一、实现

1、吴恩达提供的工具函数

这几个函数这里只是展示一下，这是吴恩达写好的工具类，在实现的部分会导入；具体查看提供的附件

sigmoid

def sigmoid(Z):A = 1/(1+np.exp(-Z))cache = Zreturn A, cache

sigmoid求导

def sigmoid_backward(dA, cache):Z = caches = 1/(1+np.exp(-Z))dZ = dA * s * (1-s)return dZ

relu

def relu(Z):A = np.maximum(0,Z)cache = Z return A, cache

relu求导

def relu_backward(dA, cache):Z = cachedZ = np.array(dA, copy=True)dZ[Z <= 0] = 0return dZ

2、实现代码

导包和配置

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'%load_ext autoreload
%autoreload 2np.random.seed(1)

初始化参数

def initialize_parameters_deep(layer_dims):"""Arguments:layer_dims -- python array (list) containing thedimensions of each layer in our networkReturns:parameters -- python dictionary containing yourparameters "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%d' % l] = np.random.randn(layer_dims[l],layer_dims[l-1]) * 0.01parameters['b%d' % l] = np.zeros((layer_dims[l], 1))return parameters  

前向运算

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) + bcache = (A, W, b)return Z, cache

def linear_activation_forward(A_prev, W, b, activation):"""Implement the forward propagation for the LINEAR->ACTIVATION layerArguments: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)cache = (linear_cache, activation_cache)return A, cache

def L_model_forward(X, parameters):"""Implement forward propagation for the [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID computationArguments:X -- data, numpy array of shape (input size, number of examples)parameters -- output of initialize_parameters_deep()Returns:AL -- last post-activation valuecaches -- 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 = XL = 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, linear_activation_cache = linear_activation_forward(A_prev,parameters['W%s' % l],  parameters['b%s' % l], activation = "relu")caches.append(linear_activation_cache)### END CODE HERE #### Implement LINEAR -> SIGMOID. Add "cache" to the "caches" list.### START CODE HERE ### (≈ 2 lines of code)AL, linear_activation_cache = linear_activation_forward(A,parameters['W%s' % L],  parameters['b%s' % L], activation = "sigmoid")caches.append(linear_activation_cache)### END CODE HERE ###return AL, caches

计算损失

def compute_cost(AL, Y):m = Y.shape[1]# Compute loss from aL and y.### START CODE HERE ### (≈ 1 lines of code)cost = -1./ m * (np.dot(np.log(AL), Y.T) + np.dot(np.log(1-AL), (1-Y).T))### END CODE HERE ###cost = np.squeeze(cost)      return cost

后向运算

def linear_backward(dZ, cache):"""Implement the linear portion of backward propagation fora 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 layerReturns:dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prevdW -- Gradient of the cost with respect to W (current layer l), same shape as Wdb -- Gradient of the cost with respect to b (current layer l), same shape as b"""A_prev, W, b = cachem = A_prev.shape[1]dA_prev = np.dot(W.T, dZ)dW = 1./ m * np.dot(dZ, A_prev.T)db = 1./m * np.sum(dZ, axis=1, keepdims=True)### END CODE HERE ###return dA_prev, dW, db

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 efficientlyactivation -- 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_prevdW -- Gradient of the cost with respect to W (current layer l), same shape as Wdb -- Gradient of the cost with respect to b (current layer l), same shape as b"""linear_cache, activation_cache = cacheif activation == "relu":### 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 ###elif activation == "sigmoid":### 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 ###return dA_prev, dW, db

def L_model_backward(AL, Y, caches):"""Implement the backward propagation for the [LINEAR->RELU] * (L-1) -> LINEAR -> SIGMOID groupArguments: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 gradientsgrads["dA" + str(l)] = ...grads["dW" + str(l)] = ...grads["db" + str(l)] = ..."""grads = {}L = len(caches) # the number of layersm = 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)grads['dA'+str(L)] = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))### END CODE HERE ###layer = Lgrads['dA'+str(layer-1)], grads['dW'+str(layer)],grads['db'+str(layer)]= linear_activation_backward(grads['dA'+str(layer)], caches[layer-1],activation = "sigmoid")for l in reversed(range(L - 1)):layer = l + 1grads['dA'+str(layer-1)],grads['dW'+str(layer)],grads['db'+str(layer)] = linear_activation_backward(grads['dA'+str(layer)], caches[layer-1], activation = "relu")return grads

