人工神经网络——学习率与收敛速度

from matplotlib import pyplot as plt from mpl_toolkits.mplot3d import Axes3D import numpy as np from sklearn.neural_network import MLPClassifier from sklearn.datasets import load_iris

生成线性可分数据集

def creat_data(n): np.random.seed(1) x_11=np.random.randint(0,100,(n,1)) x_12=np.random.randint(0,100,(n,1,)) x_13=20+np.random.randint(0,10,(n,1,)) x_21=np.random.randint(0,100,(n,1)) x_22=np.random.randint(0,100,(n,1)) x_23=10-np.random.randint(0,10,(n,1,)) new_x_12=x_12*np.sqrt(2)/2-x_13*np.sqrt(2)/2 new_x_13=x_12*np.sqrt(2)/2+x_13*np.sqrt(2)/2 new_x_22=x_22*np.sqrt(2)/2-x_23*np.sqrt(2)/2 new_x_23=x_22*np.sqrt(2)/2+x_23*np.sqrt(2)/2 plus_samples=np.hstack([x_11,new_x_12,new_x_13,np.ones((n,1))]) minus_samples=np.hstack([x_21,new_x_22,new_x_23,-np.ones((n,1))]) samples=np.vstack([plus_samples,minus_samples]) np.random.shuffle(samples) return samples

感知机学习算法的原始形式算法

def perceptron(train_data,eta,w_0,b_0): x=train_data[:,:-1] y=train_data[:,-1] length=train_data.shape[0] w=w_0 b=b_0 step_num=0 while True: i=0 while(i<length): step_num+=1 x_i=x[i].reshape((x.shape[1],1)) y_i=y[i] if y_i*(np.dot(np.transpose(w),x_i)+b)<=0: w+=eta*y_i*x_i b+=eta*y_i break else: i+=1 if(i==length): break return(w,b,step_num)

根据训练数据集和 α⃗  α → 得到 w⃗  w →

def creat_w(train_data,alpha): x=train_data[:,:-1] y=train_data[:,-1] N=train_data.shape[0] w=np.zeros((x.shape[1],1)) for i in range(0,N): w+=alpha[i][0]*y[i]*(x[i].reshape(x[i].size,1)) return w

感知机学习算法的对偶形式

def perceptron_dual(train_data,eta,alpha_0,b_0): x=train_data[:,:-1] y=train_data[:,-1] length=train_data.shape[0] alpha=alpha_0 b=b_0 step_num=0 while_num=0 while True: i=0 while(i<length): step_num+=1 x_i=x[i].reshape((x.shape[1],1)) y_i=y[i] w=creat_w(train_data,alpha) z=y_i*(np.dot(np.transpose(w),x_i)+b) if z<=0: alpha[i][0]+=eta b+=eta*y_i break else: i+=1 if(i==length): break return (alpha,b,step_num)

η参数的影响

def test_eta(data,ax,etas,w_0,alpha_0,b_0): nums1=[] nums2=[] for eta in etas: _,_,num_1=perceptron(data,eta,w_0=w_0,b_0=b_0) _,_,num_2=perceptron_dual(data,eta=0.1,alpha_0=alpha_0,b_0=b_0) nums1.append(num_1) nums2.append(num_2) ax.plot(etas,np.array(nums1),label='orignal iteraton times') ax.plot(etas,np.array(nums2),label='dual iteraton times') fig=plt.figure() fig.suptitle('perceptron') ax=fig.add_subplot(1,1,1) ax.set_xlabel(r'$\eta$') data=creat_data(20) etas=np.linspace(0.01,1,num=25,endpoint=False) w_0,b_0,alpha_0=np.ones((3,1)),0,np.zeros((data.shape[0],1)) test_eta(data,ax,etas,w_0,alpha_0,b_0) ax.legend(loc='best',framealpha=0.5) plt.show()