先說說BatchGD用整個訓練樣本進行訓練得出損失值，SGD是只用一個訓練樣本訓練就得出損失值，GD導致訓練慢，SGD導致收斂到最小值不平滑，故引入Mini-batch GD，選取部分樣本進行訓練得出損失值，

普通梯度下降算法如下：

"""" 一般梯度下降算法 """ def update_parameters_gd(parameters,grads,learning_rate):L=len(parameters)//2for i in range(L):parameters['W'+str(i+1)]=parameters['W'+str(i+1)]-learning_rate*grads['dW'+str(i+1)]parameters['b' + str(i + 1)] = parameters['b' + str(i + 1)] - learning_rate * grads['db' + str(i + 1)]return parameters

Momentum代碼：

""" Momentum初始化參數 """ def initialize_Momentum_paremeters(parameters):L=len(parameters)//2v={}for i in range(L):v['dW'+str(i+1)]=np.zeros(parameters['W'+str(i+1)].shape)v['db' + str(i + 1)] = np.zeros(parameters['b' + str(i + 1)].shape)return v """ Momentum更新權重 """ def upate_parameters_Momentum(parameters,grads,v,beta,learning_rate):L=len(parameters)//2for i in range(L):v['dW' + str(i + 1)]=beta*v['dW'+str(i+1)]+(1-beta)*grads['dW'+str(i+1)]v['db' + str(i + 1)] = beta * v['db' + str(i + 1)] + (1 - beta) * grads['db' + str(i + 1)]parameters['W'+str(i+1)]=parameters['W'+str(i+1)]-learning_rate*v['dW' + str(i + 1)]parameters['b' + str(i + 1)] = parameters['b' + str(i + 1)] - learning_rate * v['db' + str(i + 1)]return parameters,v

Adam代碼：

""" Adam初始化參數 """ def initialize_Adam_parameters(parameters):L=len(parameters)//2v={}s={}for i in range(L):v['dW' + str(i + 1)] = np.zeros(parameters['W'+str(i+1)].shape)v['db' + str(i + 1)] = np.zeros(parameters['b' + str(i + 1)].shape)s['dW' + str(i + 1)] = np.zeros(parameters['W' + str(i + 1)].shape)s['db' + str(i + 1)] = np.zeros(parameters['b' + str(i + 1)].shape)return v,s """ Adam更新權重 """ def update_parameters_Adam(parameters,grads,v,s,t,beta1,beta2,learning_rate,epsilon):L = len(parameters) // 2v_correct={}s_correct = {}for i in range(L):v['dW' + str(i + 1)] = beta1 * v['dW' + str(i + 1)] + (1 - beta1) * grads['dW' + str(i + 1)]v['db' + str(i + 1)] = beta1 * v['db' + str(i + 1)] + (1 - beta1) * grads['db' + str(i + 1)]v_correct['dW' + str(i + 1)]=v['dW' + str(i + 1)]/(1-beta1**t)v_correct['db' + str(i + 1)] = v['db' + str(i + 1)] / (1 - beta1 ** t)s['dW' + str(i + 1)] = beta2 * s['dW' + str(i + 1)] + (1 - beta2) * np.square(grads['dW' + str(i + 1)])s['db' + str(i + 1)] = beta2 * s['db' + str(i + 1)] + (1 - beta2) * np.square(grads['db' + str(i + 1)])s_correct['dW' + str(i + 1)] = s['dW' + str(i + 1)] / (1 - beta2 ** t)s_correct['db' + str(i + 1)] = s['db' + str(i + 1)] / (1 - beta2 ** t)parameters['W' + str(i + 1)] = parameters['W' + str(i + 1)] - \learning_rate * (v_correct['dW' + str(i + 1)]/(np.sqrt(s['dW' + str(i + 1)])+epsilon))parameters['b' + str(i + 1)] = parameters['b' + str(i + 1)] - \learning_rate * (v_correct['db' + str(i + 1)]/(np.sqrt(s['db' + str(i + 1)])+epsilon))return parameters, v,s

