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Question1-14

对于有明确公式和定义的不需要使用到ml

智能系统在与环境的连续互动中学习最优行为策略的机器学习问题，学习最优的序贯决策

无标签分类

从标注数据 学习预测模型

主动地提出一些标注请求，将一些经过筛选的数据提交给专家进行标注

• 解题关键是计算N+1到N+L上的偶数个数
• 0到N的偶数个数是$\frac{\lfloor N\rfloor }{2}$
• 问题转化成（0到N+L的偶数个数-0到N的偶数个数）
  
  generate了D，但是N+1到N+L上L个点没有generate。每个点都有{被generate，没被generate}两种可能，所以是$2^L$
  
  由“无免费午餐定理”可知，任何算法在没有噪声时对于未知样本期望相等
  
  $$P(5orange\&5else)=\frac{C_{10}^5}{2^{10}}$$

from scipy.special import comb
print(comb(10,5)/2**10)

$$P(9orange\&1else)=\frac{C_{10}^9}{0.9^9\times 0.1}$$

print(comb(10,9)*((0.9)**9)*0.1)

• 分v=0.1和0时讨论
  $$P=C_{10}^1(\frac{9}{10}{)}^1{(\frac{1}{10})}^9+C_{10}^0{(\frac{1}{10})}^{10}$$
  
  
  $$\begin{gathered} Hoeffding:\mathbb{P}[|\mu -v|>ϵ]\le 2e^{-2{ϵ}^{2}N} \\ \begin{array}{rl}\mathbb{P}[v\le 0.1]& =P[0.9-v\ge 0.8]\\ & =P[\mu -v\ge 0.8]\\ & \le P[|\mu -v|\ge 0.8]\\ & \le 2e^{-2\times 0.8^2\times 10}\\ & \approx 5.5215451440744015\times 10^{-6}\end{array} \end{gathered}$$
  
• A：奇数绿，偶数橙
• B：奇数橙，偶数绿
• C:1-3橙，4-6绿
• D:1-3绿，4-6橙

5个橙1，只可能是BC中，所以$\frac{1}{32}=\frac{8}{256}$

• 1全橙：BC
• 2全橙：AC
• 3全橙：BC
• 4全橙：AD
• 5全橙：BD
• 6全橙：AD
• 全A，B，C，D被重复算了一遍，要减去4
  $$P=\frac{4\times 2^5-4}{4^5}=\frac{31}{256}$$
  

Question15-PLA

data链接

代码部分：
utils函数：

import numpy as np
#判别函数，判断所有数据是否分类完成
def Judge(X, y, w):n = X.shape[0]num = np.sum(X.dot(w) * y > 0)return num == ndef PLA(X, y, eta=1, max_step=np.inf):# 获取维度n, d = X.shape# 初始化w = np.zeros(d)# 迭代次数t = 0# 元素的下标i = 0# 错误的下标last = 0while not (Judge(X, y, w)) :if np.sign(X[i, :].dot(w) * y[i]) <= 0:t += 1w += eta * y[i] * X[i, :]# 更新错误last = i# 移动到下一个元素，如果达到n，则重置为0i += 1if i == n:i = 0return t, last, w

主函数：

import numpy as np
import utils as util#读取数据
data = np.genfromtxt("hw1_15_train.dat")
#获取维度
n, d = data.shape
#分离X
X = data[:, :-1]
#添加偏置项1
X = np.c_[np.ones(n), X]
#分离y
y = data[:, -1]
print(util.PLA(X, y))

运行结果：

Question16-PLA平均迭代次数

代码部分：
utils函数：

import numpy as np
import matplotlib.pyplot as pltdef Judge(X, y, w):n = X.shape[0]num = np.sum(X.dot(w) * y > 0)return num == ndef PLA(X, y, eta=1):n, d = X.shapew = np.zeros(d)t = 0i = 0last = 0while not (Judge(X, y, w)):if np.sign(X[i, :].dot(w) * y[i]) <= 0:t += 1w += eta * y[i] * X[i, :]last = ii += 1if i == n:i = 0return t, last, w#运行g算法n次并返回平均的迭代次数
def average_of_n(g, X, y, n, eta=1):result = []data = np.c_[X, y]for i in range(n):np.random.shuffle(data)X = data[:, :-1]y = data[:, -1]result.append(g(X, y, eta=eta)[0])plt.hist(result)plt.xlabel("迭代次数")plt.title("平均运行次数为" + str(np.mean(result)))plt.show()

