正则化解决过拟合

本片举三个例子进行对比，分别是：不使用正则化、使用L2正则化、使用dropout正则化。

首先是前后向传播、加载数据、画图所需要的相关函数的reg_utils.py：

# -*- coding: utf-8 -*-import numpy as np
import matplotlib.pyplot as plt
import scipy.io as siodef 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 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(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 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 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.nansum(logprobs)return costdef 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 load_2D_dataset(is_plot=True):data = sio.loadmat('datasets/data.mat')train_X = data['X'].Ttrain_Y = data['y'].Ttest_X = data['Xval'].Ttest_Y = data['yval'].Tif is_plot:plt.scatter(train_X[0, :], train_X[1, :], c=train_Y, s=40, cmap=plt.cm.Spectral)plt.show()return train_X, train_Y, test_X, test_Ydef 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 predictions

可以先画出数据看是什么样：

train_X, train_Y, test_X, test_Y = reg_utils.load_2D_dataset(is_plot=True)

然后开始测试代码：

不使用正则化

首先我们不使用正则化，让lambd参数（删了个a不与python关键字重合）和keep_prob为默认值0和1，表示不使用这两个正则化。

import numpy as np
import matplotlib.pyplot as plt
import sklearn
import sklearn.datasets
import reg_utils  plt.rcParams['figure.figsize'] = (7.0, 4.0)  # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'# 加载数据集
train_X, train_Y, test_X, test_Y = reg_utils.load_2D_dataset(is_plot=False)def model(X, Y, learning_rate=0.3, num_iterations=30000, print_cost=True, is_plot=True, lambd=0, keep_prob=1):"""实现一个三层的神经网络：LINEAR ->RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID参数：X - 输入的数据，维度为(2, 要训练/测试的数量)Y - 标签，【0(蓝色) | 1(红色)】，维度为(1，对应的是输入的数据的标签)learning_rate - 学习速率num_iterations - 迭代的次数print_cost - 是否打印成本值，每迭代10000次打印一次，但是每1000次记录一个成本值is_polt - 是否绘制梯度下降的曲线图lambd - 正则化的超参数，实数keep_prob - 随机删除节点的概率返回parameters - 学习后的参数"""grads = {}costs = []m = X.shape[1]layers_dims = [X.shape[0], 20, 3, 1]# 初始化参数parameters = reg_utils.initialize_parameters(layers_dims)# 开始学习for i in range(0, num_iterations):# 前向传播## 是否随机删除节点if keep_prob == 1:### 不随机删除节点a3, cache = reg_utils.forward_propagation(X, parameters)elif keep_prob < 1:### 随机删除节点a3, cache = forward_propagation_with_dropout(X, parameters, keep_prob)else:print("keep_prob参数错误！程序退出。")exit# 计算成本## 是否使用二范数if lambd == 0:### 不使用L2正则化cost = reg_utils.compute_cost(a3, Y)else:### 使用L2正则化cost = compute_cost_with_regularization(a3, Y, parameters, lambd)# 反向传播## 可以同时使用L2正则化和随机删除节点，但是本次实验不同时使用。assert (lambd == 0 or keep_prob == 1)## 两个参数的使用情况if (lambd == 0 and keep_prob == 1):### 不使用L2正则化和不使用随机删除节点grads = reg_utils.backward_propagation(X, Y, cache)elif lambd != 0:### 使用L2正则化，不使用随机删除节点grads = backward_propagation_with_regularization(X, Y, cache, lambd)elif keep_prob < 1:### 使用随机删除节点，不使用L2正则化grads = backward_propagation_with_dropout(X, Y, cache, keep_prob)# 更新参数parameters = reg_utils.update_parameters(parameters, grads, learning_rate)# 记录并打印成本if i % 1000 == 0:## 记录成本costs.append(cost)if (print_cost and i % 10000 == 0):# 打印成本print("第" + str(i) + "次迭代，成本值为：" + str(cost))# 是否绘制成本曲线图if is_plot:plt.plot(costs)plt.ylabel('cost')plt.xlabel('iterations (x1,000)')plt.title("Learning rate =" + str(learning_rate))plt.show()# 返回学习后的参数return parameters# 进行模型学习，得到最终的参数
parameters = model(train_X, train_Y, is_plot=True)
print("训练集:")
predictions_train = reg_utils.predict(train_X, train_Y, parameters)
print("测试集:")
predictions_test = reg_utils.predict(test_X, test_Y, parameters)

运行后结果如下：

第0次迭代，成本值为：0.6557412523481002
第10000次迭代，成本值为：0.16329987525724213
第20000次迭代，成本值为：0.13851642423265018
训练集:
Accuracy: 0.9478672985781991
测试集:
Accuracy: 0.915

