權重初始化的 正確選擇能夠有效的避免多層神經網絡傳播過程中的梯度消失和梯度爆炸問題，下面通過三個初始化的方法來驗證：

sigmoid導數函數：最大值小于0.25，故經過多層反向傳播以后，會導致最初的層，權重無法更新。

首先看數據集，init_utils.py代碼，激活函數，數據集等等，代碼如下：

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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 表示給 1 0點不同的顏色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

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載入數據集：

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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()

打印結果：

完整代碼：

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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() """ 初始化權重為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 """ 隨機初始化權重 """ 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 """ 隨機初始化權重 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 """ 模型傳播過程 """ 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

結果1：初始化權重為0的結果

結果2：初始化權重為0~1之間的數的結果

結果3：初始化權重為0~1之間,方差為2/n的結果

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總結

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