实际场景中训练数据往往不是线性可分的,当训练数据近似线性可分时,就需要使用线性支持向量机或软间隔支持向量机。
给定训练数据集 T = { ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x N , y N ) } T=\{(x_1,y_1),(x_2,y_2),...,(x_N,y_N)\} T={(x1,y1),(x2,y2),...,(xN,yN)},其中 X = R n , y i ∈ Y = { − 1 , + 1 } , i = 1 , 2 , . . . , N \mathcal{X}=\mathcal{R}^{n},y_i \in \mathcal{Y}=\{-1,+1\},i=1,2,...,N X=Rn,yi∈Y={−1,+1},i=1,2,...,N, x i x_i xi为第 i i i个特征向量, y i y_i yi为 x i x_i xi的类标记.
由于数据中常常有噪声或特异点等,导致数据线性不可分。这就意味着,部分样本点 ( x i , y i ) (x_i,y_i) (xi,yi)不能满足函数间隔大于等于1的约束条件。为此引入松弛变量 ξ i \xi_{i} ξi,使得约束条件变为:
y i ( w ⋅ x i + b ) ⩾ 1 − ξ i y_i(w \cdot x_i +b) \geqslant 1- \xi_{i} yi(w⋅xi+b)⩾1−ξi
同时,对每一个松弛变量 ξ i \xi_{i} ξi支付一个代价 ξ i \xi_{i} ξi.则目标函数变为:
1 2 ∥ w ∥ 2 + C ∑ i = 1 N ξ i {\frac{1}{2}\|w\|^{2}+C \sum_{i=1}^{N}\xi_{i}} 21∥w∥2+Ci=1∑Nξi
其中, C > 0 C>0 C>0为惩罚参数.
由此线性不可分的线性支持向量的学习问题转化为凸二次规划问题:
min w , b , ξ 1 2 ∥ w ∥ 2 + C ∑ i = 1 N ξ i \min_{w,b,\xi} \quad \frac{1}{2}\|w\|^{2}+C\sum_{i=1}^{N}\xi_{i} w,b,ξmin21∥w∥2+Ci=1∑Nξi
s . t y i ( w ⋅ x i + b ) ⩾ 1 − ξ i , i = 1 , 2 , . . . , N s.t \quad y_i(w \cdot x_i+b) \geqslant 1- \xi_{i}, \quad i=1,2,...,N s.tyi(w⋅xi+b)⩾1−ξi,i=1,2,...,N
ξ i ⩾ 0 , i = 1 , 2 , . . . , N \xi_{i} \geqslant 0,\quad i=1,2,...,N ξi⩾0,i=1,2,...,N
对于给定的线性不可分的训练数据集,通过求解凸二次规划问题,即软间隔最大化问题,可得到分离的超平面: w ∗ ⋅ x + b ∗ w^{*} \cdot x+b^{*} w∗⋅x+b∗
以及相应的分类决策函数: f ( x ) = sign ( w ∗ ⋅ x + b ∗ ) f(x)=\operatorname{sign}(w^{*} \cdot x+b^{*}) f(x)=sign(w∗⋅x+b∗),称为线性支持向量机.
输入:训练数据集 T = { ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x N , y N ) } T=\{(x_1,y_1),(x_2,y_2),...,(x_N,y_N)\} T={(x1,y1),(x2,y2),...,(xN,yN)},其中 X = R n , y i ∈ Y = { − 1 , + 1 } , i = 1 , 2 , . . . , N \mathcal{X}=\mathcal{R}^{n},y_i \in \mathcal{Y}=\{-1,+1\},i=1,2,...,N X=Rn,yi∈Y={−1,+1},i=1,2,...,N;
输出:分离超平面和分类决策函数。
第一步:选择惩罚参数 C > 0 C>0 C>0,构造并求解凸二次规划问题
min α 1 2 ∑ i = 1 N ∑ j = 1 N α i α j y i y j ( x i ⋅ x j ) − ∑ i = 1 N α i {\min_{\alpha} \quad \frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}\alpha_{i}\alpha_{j}y_{i}y_{j}(x_i \cdot x_j)-\sum_{i=1}^{N}\alpha_{i}} αmin21i=1∑Nj=1∑Nαiαjyiyj(xi⋅xj)−i=1∑Nαi
s . t . ∑ i = 1 N α i y i = 0 {s.t. \quad \sum_{i=1}^{N}\alpha_{i}y_i=0} s.t.i=1∑Nαiyi=0
0 ⩽ α i ⩽ C , i = 1 , 2 , ⋯   , N {0 \leqslant \alpha_{i} \leqslant C,\quad i=1,2, \cdots ,N} 0⩽αi⩽C,i=1,2,⋯,N
求得最优化问题的解 α ∗ = ( α 1 ∗ , α 2 ∗ , ⋯   , α N ∗ ) T \alpha^{*}=(\alpha_{1}^{*},\alpha_{2}^{*}, \cdots,\alpha_{N}^{*})^{T} α∗=(α1∗,α2∗,⋯,αN∗)T
第二步:计算 w ∗ = ∑ i = 1 N α i ∗ y i x i {w^{*}=\sum_{i=1}^{N}\alpha_{i}^{*} y_{i} x_{i}} w∗=∑i=1Nαi∗yixi,并选择 α ∗ \alpha^{*} α∗的一个分量 a j ∗ a_{j}^{*} aj∗适合条件 0 < α j ∗ < C 0< \alpha_{j}^{*}<C 0<αj∗<C,计算
b ∗ = y i − ∑ i = 1 N y i α i ∗ ( x i ⋅ x j ) b^{*}=y_i-\sum_{i=1}^{N}y_{i}\alpha_{i}^{*}(x_i \cdot x_j) b∗=yi−i=1∑Nyiαi∗(xi⋅xj)
第三步:求得分离超平面 w ∗ ⋅ x + b ∗ w^{*} \cdot x+b^{*} w∗⋅x+b∗,分类决策函数为: f ( x ) = sign ( w ∗ ⋅ x + b ∗ ) f(x)=\operatorname{sign}(w^{*} \cdot x+b^{*}) f(x)=sign(w∗⋅x+b∗)
案例代码已上传:Github地址
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()
# 创建计算图
sess = tf.Session()
第一步:加载数据并划分数据,80%数据用来训练,20%的数据用来做测试
#加载数据
# iris数据 [(Sepal Length, Sepal Width, Petal Length, Petal Width)]
iris = datasets.load_iris()
x_vals = np.array([[x[0], x[3]] for x in iris.data])
