Python实现机器学习算法——Ridge岭回归

今天我们来看基于L2正则化的线性回归模型。

L2正则化

相较于L0和L1，其实L2才是正则化中的天选之子。在各种防止过拟合和正则化处理过程中，L2正则化可谓第一候选。

L2范数是指矩阵中各元素的平方和后的求根结果。采用L2范数进行正则化的原理在于最小化参数矩阵的每个元素，使其无限接近于0但又不像L1那样等于0，那么为什么参数矩阵中每个元素变得很小就能防止过拟合？

这里我们就拿深度神经网络来举例说明。在L2正则化中，如何正则化系数变得比较大，参数矩阵W中的每个元素都在变小，线性计算的和Z也会变小，激活函数在此时相对呈线性状态，这样就大大简化了深度神经网络的复杂性，因而可以防止过拟合。

加入L2正则化的线性回归损失函数如下所示。其中第一项为MSE损失，第二项就是L2正则化项。

L2正则化相比于L1正则化在计算梯度时更加简单。直接对损失函数关于w求导即可。这种基于L2正则化的回归模型便是著名的岭回归(Ridge Regression)。

Ridge

有了上一讲的代码框架，我们直接在原基础上对损失函数和梯度计算公式进行修改即可。下面来看具体代码。

导入相关模块：

import numpy as npimport pandas as pdimport matplotlib.pyplot as pltfrom sklearn.model_selection import train_test_split

读入示例数据并划分：

data = pd.read_csv('./abalone.csv')data['Sex'] = data['Sex'].map({'M':0, 'F':1, 'I':2})X = data.drop(['Rings'], axis=1)y = data[['Rings']]X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.25)X_train, X_test, y_train, y_test = X_train.values, X_test.values, y_train.values, y_test.valuesprint(X_train.shape, y_train.shape, X_test.shape, y_test.shape)

模型参数初始化：

# 定义参数初始化函数def initialize(dims): w = np.zeros((dims, 1)) b = 0 return w, b

定义L2损失函数和梯度计算：

# 定义ridge损失函数def l2_loss(X, y, w, b, alpha):    num_train = X.shape[0]    num_feature = X.shape[1]    y_hat = np.dot(X, w) + b    loss = np.sum((y_hat-y)**2)/num_train + alpha*(np.sum(np.square(w)))    dw = np.dot(X.T, (y_hat-y)) /num_train + 2*alpha*w    db = np.sum((y_hat-y)) /num_train    return y_hat, loss, dw, db

定义Ridge训练过程：

# 定义训练过程def ridge_train(X, y, learning_rate=0.001, epochs=5000): loss_list = [] w, b = initialize(X.shape[1]) for i in range(1, epochs): y_hat, loss, dw, db = l2_loss(X, y, w, b, 0.1) w += -learning_rate * dw b += -learning_rate * db loss_list.append(loss) if i % 100 == 0: print('epoch %d loss %f' % (i, loss)) params = { 'w': w, 'b': b } grads = { 'dw': dw, 'db': db } return loss, loss_list, params, grads

执行示例训练：

# 执行训练示例loss, loss_list, params, grads = ridge_train(X_train, y_train, 0.01, 1000)

模型参数：

定义模型预测函数：

# 定义预测函数def predict(X, params): w = params['w'] b = params['b'] y_pred = np.dot(X, w) + b return y_predy_pred = predict(X_test, params)y_pred[:5]

测试集数据和模型预测数据的绘图展示：

# 简单绘图import matplotlib.pyplot as pltf = X_test.dot(params['w']) + params['b']plt.scatter(range(X_test.shape[0]), y_test)plt.plot(f, color = 'darkorange')plt.xlabel('X')plt.ylabel('y')plt.show();

可以看到模型预测对于高低值的拟合较差，但能拟合大多数值。这样的模型相对具备较强的泛化能力，不会产生严重的过拟合问题。

最后进行简单的封装：

import numpy as npimport pandas as pdfrom sklearn.model_selection import train_test_splitclass Ridge(): def __init__(self): pass def prepare_data(self): data = pd.read_csv('./abalone.csv') data['Sex'] = data['Sex'].map({'M': 0, 'F': 1, 'I': 2}) X = data.drop(['Rings'], axis=1) y = data[['Rings']] X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.25) X_train, X_test, y_train, y_test = X_train.values, X_test.values, y_train.values, y_test.values return X_train, y_train, X_test, y_test def initialize(self, dims): w = np.zeros((dims, 1)) b = 0 return w, b def l2_loss(self, X, y, w, b, alpha): num_train = X.shape[0] num_feature = X.shape[1] y_hat = np.dot(X, w) + b loss = np.sum((y_hat - y) ** 2) / num_train + alpha * (np.sum(np.square(w))) dw = np.dot(X.T, (y_hat - y)) / num_train + 2 * alpha * w db = np.sum((y_hat - y)) / num_train return y_hat, loss, dw, db def ridge_train(self, X, y, learning_rate=0.01, epochs=1000): loss_list = [] w, b = self.initialize(X.shape[1]) for i in range(1, epochs): y_hat, loss, dw, db = self.l2_loss(X, y, w, b, 0.1) w += -learning_rate * dw b += -learning_rate * db loss_list.append(loss) if i % 100 == 0: print('epoch %d loss %f' % (i, loss)) params = { 'w': w, 'b': b } grads = { 'dw': dw, 'db': db } return loss, loss_list, params, grads def predict(self, X, params): w = params['w'] b = params['b'] y_pred = np.dot(X, w) + b return y_pred if __name__ == '__main__': ridge = Ridge() X_train, y_train, X_test, y_test = ridge.prepare_data() loss, loss_list, params, grads = ridge.ridge_train(X_train, y_train, 0.01, 1000) print(params)

sklearn中也提供了Ridge的实现方式：

# 导入线性模型模块from sklearn.linear_model import Ridge# 创建Ridge模型实例clf = Ridge(alpha=1.0)# 对训练集进行拟合clf.fit(X_train, y_train)# 打印模型相关系数print('sklearn Ridge intercept :', clf.intercept_)print('\nsklearn Ridge coefficients :\n', clf.coef_)

参考资料：

https://mp.weixin.qq.com/s/nI-ps1vPCQe2AS6LGz2Csw