作为自学练习，我尝试从头开始在线性回归问题上实现梯度下降算法，并在等高线图上绘制结果迭代。
我的梯度下降实现给出了正确的结果（使用Sklearn测试过），然而，梯度下降图似乎并不与等高线垂直。这是预期的，还是我的代码/理解有误？
算法
成本函数和梯度下降
import numpy as npimport pandas as pdfrom matplotlib import pyplot as pltfrom mpl_toolkits.mplot3d import Axes3Ddef costfunction(X,y,theta): m = np.size(y) #成本函数的向量化形式 h = X @ theta J = float((1./(2*m)) * (h - y).T @ (h - y)); return J;def gradient_descent(X,y,theta,alpha = 0.0005,num_iters=1000): #初始化有用的值 m = np.size(y) J_history = np.zeros(num_iters) theta_0_hist, theta_1_hist = [], [] #用于后续绘图 for i in range(num_iters): #梯度函数的向量化形式 h = X @ theta theta = theta - alpha * (1/m)* (X.T @ (h-y)) #每轮迭代的成本和中间值 J_history[i] = costfunction(X,y,theta) theta_0_hist.append(theta[0,0]) theta_1_hist.append(theta[1,0]) return theta,J_history, theta_0_hist, theta_1_hist绘图
#创建数据集（如前所述）x = np.linspace(0,1,40)noise = 1*np.random.uniform( size = 40)y = np.sin(x * 1.5 * np.pi ) y_noise = (y + noise).reshape(-1,1)X = np.vstack((np.ones(len(x)),x)).T#设置theta值的网格T0, T1 = np.meshgrid(np.linspace(-1,3,100),np.linspace(-6,2,100))#计算每个theta组合的成本函数zs = np.array( [costfunction(X, y_noise.reshape(-1,1),np.array([t0,t1]).reshape(-1,1)) for t0, t1 in zip(np.ravel(T0), np.ravel(T1)) ] )#重塑成本值 Z = zs.reshape(T0.shape)#计算梯度下降theta_result,J_history, theta_0, theta_1 = gradient_descent(X,y_noise,np.array([0,-6]).reshape(-1,1),alpha = 0.3,num_iters=1000)#用于箭头图的角度anglesx = np.array(theta_0)[1:] - np.array(theta_0)[:-1]anglesy = np.array(theta_1)[1:] - np.array(theta_1)[:-1]%matplotlib inlinefig = plt.figure(figsize = (16,8))#表面图ax = fig.add_subplot(1, 2, 1, projection='3d')ax.plot_surface(T0, T1, Z, rstride = 5, cstride = 5, cmap = 'jet', alpha=0.5)ax.plot(theta_0,theta_1,J_history, marker = '*', color = 'r', alpha = .4, label = 'Gradient descent')ax.set_xlabel('theta 0')ax.set_ylabel('theta 1')ax.set_zlabel('成本函数')ax.set_title('梯度下降: 根在{}'.format(theta_result.ravel()))ax.view_init(45, 45)#等高线图ax = fig.add_subplot(1, 2, 2)ax.contour(T0, T1, Z, 70, cmap = 'jet')ax.quiver(theta_0[:-1], theta_1[:-1], anglesx, anglesy, scale_units = 'xy', angles = 'xy', scale = 1, color = 'r', alpha = .9)plt.show()表面和等高线图
评论
我的理解是梯度下降应垂直于等高线进行。这不是这样的吗？谢谢
回答：
等高线图的问题在于theta0和theta1的尺度不同。只需在等高线图指令中添加”plt.axis(‘equal’)”，你就会发现梯度下降实际上是垂直于等高线的。