數據集 放在opt_utils.py?? 代碼如下：還包含激活函數 前向傳播 后向傳播等函數

import numpy as np import matplotlib.pyplot as plt import h5py import scipy.io 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 load_params_and_grads(seed=1):np.random.seed(seed)W1 = np.random.randn(2,3)b1 = np.random.randn(2,1)W2 = np.random.randn(3,3)b2 = np.random.randn(3,1)dW1 = np.random.randn(2,3)db1 = np.random.randn(2,1)dW2 = np.random.randn(3,3)db2 = np.random.randn(3,1)return W1, b1, W2, b2, dW1, db1, dW2, db2def initialize_parameters(layer_dims):"""Arguments:layer_dims -- python array (list) containing the dimensions of each layer in our networkReturns:parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":W1 -- weight matrix of shape (layer_dims[l], layer_dims[l-1])b1 -- bias vector of shape (layer_dims[l], 1)Wl -- weight matrix of shape (layer_dims[l-1], layer_dims[l])bl -- bias vector of shape (1, layer_dims[l])Tips:- For example: the layer_dims for the "Planar Data classification model" would have been [2,2,1]. This means W1's shape was (2,2), b1 was (1,2), W2 was (2,1) and b2 was (1,1). Now you have to generalize it!- In the for loop, use parameters['W' + str(l)] to access Wl, where l is the iterative integer."""np.random.seed(3)parameters = {}L = len(layer_dims) # number of layers in the networkfor l in range(1, L):parameters['W' + str(l)] = np.random.randn(layer_dims[l], layer_dims[l-1])* np.sqrt(2 / layer_dims[l-1])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['W' + str(l)].shape == layer_dims[l], 1)return parametersdef compute_cost(a3, Y):"""Implement the cost functionArguments:a3 -- post-activation, output of forward propagationY -- "true" labels vector, same shape as a3Returns:cost - value of the cost function"""m = Y.shape[1]logprobs = np.multiply(-np.log(a3),Y) + np.multiply(-np.log(1 - a3), 1 - Y)cost = 1./m * np.sum(logprobs)return costdef 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)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 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 results#print ("predictions: " + str(p[0,:]))#print ("true labels: " + str(y[0,:]))print("Accuracy: " + str(np.mean((p[0,:] == y[0,:]))))return pdef load_2D_dataset():data = scipy.io.loadmat('datasets/data.mat')train_X = data['X'].Ttrain_Y = data['y'].Ttest_X = data['Xval'].Ttest_Y = data['yval'].Tplt.scatter(train_X[0, :], train_X[1, :], c=train_Y, s=40, cmap=plt.cm.Spectral);return train_X, train_Y, test_X, test_Ydef 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(3)#(300,2) (300,)train_X, train_Y = sklearn.datasets.make_moons(n_samples=300, noise=.2) #300 #0.2 #print(train_X,train_Y)# Visualize the data#plt.scatter(train_X[:, 0], train_X[:, 1], c=train_Y, s=40, cmap=plt.cm.Spectral);train_X = train_X.Ttrain_Y = train_Y.reshape((1, train_Y.shape[0]))return train_X, train_Y

