对于二属性样本分类，L2和L1正则化基本相同，仅仅正则化项不同
LASSO回归为在损失函数加入\(||\omega||_1\) ,\(\omega\) 的1范数 而 岭回归为\(||\omega||_2^2\)，\(\omega\) 的2范数
*矩阵、向量范数
*L1正则化（岭回归）

LASSO Regression

Loss Function

\[J(\omega)= (X \omega - Y)^T(X \omega - Y) + \lambda ||\omega||_1 \]

\(||\omega||_1\)导数不连续，采用坐标下降法求\(\omega\)
坐标下降法推导过程

import numpy as np 
import matplotlib.pyplot as plt 
from mpl_toolkits.mplot3d import Axes3D
M = 3 #属性个数+1 属性加 偏移项b， 一个3个参数
N = 50 #样本个数
#随机生成两个属性的N个样本
feature1 = np.random.rand(N)*10
feature2 = np.random.rand(N)*10
splt = np.ones((1, N))
#
temp_X1 = np.row_stack((feature1, feature2))
temp_X = np.vstack((temp_X1, splt))
X_t = np.mat(temp_X)
X = X_t.T 
temp_Y = np.random.rand(N)*10
Y_t = np.mat(temp_Y)
Y = Y_t.T 
#画样本散点图
fig = plt.figure()
ax1 = Axes3D(fig)
ax1.scatter(feature1, feature2, temp_Y)
######
def errors(X, Y, Omega) :
    err = (X*Omega - Y).T*(X*Omega - Y)
    return err
#坐标下降算法
def lasso_regression(X, Y, lambd, threshold):
    #
    Omega = np.mat(np.zeros((M, 1)))
    err = errors(X, Y, Omega)
    counts = 0          #统计迭代次数
    # 使用坐标下降法优化回归系数Omega
    while err > threshold:
        counts += 1
        for k in range(M):
            # 计算常量值z_k和p_k
            z_k = (X[:, k].T*X[:, k])[0, 0]
            p_k = 0
            for i in range(N):
                p_k += X[i, k]*(Y[i, 0] - sum([X[i, j]*Omega[j, 0] for j in range(M) if j != k]))
            if p_k < -lambd/2:
                w_k = (p_k + lambd/2)/z_k
            elif p_k > lambd/2:
                w_k = (p_k - lambd/2)/z_k
            else:
                w_k = 0
            Omega[k, 0] = w_k
        err_prime = errors(X, Y, Omega)
        delta = abs(err_prime - err)[0, 0]
        err = err_prime
        print('Iteration: {}, delta = {}'.format(counts, delta))
        if delta < threshold:
            break
    return Omega
#求Omega
lambd = 10.0
threshold = 0.1
Omega = lasso_regression(X, Y, lambd, threshold)
#画分类面
xx = np.linspace(0,10, num=50)
yy = np.linspace(0,10, num=50)
xx_1, yy_1 = np.meshgrid(xx, yy)
Omega_h = np.array(Omega.T)
zz_1 = Omega_h[0, 0]*xx_1 + Omega_h[0, 1]*yy_1 + Omega_h[0, 2]
ax1.plot_surface(xx_1, yy_1, zz_1, alpha= 0.6, color= "r")
plt.show()

效果

参考资料

机器学习算法实践-岭回归和LASSO