更新参数

def update_parameters(parameters, grads, learning_rate):"""Update parameters using gradient descentArguments:parameters -- python dictionary containing your parameters grads -- python dictionary containing your gradients, output of L_model_backwardReturns: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 networkfor l in range(1, L+1):parameters['W'+str(l)] = parameters['W'+str(l)] - learning_rate * grads['dW'+str(l)]parameters['b'+str(l)] = parameters['b'+str(l)] - learning_rate * grads['db'+str(l)]return parameters

组装模型

def L_layer_model(X, Y, layers_dims, learning_rate = 0.0075,num_iterations = 3000, print_cost=False):#lr was 0.009"""Implements a L-layer neural network: [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID.Arguments:X -- data, numpy array of shape (number of examples, num_px * num_px * 3)Y -- true "label" vector (containing 0 if cat, 1 if non-cat),of shape (1, number of examples)layers_dims -- list containing the input size and eachlayer size, of length (number of layers + 1).learning_rate -- learning rate of the gradient descent update rulenum_iterations -- number of iterations of the optimization loopprint_cost -- if True, it prints the cost every 100 stepsReturns:parameters -- parameters learnt by the model.They can then be used to predict."""np.random.seed(1)costs = []# Parameters initialization.### START CODE HERE ###parameters = initialize_parameters_deep(layers_dims)### END CODE HERE #### Loop (gradient descent)for i in range(0, num_iterations):# Forward propagation: [LINEAR -> RELU]*(L-1) -> LINEAR -> SIGMOID.### START CODE HERE ### (≈ 1 line of code)AL, caches = L_model_forward(X, parameters)### END CODE HERE #### Compute cost.### START CODE HERE ### (≈ 1 line of code)cost = compute_cost(AL, Y)### END CODE HERE #### Backward propagation.### START CODE HERE ### (≈ 1 line of code)grads = L_model_backward(AL, Y, caches)### END CODE HERE #### Update parameters.### START CODE HERE ### (≈ 1 line of code)parameters = update_parameters(parameters, grads, learning_rate)### END CODE HERE #### Print the cost every 100 training exampleif print_cost and i % 100 == 0:print ("Cost after iteration %i: %f" %(i, cost))if print_cost and i % 100 == 0:costs.append(cost)# plot the costplt.plot(np.squeeze(costs))plt.ylabel('cost')plt.xlabel('iterations (per tens)')plt.title("Learning rate =" + str(learning_rate))plt.show()return parameters

3、问题及思考

除了L层的后向运算的测试用例外，其余各个环节及最后的结果都是正确的。
我的代码的运行结果和测试用例的对比如下图所示：

可以看到运行结果是完全对不上的！
所以网上找了很多答案，他们的代码与我的代码的不同之处都在L层的反向传播处：
（1）来源

(2)来源

就这两个答案来看，他们的写法在我看来是错误的；但是他们能对上答案，而对不上；我改成了它们的样子也对不上~所以对不上答案的问题可能在于我的测试用例？我也没有去看他们的测试用例和我的是否一样！反正就是这一个测试用例过不去，后面的全对。证明我的实现是没有问题的。
然后要说为什么它们的代码在我看来是不正确的，我的代码如下

很明显，1处的公式得到的就是输出层（L）的激活值的导数，而2和3每次求导，都应该得到前一层的激活值的导数与当前层的W和b的导数！如下图所示