打印數據集看看：

全部代碼：

import numpy as np import sklearn import matplotlib.pyplot as plt import sklearn import sklearn.datasets import scipy.io import math import opt_utils import testCases1 """" 一般梯度下降算法 """ def update_parameters_gd(parameters,grads,learning_rate):L=len(parameters)//2for i in range(L):parameters['W'+str(i+1)]=parameters['W'+str(i+1)]-learning_rate*grads['dW'+str(i+1)]parameters['b' + str(i + 1)] = parameters['b' + str(i + 1)] - learning_rate * grads['db' + str(i + 1)]return parameters """" 制作樣本 mini-batch """ def random_mini_batches(X,Y,mini_batch_size):m=X.shape[1]###3mini_batchs=[]permutation = list(np.random.permutation(m))#[2,1,0]shuffled_X = X[:,permutation]##X[:,[2,1,0]] 洗牌shuffled_Y = Y[:, permutation] ##X[:,[2,1,0]]num_mini_batch=math.floor(m/mini_batch_size)for i in range(num_mini_batch):mini_batch_X=shuffled_X[:,i*mini_batch_size:(i+1)*mini_batch_size]mini_batch_Y=shuffled_Y[:,i*mini_batch_size:(i+1)*mini_batch_size]mini_batch=(mini_batch_X,mini_batch_Y)mini_batchs.append(mini_batch)if m/mini_batch_size!=0:mini_batch_X = shuffled_X[:, (i + 1) * mini_batch_size:]mini_batch_Y = shuffled_Y[:, (i + 1) * mini_batch_size:]mini_batch = (mini_batch_X, mini_batch_Y)mini_batchs.append(mini_batch)return mini_batchs """ Momentum初始化參數 """ def initialize_Momentum_paremeters(parameters):L=len(parameters)//2v={}for i in range(L):v['dW'+str(i+1)]=np.zeros(parameters['W'+str(i+1)].shape)v['db' + str(i + 1)] = np.zeros(parameters['b' + str(i + 1)].shape)return v """ Momentum更新權重 """ def upate_parameters_Momentum(parameters,grads,v,beta,learning_rate):L=len(parameters)//2for i in range(L):v['dW' + str(i + 1)]=beta*v['dW'+str(i+1)]+(1-beta)*grads['dW'+str(i+1)]v['db' + str(i + 1)] = beta * v['db' + str(i + 1)] + (1 - beta) * grads['db' + str(i + 1)]parameters['W'+str(i+1)]=parameters['W'+str(i+1)]-learning_rate*v['dW' + str(i + 1)]parameters['b' + str(i + 1)] = parameters['b' + str(i + 1)] - learning_rate * v['db' + str(i + 1)]return parameters,v """ Adam初始化參數 """ def initialize_Adam_parameters(parameters):L=len(parameters)//2v={}s={}for i in range(L):v['dW' + str(i + 1)] = np.zeros(parameters['W'+str(i+1)].shape)v['db' + str(i + 1)] = np.zeros(parameters['b' + str(i + 1)].shape)s['dW' + str(i + 1)] = np.zeros(parameters['W' + str(i + 1)].shape)s['db' + str(i + 1)] = np.zeros(parameters['b' + str(i + 1)].shape)return v,s """ Adam更新權重 """ def update_parameters_Adam(parameters,grads,v,s,t,beta1,beta2,learning_rate,epsilon):L = len(parameters) // 2v_correct={}s_correct = {}for i in range(L):v['dW' + str(i + 1)] = beta1 * v['dW' + str(i + 1)] + (1 - beta1) * grads['dW' + str(i + 1)]v['db' + str(i + 1)] = beta1 * v['db' + str(i + 1)] + (1 - beta1) * grads['db' + str(i + 1)]v_correct['dW' + str(i + 1)]=v['dW' + str(i + 1)]/(1-beta1**t)v_correct['db' + str(i + 1)] = v['db' + str(i + 1)] / (1 - beta1 ** t)s['dW' + str(i + 1)] = beta2 * s['dW' + str(i + 1)] + (1 - beta2) * np.square(grads['dW' + str(i + 1)])s['db' + str(i + 1)] = beta2 * s['db' + str(i + 1)] + (1 - beta2) * np.square(grads['db' + str(i + 1)])s_correct['dW' + str(i + 1)] = s['dW' + str(i + 1)] / (1 - beta2 ** t)s_correct['db' + str(i + 1)] = s['db' + str(i + 1)] / (1 - beta2 ** t)parameters['W' + str(i + 