最优化理论实践——支持向量机

Doris ·

更新时间:2024-09-21

· 611 次阅读

约束优化算法实现SVM 约束优化算法概述

阅读文章前，我希望你应该知道约束优化问题的KKT条件，KKT条件能够给出一组方程，并且是最优解的必要条件，在这些解里做遍历并用二阶条件判断是一种解决问题的方式，但对一些并不容易计算的非线性方程组和hessian矩阵，这种方法并不高效。因此我们来介绍一些简单的解决一般约束优化问题的算法。

支持向量机算法推导

Python实现

import numpy as np
import random
from copy import deepcopy
from matplotlib import pyplot as plt

这里使用0.1的起始γ，每次迭代倍增，最后将有3000以上的惩罚因子大小。学习率随γ的变化自动调整。


def Dot(x,y):
    x = x.reshape(1,-1)
    y = y.reshape(-1,1)
    return x.dot(y)[0]
class SVM:
    def __init__(self, dim):
        '''
        给定数据维度dim进行初始化
        参数的初始化一般随机
        '''
        self.w = np.zeros(dim)
        self.b = 0
        self.support = []
    def partial(self, X, Y, gamma, printing=False):
        N,dim = X.shape;
        dw = deepcopy(self.w)
        db = 0;
        for i in range(N):
            if (printing):
                print("x:",X[i])
                print("y:",Y[i])
                print(1-Y[i]*(Dot(self.w,X[i])+self.b))
            if 1-Y[i]*(Dot(self.w,X[i])+self.b)=0)
        return y*2-1
    def show(self,a,b,X,Y):
        #以现有的w和b打印划分平面，要求dim=2
        if self.w.size!=2:
            print("Dimension is not 2!")
            return
        x = np.linspace(a,b,num=100)
        y = -self.w[0]/self.w[1]*x-self.b/self.w[1]
        plt.plot(x,y,c='black')
        plt.xlabel("x1")
        plt.xlabel("x2")
        plt.title("Support vector mechine by penalty method")
        #同时，展示支撑向量，用+表示
        sv = [None,None]
        svdis = [0,0]
        for i in range(len(X)):
            if Y[i]==1:
                cls = 0
            else:
                cls = 1
            f = abs(Dot(self.w,X[i])+self.b)
            if type(sv[cls])==type(None) or f<svdis[cls]:
                sv[cls] = X[i]
                svdis[cls] = f
        plt.scatter(sv[0][0],sv[0][1],edgecolors='black',c='r',marker='+',linewidths=15)
        plt.scatter(sv[1][0],sv[1][1],edgecolors='black',c='r',marker='+',linewidths=15)

在均匀分布上使用高斯分布制造样本点，训练后的分类平面如下。

X11 = np.random.rand(20,1)
X12 = np.random.rand(20,1)
X11+=2;X12+=2
X11+=np.random.randn(20,1)
X12+=np.random.randn(20,1)
X1 = np.concatenate([X11,X12],axis =1)
X21 = np.random.rand(20,1)
X22 = np.random.rand(20,1)
X21-=1.5;X22-=1
X21+=np.random.randn(20,1)
X22+=np.random.randn(20,1)
X2 = np.concatenate([X21,X22],axis =1)

plt.scatter(X1[:,0].reshape(-1),X1[:,1].reshape(-1),c='b')
plt.scatter(X2[:,0].reshape(-1),X2[:,1].reshape(-1),c='r')
plt.show()

X = np.concatenate((X1,X2))
Y = np.array([1 for i in range(20)]+[-1 for i in range(20)])

svm = SVM(2)
svm.fit(X,Y)

Iteration: 0
W: [0.55302791 0.44010954]
b: -0.9099999999999999
Iteration: 1
W: [0.63101341 0.45857024]
b: -1.09
Iteration: 2
W: [0.72082093 0.57861661]
b: -1.3000000000000003
Iteration: 3
W: [0.8583813  0.65772783]
b: -1.7600000000000007
Iteration: 4
W: [0.94732829 0.7313659 ]
b: -1.8400000000000007
Iteration: 5
W: [1.18942914 0.91113098]
b: -2.05
Iteration: 6
W: [1.50525949 1.17111112]
b: -2.329999999999994
Iteration: 7
W: [1.82098084 1.41716943]
b: -2.639999999999989
Iteration: 8
W: [1.83692313 1.43355505]
b: -2.63999999999999
Iteration: 9
W: [1.83612493 1.43161008]
b: -2.6399999999999904
Iteration: 10
W: [1.82989512 1.43103962]
b: -2.6399999999999904
Iteration: 11
W: [1.82679169 1.43074295]
b: -2.6399999999999904
Iteration: 12
W: [1.83980108 1.43317587]
b: -2.639999999999991
Iteration: 13
W: [1.83755659 1.43142744]
b: -2.639999999999991
Iteration: 14
W: [1.83643537 1.43055403]
b: -2.639999999999991

plt.figure(figsize=(10, 6.5))
plt.scatter(X1[:,0].reshape(-1),X1[:,1].reshape(-1),c='w',edgecolors='g')
plt.scatter(X2[:,0].reshape(-1),X2[:,1].reshape(-1),c='w',edgecolors='b')
svm.show(-5,5,X,Y)