 在新建的
L_model_forward函数中，将LINEAR->RELU前向函数执行L-1 次（1~L-1层），然后再执行一次LINEAR->SIGMOID前向函数（最后的L层）
- 计算损失
- 建立反向传播模型
- 完成一层中反向传播的线性部分
- 激活函数的导数
relu_backward/sigmoid_backward - 将前两步结合到一个新的
LINEAR->ACTIVATION反向函数中 - 在新建的
L_model_backward函数中，将LINEAR->SIGMOID反向函数执行1次，然后再执行L-1次LINEAR->RELU反向函数
- 更新参数
3 初始化参数
3.1 2层神经网络
该模型的结构是： LINEAR -> RELU -> LINEAR -> SIGMOID
# GRADED FUNCTION: initialize_parameters
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
3.2 L层神经网络
该模型的结构是： [LINEAR -> RELU] × (L-1) -> LINEAR -> SIGMOID
# GRADED FUNCTION: initialize_parameters_deep
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))
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
4 前向传播模型
4.1 线性部分（Linear）
线性部分要计算的公式如下：
\[Z^{[l]}=W^{[l]}A^{[l-1]}+b^{[l]}\]此处\(A^{[0]}=X\)
# GRADED FUNCTION: linear_forward
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
4.2 线性激活部分（Linear-Activation）
为了更方便，将把两个函数（
LINEAR和ACTIVATION）放入一个函数（LINEAR-> ACTIVATION）当中。因此，该函数将首先执行LINEAR步骤，然后执行ACTIVATION步骤。
# GRADED FUNCTION: linear_activation_forward
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":
Z, linear_cache = linear_forward(A_prev, W, b)
A, activation_cache = sigmoid(Z)
elif activation == "relu":
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
4.3 L层前向模型
该模型需执行使用RELU的
linear_activation_forward函数L-1次，然后执行使用SIGMOID的linear_activation_forward函数一次
# GRADED FUNCTION: L_model_forward
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)], "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)], "sigmoid")
caches.append(cache)
assert (AL.shape == (1, X.shape[1]))
return AL, caches
5 计算成本
计算成本的公式如下：
\[J=-\frac{1}{m}\sum_{i=1}^m(y^{(i)}log(a^{[L](i)})+(1-y^{(i)})log(1-a^{[L](i)}))\]
# GRADED FUNCTION: compute_cost
def compute_cost(AL, Y):
"""
Implement the cost function
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 = -np.sum(np.multiply(Y, np.log(AL)) + np.multiply(1 - Y, np.log(1 - AL))) / m
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
6 反向传播模型
6.1 线性部分（Linear）
假设已经计算了导数\(dZ^{[l]}= \frac{∂L}{∂Z^{[l]}}\)
现在需要求得参数\(dW^{[l]},db^{[l]},dA^{[l-1]}\)
可使用以下公式：
\[dW^{[l]}=\frac{∂L}{dW^{[l]}}=\frac{1}{m}dZ^{[l]}A^{[l-1]T}\] \[db^{[l]}=\frac{∂L}{db^{[l]}}=\frac{1}{m}\sum_{i=1}^mdZ^{[l](i)}\] \[dA^{[l-1]}=\frac{∂L}{dA^{[l-1]}}=W^{[l]T}dZ^{[l]}\]
# GRADED FUNCTION: linear_backward
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 = np.dot(dZ, A_prev.T) / m
db = np.sum(dZ, axis=1, keepdims=True) / m
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
6.2 线性激活部分（Linear-Activation）
为了更方便，将把两个函数（
LINEAR和ACTIVATION）放入一个函数（LINEAR-> ACTIVATION）当中。因此，该函数将首先执行ACTIVATION步骤，然后执行LINEAR步骤。
其中 \(dZ^{[l]}=dA^{[l]}∗g′(Z^{[l]})\) ， \(g′()\) 为激活函数的导数
# GRADED FUNCTION: linear_activation_backward
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
6.3 L层反向模型
首先计算\(dA^{[L]}=\frac{∂L}{dA^{[L]}}\)
即
dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))紧接着，便可以利用这个导数
dAL继续进行反向计算首先进行一次
LINEAR->SIGMOID反向函数的计算，然后再循环计算LINEAR->RELU反向函数L-1次应将每个
dA，dW和db存储在grads字典中，以便进行后续参数的更新
# GRADED FUNCTION: L_model_backward
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
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,
"sigmoid")
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)]
current_cache = caches[l]
dA_prev_temp, dW_temp, db_temp = linear_activation_backward(grads["dA" + str(l + 2)], current_cache, "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
6.4 更新参数
使用梯度下降来更新参数：
\[W^{[l]} = W^{[l]} - \alpha dW^{[l]}\] \[b^{[l]} = b^{[l]} - \alpha db^{[l]}\]此处的 \(\alpha\) 是学习率
# GRADED FUNCTION: update_parameters
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