这样的结果看起来还算正常（因为数据集的问题，过拟合的特征还看不太出来不是很明显），接下来绘制决策边界分割曲线会看得比较明显：

plt.title("Model without regularization")
axes = plt.gca()
axes.set_xlim([-0.75, 0.40])
axes.set_ylim([-0.75, 0.65])
reg_utils.plot_decision_boundary(lambda x: reg_utils.predict_dec(parameters, x.T), train_X, train_Y)

运行结果如下：

可以很明显的看到过拟合了，钻牛角尖过分学习那几个局部特征了。
接下来试验一下引入正则化的效果。

使用L2正则化

L2正则化公式如下（L2正则化主要体现在loss的公式上面）：

L2正则化成本其实就是每一层的权重的平方和，用代码np.sum(np.square(Wl))来计算。
$$d{W}^{[l]}=(frombackprop)+\frac{\lambda }{m}{W}^{[l]}，frombackprop就是d{W}^{[l]}$$

$$更新参数时，W^{[l]}=W^{[l]}-\alpha d{W}^{[l]}$$
$$最终合并同类项为：W^{[l]}=(1-\frac{\lambda }{m})W^{[l]}-\alpha d{W}^{[l]}$$
通过更新参数的公式可以看到，L2正则化是通过加入正则化参数 $\lambda$ 使得网络的权重变小（重量衰减），从而削弱众多神经元的影响来解决过拟合问题。
加入如下代码，计算L2正则化的loss和反向的梯度：

def compute_cost_with_regularization(A3, Y, parameters, lambd):"""实现公式2的L2正则化计算成本参数：A3 - 正向传播的输出结果，维度为（输出节点数量，训练/测试的数量）Y - 标签向量，与数据一一对应，维度为(输出节点数量，训练/测试的数量)parameters - 包含模型学习后的参数的字典返回：cost - 使用公式2计算出来的正则化损失的值"""m = Y.shape[1]W1 = parameters["W1"]W2 = parameters["W2"]W3 = parameters["W3"]# 无正则化losscross_entropy_cost = reg_utils.compute_cost(A3, Y)# L2正则化loss，lambd*每层权重的平方和的和/(2*m)L2_regularization_cost = lambd * (np.sum(np.square(W1)) + np.sum(np.square(W2)) + np.sum(np.square(W3))) / (2 * m)cost = cross_entropy_cost + L2_regularization_costreturn cost# 当然，因为改变了成本函数，我们也必须改变向后传播的函数， 所有的梯度都必须根据这个新的成本值来计算。
def backward_propagation_with_regularization(X, Y, cache, lambd):"""实现我们添加了L2正则化的模型的后向传播。参数：X - 输入数据集，维度为（输入节点数量，数据集里面的数量）Y - 标签，维度为（输出节点数量，数据集里面的数量）cache - 来自forward_propagation（）的cache输出lambda - regularization超参数，实数返回：gradients - 一个包含了每个参数、激活值和预激活值变量的梯度的字典"""m = X.shape[1](Z1, A1, W1, b1, Z2, A2, W2, b2, Z3, A3, W3, b3) = cachedZ3 = A3 - YdW3 = (1 / m) * np.dot(dZ3, A2.T) + ((lambd * W3) / m)    # 前一项为frombackprop，即原来的dW3db3 = (1 / m) * np.sum(dZ3, axis=1, keepdims=True)dA2 = np.dot(W3.T, dZ3)dZ2 = np.multiply(dA2, np.int64(A2 > 0))dW2 = (1 / m) * np.dot(dZ2, A1.T) + ((lambd * W2) / m)db2 = (1 / m) * np.sum(dZ2, axis=1, keepdims=True)dA1 = np.dot(W2.T, dZ2)dZ1 = np.multiply(dA1, np.int64(A1 > 0))dW1 = (1 / m) * np.dot(dZ1, X.T) + ((lambd * W1) / m)db1 = (1 / m) * 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 gradients

调用model函数时加入lambd参数：

parameters = model(train_X, train_Y, lambd=0.7,is_plot=True)

运行代码结果如下：

第0次迭代，成本值为：0.6974484493131264
第10000次迭代，成本值为：0.2684918873282239
第20000次迭代，成本值为：0.2680916337127301
训练集:
Accuracy: 0.9383886255924171
测试集:
Accuracy: 0.93

loss走势曲线：

绘制决策边界：
这里的标题可以改一下：

plt.title("Model with L2-regularization")

可以看到训练集和测试集上的准确率几乎没有差距，或者说比无正则化的差距要小，从绘制边界可以看到没有过拟合的特征。
L2正则化会使决策边界更加平滑。但要注意，如果λ太大，也可能会“过度平滑”，从而导致模型高偏差，从而变成欠拟合的状态。

使用dropout正则化

原理是在某层当中设置保留某个神经元的概率keep-prob，在这层中随机失活1 - keep-prob概率的节点。则这层当中失活的节点在本轮迭代中的正向传播和反向传播均不参与，即失活的节点的参数在本轮训练中不作更新，没失火的节点的参数进行更新。
假设在第3层进行随机失活，在正向传播时需要进行以下三步(假设在第三层的失活)：

