权重初始化的 正确选择能够有效的避免多层神经网络传播过程中的梯度消失和梯度爆炸问题，下面通过三个初始化的方法来验证：
sigmoid导数函数：最大值小于0.25，故经过多层反向传播以后，会导致最初的层，权重无法更新。
首先看数据集，init_utils.py代码，激活函数，数据集等等，代码如下：
import numpy as np
import matplotlib.pyplot as plt
import h5py
import sklearn
import sklearn.datasetsdef sigmoid(x):"""Compute the sigmoid of xArguments:x -- A scalar or numpy array of any size.Return:s -- sigmoid(x)"""s = 1/(1+np.exp(-x))return sdef relu(x):"""Compute the relu of xArguments:x -- A scalar or numpy array of any size.Return:s -- relu(x)"""s = np.maximum(0,x)return sdef forward_propagation(X, parameters):"""Implements the forward propagation (and computes the loss) presented in Figure 2.Arguments:X -- input dataset, of shape (input size, number of examples)Y -- true "label" vector (containing 0 if cat, 1 if non-cat)parameters -- python dictionary containing your parameters "W1", "b1", "W2", "b2", "W3", "b3":W1 -- weight matrix of shape ()b1 -- bias vector of shape ()W2 -- weight matrix of shape ()b2 -- bias vector of shape ()W3 -- weight matrix of shape ()b3 -- bias vector of shape ()Returns:loss -- the loss function (vanilla logistic loss)"""# retrieve parametersW1 = parameters["W1"]b1 = parameters["b1"]W2 = parameters["W2"]b2 = parameters["b2"]W3 = parameters["W3"]b3 = parameters["b3"]# LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOIDz1 = np.dot(W1, X) + b1a1 = relu(z1)z2 = np.dot(W2, a1) + b2a2 = relu(z2)z3 = np.dot(W3, a2) + b3a3 = sigmoid(z3)cache = (z1, a1, W1, b1, z2, a2, W2, b2, z3, a3, W3, b3)return a3, cachedef backward_propagation(X, Y, cache):"""Implement the backward propagation presented in figure 2.Arguments:X -- input dataset, of shape (input size, number of examples)Y -- true "label" vector (containing 0 if cat, 1 if non-cat)cache -- cache output from forward_propagation()Returns:gradients -- A dictionary with the gradients with respect to each parameter, activation and pre-activation variables"""m = X.shape[1](z1, a1, W1, b1, z2, a2, W2, b2, z3, a3, W3, b3) = cachedz3 = 1./m * (a3 - Y)dW3 = np.dot(dz3, a2.T)db3 = np.sum(dz3, axis=1, keepdims = True)da2 = np.dot(W3.T, dz3)dz2 = np.multiply(da2, np.int64(a2 > 0))dW2 = np.dot(dz2, a1.T)db2 = np.sum(dz2, axis=1, keepdims = True)da1 = np.dot(W2.T, dz2)dz1 = np.multiply(da1, np.int64(a1 > 0))dW1 = np.dot(dz1, X.T)db1 = np.sum(dz1, axis=1, keepdims = True)gradients = {"dz3": dz3, "dW3": dW3, "db3": db3,"da2": da2, "dz2": dz2, "dW2": dW2, "db2": db2,"da1": da1, "dz1": dz1, "dW1": dW1, "db1": db1}return gradientsdef 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 n_model_backwardReturns:parameters -- python dictionary containing your updated parameters parameters['W' + str(i)] = ... parameters['b' + str(i)] = ..."""L = len(parameters) // 2 # number of layers in the neural networks# Update rule for each parameterfor k in range(L):parameters["W" + str(k+1)] = parameters["W" + str(k+1)] - learning_rate * grads["dW" + str(k+1)]parameters["b" + str(k+1)] = parameters["b" + str(k+1)] - learning_rate * grads["db" + str(k+1)]return parametersdef compute_loss(a3, Y):"""Implement the loss functionArguments:a3 -- post-activation, output of forward propagationY -- "true" labels vector, same shape as a3Returns:loss - value of the loss function"""m = Y.shape[1]logprobs = np.multiply(-np.log(a3),Y) + np.multiply(-np.log(1 - a3), 1 - Y)loss = 1./m * np.nansum(logprobs)return lossdef load_cat_dataset():train_dataset = h5py.File('datasets/train_catvnoncat.h5', "r")train_set_x_orig = np.array(train_dataset["train_set_x"][:]) # your train set featurestrain_set_y_orig = np.array(train_dataset["train_set_y"][:]) # your train set labelstest_dataset = h5py.File('datasets/test_catvnoncat.h5', "r")test_set_x_orig = np.array(test_dataset["test_set_x"][:]) # your test set featurestest_set_y_orig = np.array(test_dataset["test_set_y"][:]) # your test set labelsclasses = np.array(test_dataset["list_classes"][:]) # the list of classestrain_set_y = train_set_y_orig.reshape((1, train_set_y_orig.shape[0]))test_set_y = test_set_y_orig.reshape((1, test_set_y_orig.shape[0]))train_set_x_orig = train_set_x_orig.reshape(train_set_x_orig.shape[0], -1).Ttest_set_x_orig = test_set_x_orig.reshape(test_set_x_orig.shape[0], -1).Ttrain_set_x = train_set_x_orig/255test_set_x = test_set_x_orig/255return train_set_x, train_set_y, test_set_x, test_set_y, classesdef predict(X, y, parameters):"""This function is used to predict the results of a n-layer neural network.Arguments:X -- data set of examples you would like to labelparameters -- parameters of the trained modelReturns:p -- predictions for the given dataset X"""m = X.shape[1]p = np.zeros((1,m), dtype = np.int)# Forward propagationa3, caches = forward_propagation(X, parameters)# convert probas to 0/1 predictionsfor i in range(0, a3.shape[1]):if a3[0,i] > 0.5:p[0,i] = 1else:p[0,i] = 0# print resultsprint("Accuracy: " + str(np.mean((p[0,:] == y[0,:]))))return pdef plot_decision_boundary(model, X, y):# Set min and max values and give it some paddingx_min, x_max = X[0, :].min() - 1, X[0, :].max() + 1y_min, y_max = X[1, :].min() - 1, X[1, :].max() + 1h = 0.01# Generate a grid of points with distance h between themxx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))# Predict the function value for the whole gridZ = model(np.c_[xx.ravel(), yy.ravel()])Z = Z.reshape(xx.shape)# Plot the contour and training examplesplt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)plt.ylabel('x2')plt.xlabel('x1')plt.scatter(X[0, :], X[1, :], c=y, cmap=plt.cm.Spectral)plt.show()def predict_dec(parameters, X):"""Used for plotting decision boundary.Arguments:parameters -- python dictionary containing your parameters X -- input data of size (m, K)Returnspredictions -- vector of predictions of our model (red: 0 / blue: 1)"""# Predict using forward propagation and a classification threshold of 0.5a3, cache = forward_propagation(X, parameters)predictions = (a3>0.5)return predictionsdef load_dataset():np.random.seed(1)train_X, train_Y = sklearn.datasets.make_circles(n_samples=300, noise=.05)#print(train_X.shape)(300,2)#print(train_Y) (300,)np.random.seed(2)test_X, test_Y = sklearn.datasets.make_circles(n_samples=100, noise=.05)# Visualize the data cmap = plt.cm.Spectral 表示给 1 0点不同的颜色plt.scatter(train_X[:, 0], train_X[:, 1], c=train_Y, s=40, cmap=plt.cm.Spectral)train_X = train_X.T #(2,300)#print(train_X)train_Y = train_Y.reshape((1, train_Y.shape[0])) #(1,300)#print(train_Y)test_X = test_X.T #(2,100)test_Y = test_Y.reshape((1, test_Y.shape[0])) #(1,100)return train_X, train_Y, test_X, test_Y
载入数据集：
import numpy as np
import init_utils
import matplotlib.pyplot as plt
import sklearn
import sklearn.datasets
#(2, 300)(1, 300)(2, 100)(1, 100)
train_X, train_Y, test_X, test_Y=init_utils.load_dataset()
print(train_X.shape)
print(train_Y.shape)
