Tensorflow使用支持向量机拟合线性回归

支持向量机可以用来拟合线性回归。

相同的最大间隔（maximum margin）的概念应用到线性回归拟合。代替最大化分割两类目标是，最大化分割包含大部分的数据点（x，y）。我们将用相同的iris数据集，展示用刚才的概念来进行花萼长度与花瓣宽度之间的线性拟合。

相关的损失函数类似于max（0，|yi-（Axi+b）|-ε）。ε这里，是间隔宽度的一半，这意味着如果一个数据点在该区域，则损失等于0。

# SVM Regression #---------------------------------- # # This function shows how to use TensorFlow to # solve support vector regression. We are going # to find the line that has the maximum margin # which INCLUDES as many points as possible # # We will use the iris data, specifically: # y = Sepal Length # x = Pedal Width import matplotlib.pyplot as plt import numpy as np import tensorflow as tf from sklearn import datasets from tensorflow.python.framework import ops ops.reset_default_graph() # Create graph sess = tf.Session() # Load the data # iris.data = [(Sepal Length, Sepal Width, Petal Length, Petal Width)] iris = datasets.load_iris() x_vals = np.array([x[3] for x in iris.data]) y_vals = np.array([y[0] for y in iris.data]) # Split data into train/test sets train_indices = np.random.choice(len(x_vals), round(len(x_vals)*0.8), replace=False) test_indices = np.array(list(set(range(len(x_vals))) - set(train_indices))) x_vals_train = x_vals[train_indices] x_vals_test = x_vals[test_indices] y_vals_train = y_vals[train_indices] y_vals_test = y_vals[test_indices] # Declare batch size batch_size = 50 # Initialize placeholders x_data = tf.placeholder(shape=[None, 1], dtype=tf.float32) y_target = tf.placeholder(shape=[None, 1], dtype=tf.float32) # Create variables for linear regression A = tf.Variable(tf.random_normal(shape=[1,1])) b = tf.Variable(tf.random_normal(shape=[1,1])) # Declare model operations model_output = tf.add(tf.matmul(x_data, A), b) # Declare loss function # = max(0, abs(target - predicted) + epsilon) # 1/2 margin width parameter = epsilon epsilon = tf.constant([0.5]) # Margin term in loss loss = tf.reduce_mean(tf.maximum(0., tf.subtract(tf.abs(tf.subtract(model_output, y_target)), epsilon))) # Declare optimizer my_opt = tf.train.GradientDescentOptimizer(0.075) train_step = my_opt.minimize(loss) # Initialize variables init = tf.global_variables_initializer() sess.run(init) # Training loop train_loss = [] test_loss = [] for i in range(200): rand_index = np.random.choice(len(x_vals_train), size=batch_size) rand_x = np.transpose([x_vals_train[rand_index]]) rand_y = np.transpose([y_vals_train[rand_index]]) sess.run(train_step, feed_dict={x_data: rand_x, y_target: rand_y}) temp_train_loss = sess.run(loss, feed_dict={x_data: np.transpose([x_vals_train]), y_target: np.transpose([y_vals_train])}) train_loss.append(temp_train_loss) temp_test_loss = sess.run(loss, feed_dict={x_data: np.transpose([x_vals_test]), y_target: np.transpose([y_vals_test])}) test_loss.append(temp_test_loss) if (i+1)%50==0: print('-----------') print('Generation: ' + str(i+1)) print('A = ' + str(sess.run(A)) + ' b = ' + str(sess.run(b))) print('Train Loss = ' + str(temp_train_loss)) print('Test Loss = ' + str(temp_test_loss)) # Extract Coefficients [[slope]] = sess.run(A) [[y_intercept]] = sess.run(b) [width] = sess.run(epsilon) # Get best fit line best_fit = [] best_fit_upper = [] best_fit_lower = [] for i in x_vals: best_fit.append(slope*i+y_intercept) best_fit_upper.append(slope*i+y_intercept+width) best_fit_lower.append(slope*i+y_intercept-width) # Plot fit with data plt.plot(x_vals, y_vals, 'o', label='Data Points') plt.plot(x_vals, best_fit, 'r-', label='SVM Regression Line', linewidth=3) plt.plot(x_vals, best_fit_upper, 'r--', linewidth=2) plt.plot(x_vals, best_fit_lower, 'r--', linewidth=2) plt.ylim([0, 10]) plt.legend(loc='lower right') plt.title('Sepal Length vs Pedal Width') plt.xlabel('Pedal Width') plt.ylabel('Sepal Length') plt.show() # Plot loss over time plt.plot(train_loss, 'k-', label='Train Set Loss') plt.plot(test_loss, 'r--', label='Test Set Loss') plt.title('L2 Loss per Generation') plt.xlabel('Generation') plt.ylabel('L2 Loss') plt.legend(loc='upper right') plt.show()

输出结果：

----------- Generation: 50 A = [[ 2.91328382]] b = [[ 1.18453276]] Train Loss = 1.17104 Test Loss = 1.1143 ----------- Generation: 100 A = [[ 2.42788291]] b = [[ 2.3755331]] Train Loss = 0.703519 Test Loss = 0.715295 ----------- Generation: 150 A = [[ 1.84078252]] b = [[ 3.40453291]] Train Loss = 0.338596 Test Loss = 0.365562 ----------- Generation: 200 A = [[ 1.35343242]] b = [[ 4.14853334]] Train Loss = 0.125198 Test Loss = 0.16121

基于iris数据集（花萼长度和花瓣宽度）的支持向量机回归，间隔宽度为0.5

每次迭代的支持向量机回归的损失值（训练集和测试集）

直观地讲，我们认为SVM回归算法试图把更多的数据点拟合到直线两边2ε宽度的间隔内。这时拟合的直线对于ε参数更有意义。如果选择太小的ε值，SVM回归算法在间隔宽度内不能拟合更多的数据点；如果选择太大的ε值，将有许多条直线能够在间隔宽度内拟合所有的数据点。作者更倾向于选取更小的ε值，因为在间隔宽度附近的数据点比远处的数据点贡献更少的损失。

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