1. d3 = np.random.rand(a3.shape[0], a3.shape[1]) < keep-prob 。这句话的意思是创建一个跟a3相同shape的随机矩阵，每个值与keep-prob进行对比，小于keep-prob的为True（python计算时自动变为1），大于keep-prob即不符合的为False即0。
2. a3 = np.multiply(a3, d3) 。通过和d3相乘，来失活1 - keep-prob的节点不参与计算（与0相乘为0）。
3. a3 /= keep-prob 。通过缩放就在计算成本的时候仍然大致具有相同的期望值，这叫做反向dropout。
   在反向传播时需要进行以下两步(假设在第三层的失活)：
4. dA3 = dA3 * D3 。舍弃正向传播中舍弃的节点，不进行计算梯度即不进行更新。
5. dA2 /= keep_prob 。进行缩放，保持大致期望。
   加入以下代码进行dropout的正反向传播：

def forward_propagation_with_dropout(X, parameters, keep_prob=0.5):"""实现具有随机舍弃节点的前向传播。LINEAR -> RELU + DROPOUT -> LINEAR -> RELU + DROPOUT -> LINEAR -> SIGMOID.参数：X  - 输入数据集，维度为（2，示例数）parameters - 包含参数“W1”，“b1”，“W2”，“b2”，“W3”，“b3”的python字典：W1  - 权重矩阵，维度为（20,2）b1  - 偏向量，维度为（20,1）W2  - 权重矩阵，维度为（3,20）b2  - 偏向量，维度为（3,1）W3  - 权重矩阵，维度为（1,3）b3  - 偏向量，维度为（1,1）keep_prob  - 随机删除的概率，实数返回：A3  - 最后的激活值，维度为（1,1），正向传播的输出cache - 存储了一些用于计算反向传播的数值的元组"""np.random.seed(1)W1 = 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 = reg_utils.relu(Z1)D1 = np.random.rand(A1.shape[0], A1.shape[1])D1 = D1 < keep_prob  # 步骤1A1 = A1 * D1  # 步骤2A1 = A1 / keep_prob  # 步骤3Z2 = np.dot(W2, A1) + b2A2 = reg_utils.relu(Z2)D2 = np.random.rand(A2.shape[0], A2.shape[1])D2 = D2 < keep_prob  # 步骤1A2 = A2 * D2  # 步骤2A2 = A2 / keep_prob  # 步骤3Z3 = np.dot(W3, A2) + b3A3 = reg_utils.sigmoid(Z3)cache = (Z1, D1, A1, W1, b1, Z2, D2, A2, W2, b2, Z3, A3, W3, b3)return A3, cachedef backward_propagation_with_dropout(X, Y, cache, keep_prob):"""实现我们随机删除的模型的后向传播。参数：X  - 输入数据集，维度为（2，示例数）Y  - 标签，维度为（输出节点数量，示例数量）cache - 来自forward_propagation_with_dropout（）的cache输出keep_prob  - 随机删除的概率，实数返回：gradients - 一个关于每个参数、激活值和预激活变量的梯度值的字典"""m = X.shape[1](Z1, D1, A1, W1, b1, Z2, D2, A2, W2, b2, Z3, A3, W3, b3) = cachedZ3 = A3 - YdW3 = (1 / m) * np.dot(dZ3, A2.T)db3 = 1. / m * np.sum(dZ3, axis=1, keepdims=True)dA2 = np.dot(W3.T, dZ3)dA2 = dA2 * D2  # 步骤1dA2 = dA2 / keep_prob  # 步骤2dZ2 = np.multiply(dA2, np.int64(A2 > 0))dW2 = 1. / m * np.dot(dZ2, A1.T)db2 = 1. / m * np.sum(dZ2, axis=1, keepdims=True)dA1 = np.dot(W2.T, dZ2)dA1 = dA1 * D1  # 步骤1dA1 = dA1 / keep_prob  # 步骤2dZ1 = np.multiply(dA1, np.int64(A1 > 0))dW1 = 1. / m * np.dot(dZ1, X.T)db1 = 1. / m * 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 gradients

调用model函数时加入keep_prob参数，设为0.86，即在每次迭代中第1层和第2层的14%的节点将不参与计算：

parameters = model(train_X, train_Y, keep_prob=0.86, learning_rate=0.3,is_plot=True)

运行代码结果如下：

第10000次迭代，成本值为：0.061016986574905605
第20000次迭代，成本值为：0.060582435798513114
训练集:
Accuracy: 0.9289099526066351
测试集:
Accuracy: 0.95

这里的标题可以改一下：

plt.title("Model with dropout")

可以看到使用dropout让训练集的准确率稍微降低了些，但测试集上的准确率提升了，提高了泛化能力，还是很成功的。

dropout防止过拟合的原因：每个神经元都不依赖于任何特征，因为任意一个特征都有可能被清除。
注意，测试阶段不使用dropout，因为要保证测试结果的稳定。