print(test_X.shape)
print(test_Y.shape)
plt.show()打印结果：
完整代码：
import numpy as np
import init_utils
import matplotlib.pyplot as plt
import sklearn
import sklearn.datasets
#(2, 300)(1, 300)(2, 100)(1, 100)
train_X, train_Y, test_X, test_Y=init_utils.load_dataset()
# print(train_X.shape)
# print(train_Y.shape)
# print(test_X.shape)
# print(test_Y.shape)
plt.show()
"""
初始化权重为0
"""
def initialize_parameters_zeros(layers_dims):L=len(layers_dims)parameters={}for i in range(1,L):parameters['W'+str(i)]=np.zeros((layers_dims[i],layers_dims[i-1]))parameters['b' + str(i)]=np.zeros((layers_dims[i],1))return parameters
"""
随机初始化权重
"""
def initialize_parameters_random(layers_dims):L=len(layers_dims)parameters={}for i in range(1,L):parameters['W'+str(i)]=np.random.randn(layers_dims[i],layers_dims[i-1])parameters['b' + str(i)]=np.zeros((layers_dims[i],1))return parameters
"""
随机初始化权重 方差2/n
"""
def initialize_parameters_he(layers_dims):L=len(layers_dims)parameters={}for i in range(1,L):parameters['W'+str(i)]=np.random.randn(layers_dims[i],layers_dims[i-1])\*np.sqrt(2.0/layers_dims[i-1])parameters['b' + str(i)]=np.zeros((layers_dims[i],1))return parameters
"""
模型传播过程
"""
def model(X,Y,initialization,num_iterations,learning_rate):#m=X.shape[1]costs=[]layers_dims=[X.shape[0],10,5,1]if initialization=='zeros':parameters=initialize_parameters_zeros(layers_dims)elif initialization=='random':parameters = initialize_parameters_random(layers_dims)elif initialization == 'he':parameters = initialize_parameters_he(layers_dims)for i in range(num_iterations):a3, cache=init_utils.forward_propagation(X, parameters) #cache (z1, a1, W1, b1, z2, a2, W2, b2, z3, a3, W3, b3)cost=init_utils.compute_loss(a3, Y)grads=init_utils.backward_propagation(X, Y, cache)parameters=init_utils.update_parameters(parameters, grads, learning_rate)if i%1000==0:print('cost after number iterations {} cost is {}'.format(i,cost))costs.append(cost)plt.plot(costs)plt.xlabel('num_iterations')plt.ylabel('cost')plt.show()return parameters
def test_initialize_parameters():parameters=initialize_parameters_zeros([2,4,1])print(parameters)parameters = initialize_parameters_random([2, 4, 1])print(parameters)parameters = initialize_parameters_he([2, 4, 1])print(parameters)
def test_model():# model(X, Y, initialization, layers_dims, num_iterations, learning_rate):parameters = model(train_X, train_Y, 'zeros', 15000, 0.01)#print(parameters)predictions_train=init_utils.predict(train_X, train_Y,parameters)print('predictions_train'.format(predictions_train))init_utils.plot_decision_boundary(lambda x:init_utils.predict_dec(parameters,x.T),train_X, np.squeeze(train_Y))parameters = model(train_X, train_Y, 'random', 15000, 0.01)#print(parameters)predictions_train = init_utils.predict(train_X, train_Y, parameters)print('predictions_train'.format(predictions_train))init_utils.plot_decision_boundary(lambda x: init_utils.predict_dec(parameters, x.T), train_X, np.squeeze(train_Y))parameters = model(train_X, train_Y, 'he', 15000, 0.01)#print(parameters)predictions_train = init_utils.predict(train_X, train_Y, parameters)print('predictions_train'.format(predictions_train))init_utils.plot_decision_boundary(lambda x: init_utils.predict_dec(parameters, x.T), train_X, np.squeeze(train_Y))
if __name__=='__main__':#test_initialize_parameters()test_model()#pass结果1：初始化权重为0的结果
结果2：初始化权重为0~1之间的数的结果
结果3：初始化权重为0~1之间,方差为2/n的结果