1)] = parameters['W' + str(i + 1)] - \learning_rate * (v_correct['dW' + str(i + 1)]/(np.sqrt(s['dW' + str(i + 1)])+epsilon))parameters['b' + str(i + 1)] = parameters['b' + str(i + 1)] - \learning_rate * (v_correct['db' + str(i + 1)]/(np.sqrt(s['db' + str(i + 1)])+epsilon))return parameters, v,s def model(X,Y,layer_dims,optimizer,learning_rate,mini_batch_size,beta,beta1,beta2,epsilon,num_pochs):t=0costs=[]parameters=opt_utils.initialize_parameters(layer_dims)if optimizer=='gd':passelif optimizer=='Momentum':v=initialize_Momentum_paremeters(parameters)elif optimizer=='Adam':v, s=initialize_Adam_parameters(parameters)for i in range(num_pochs):mini_batchs=random_mini_batches(X,Y,mini_batch_size) ###[([X],[Y]),([X2],[Y2])]for minibatch in mini_batchs:(minibatch_X,minibatch_Y)=minibatchA3, cache=opt_utils.forward_propagation(minibatch_X,parameters)cost=opt_utils.compute_cost(A3,minibatch_Y)gradients=opt_utils.backward_propagation(minibatch_X, minibatch_Y, cache)if optimizer=='gd':parameters=update_parameters_gd(parameters,gradients,learning_rate)elif optimizer=='Momentum':parameters, v=upate_parameters_Momentum(parameters, gradients, v, beta, learning_rate)elif optimizer=='Adam':t=t+1parameters, v, s=update_parameters_Adam(parameters, gradients, v, s, t, beta1, beta2, learning_rate, epsilon)if i%1000==0:costs.append(cost)print('after {} epochs cost={}'.format(i,cost) )plt.plot(costs)plt.xlabel('num_pochs(per 100)')plt.ylabel('costs')plt.title('learning_rate={}'.format(learning_rate))plt.savefig('Adam.jpg')plt.show()return parameters def test(): ############test mini_batch# X, Y, mini_batch_size=testCases1.random_mini_batches_test_case()# mini_batchs=random_mini_batches(X, Y, mini_batch_size=64)# print('first x shape={}'.format(mini_batchs[0][0].shape))# print('second x shape={}'.format(mini_batchs[1][0].shape))# print('third x shape={}'.format(mini_batchs[2][0].shape))# print('first y shape={}'.format(mini_batchs[0][1].shape))# print('second y shape={}'.format(mini_batchs[1][1].shape))# print('third y shape={}'.format(mini_batchs[2][1].shape)) ############### #######test initialize_vecolity# parameters=testCases1.initialize_velocity_test_case()# v=initialize_velocity(parameters)# print(v) #################### #######test upate_parameters_Momentum# parameters, grads, v=testCases1.update_parameters_with_momentum_test_case()# parameters, v=upate_parameters_Momentum(parameters,grads,v,beta=0.9,learning_rate=0.01)# print(parameters)# print(v) ############### ########test upate_parameters_Adamparameters, grads, v, s=testCases1.update_parameters_with_adam_test_case()parameters, v, s=update_parameters_Adam(parameters,grads,v,s,t=2,beta1=0.9,beta2=0.999,learning_rate=0.01,epsilon=1e-8)print(parameters,v,s) def test_model():train_X, train_Y=opt_utils.load_dataset()layer_dims=[train_X.shape[0],5,2,1]parameters=model(train_X,train_Y,layer_dims,optimizer='gd',learning_rate=0.0007,mini_batch_size=64,beta=0.9,beta1=0.9,beta2=0.999,epsilon=1e-8,num_pochs=10000)opt_utils.predict(train_X, train_Y, parameters) if __name__=='__main__':#test()test_model()

更改model()里的optimizer即可，普通梯度下降法結果：

Momentum下降結果和上面結果差不多可能是學習率太小，數據集太簡單導致的吧

Adam下降結果，能夠更快的收斂

總結

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