y_vals = np.array([1 if y==0 else -1 for y in iris.target])
#加载数据
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]
# 声明参数
batch_size = 100 #批量大小
max_iter=500 #最大迭代次数
# 初始化占位符
x_data = tf.placeholder(shape=[None, 2], dtype=tf.float32)
y_target = tf.placeholder(shape=[None, 1], dtype=tf.float32)
#创建变量
A = tf.Variable(tf.random_normal(shape=[2,1]))
b = tf.Variable(tf.random_normal(shape=[1,1]))
#声明操作 模型
model_output = tf.subtract(tf.matmul(x_data, A), b)
# 声明L2正则化
l2_norm = tf.reduce_sum(tf.square(A))
声明损失函数
l o s s = max ( 0 , 1 − p r e d ∗ a c t u a l ) + a l p h a ∗ L 2 n o r m ( A ) 2 loss=\max(0, 1-pred*actual) + alpha * L2_{norm}(A)^2 loss=max(0,1−pred∗actual)+alpha∗L2norm(A)2
第二步:声明损失函数并设置优化器、初始化变量
# 声明损失函数
# = max(0, 1-pred*actual) + alpha * L2_norm(A)^2
# L2 正则化参数 alpha
alpha = tf.constant([0.01])
# Margin term in loss
classification_term = tf.reduce_mean(tf.maximum(0., tf.subtract(1., tf.multiply(model_output, y_target))))
# 损失函数
loss = tf.add(classification_term, tf.multiply(alpha, l2_norm))
# 声明预测函数
prediction = tf.sign(model_output)
accuracy = tf.reduce_mean(tf.cast(tf.equal(prediction, y_target), tf.float32))
# 声明优化器
my_opt = tf.train.GradientDescentOptimizer(0.01)
train_step = my_opt.minimize(loss)
# 初始化变量
init = tf.global_variables_initializer()
sess.run(init)
第三步:开始训练
# 开始训练
loss_vec = []
train_accuracy = []
test_accuracy = []
for i in range(max_iter):
rand_index = np.random.choice(len(x_vals_train), size=batch_size)
rand_x = 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_loss = sess.run(loss, feed_dict={x_data: rand_x, y_target: rand_y})
loss_vec.append(temp_loss)
train_acc_temp = sess.run(accuracy, feed_dict={x_data: x_vals_train, y_target: np.transpose([y_vals_train])})
train_accuracy.append(train_acc_temp)
test_acc_temp = sess.run(accuracy, feed_dict={x_data: x_vals_test, y_target: np.transpose([y_vals_test])})
test_accuracy.append(test_acc_temp)
if (i+1)%100==0:
print('Step #' + str(i+1) + ' A = ' + str(sess.run(A)) + ' b = ' + str(sess.run(b)))
print('损失为 ' + str(temp_loss))
# 提取系数
[[a1], [a2]] = sess.run(A)
[[b]] = sess.run(b)
slope = -a2/a1
y_intercept = b/a1
# 提取 x1 and x2 vals
x1_vals = [d[1] for d in x_vals]
# Get best fit line
best_fit = []
for i in x1_vals:
best_fit.append(slope*i+y_intercept)
# 分离数据
setosa_x = [d[1] for i,d in enumerate(x_vals) if y_vals[i]==1]
setosa_y = [d[0] for i,d in enumerate(x_vals) if y_vals[i]==1]
not_setosa_x = [d[1] for i,d in enumerate(x_vals) if y_vals[i]==-1]
not_setosa_y = [d[0] for i,d in enumerate(x_vals) if y_vals[i]==-1]
# 分类效果展示
plt.plot(setosa_x, setosa_y, 'o', label='I. setosa')
plt.plot(not_setosa_x, not_setosa_y, 'x', label='Non-setosa')
plt.plot(x1_vals, best_fit, 'r-', label='Linear Separator', linewidth=3)
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()
# 训练和测试精度展示
plt.plot(train_accuracy, 'k-', label='Training Accuracy')
plt.plot(test_accuracy, 'r--', label='Test Accuracy')
plt.title('Train and Test Set Accuracies')
plt.xlabel('Generation')
plt.ylabel('Accuracy')
plt.legend(loc='lower right')
plt.show()
案例代码已上传:Github地址
参考资料:
[1] 《统计学习方法》
[2]: 《TensorFlow machine learning cookbook》
Github地址https://github.com/Vambooo/lihang-dl
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