主函数：

import numpy as np
import utils as util
import matplotlib.pyplot as plt
plt.rcParams['font.sans-serif']=['SimHei'] #显示中文标签
plt.rcParams['axes.unicode_minus']=False #显示负号data = np.genfromtxt("hw1_15_train.dat")
#获取维度
n, d = data.shape
#分离X
X = data[:, :-1]
#添加偏置项1
X = np.c_[np.ones(n), X]
#分离y
y = data[:, -1]
util.average_of_n(util.PLA, X, y, 2000, 1)

Question17-不同迭代系数的PLA

修改迭代系数即可：

util.average_of_n(util.PLA, X, y, 2000, 0.5)

Question18-Pocket_PLA

utils函数：

import matplotlib.pyplot as plt
import numpy as np
#统计错误数量
def count(X, y, w):num = np.sum(X.dot(w) * y <= 0)return np.sum(num)#预处理
def preprocess(data):# 获取维度n, d = data.shape# 分离XX = data[:, :-1]# 添加偏置项1X = np.c_[np.ones(n), X]# 分离yy = data[:, -1]return X, ydef Pocket_PLA(X, y, eta=1, max_step=np.inf):#max_step 限制迭代次数#获得数据维度n, d = X.shape#初始化w = np.zeros(d)#记录最优向量w0 = np.zeros(d)#记录次数t = 0#记录最少错误数量error = count(X, y, w0)#记录元素的下标i = 0while (error != 0 and t < max_step):if np.sign(X[i, :].dot(w) * y[i]) <= 0:w += eta * y[i] * X[i, :]#迭代次数增加t += 1#记录当前错误error_now = count(X, y, w)if error_now < error:error = error_noww0 = np.copy(w)#移动到下一个元素i += 1#如果达到n，则重置为0if i == n:i = 0return error, w0#运行g算法n次，1代表训练集，2代表测试集
def average_of_n(g, X1, y1, X2, y2, n, eta=1, max_step=np.inf):result = []data = np.c_[X1, y1]m = X2.shape[0]for i in range(n):np.random.shuffle(data)X = data[:, :-1]y = data[:, -1]w = g(X, y, eta=eta, max_step=max_step)[-1]result.append(count(X2, y2, w) / m)plt.hist(result)plt.xlabel("错误率")plt.title("平均错误率为"+str(np.mean(result)))plt.show()

主函数：

import matplotlib.pyplot as plt
import numpy as np
import utils as util
plt.rcParams['font.sans-serif']=['SimHei'] #用来正常显示中文标签
plt.rcParams['axes.unicode_minus']=False #用来正常显示负号data_train = np.genfromtxt("hw1_18_train.dat")
data_test = np.genfromtxt("hw1_18_test.dat")X_train, y_train = util.preprocess(data_train)
X_test, y_test = util.preprocess(data_test)util.average_of_n(util.Pocket_PLA, X_train, y_train, X_test, y_test, 2000, max_step=50)

Question19-PLA的错误率

utils函数：

import matplotlib.pyplot as plt
import numpy as npdef count(X, y, w):#判断是否同号num = np.sum(X.dot(w) * y <= 0)return np.sum(num)def Judge(X, y, w):n = X.shape[0]#判断是否同号num = np.sum(X.dot(w) * y > 0)return num == ndef preprocess(data):"""数据预处理"""# 获取维度n, d = data.shape# 分离XX = data[:, :-1]# 添加偏置项1X = np.c_[np.ones(n), X]# 分离yy = data[:, -1]return X, ydef PLA(X, y, eta=1,max_step=np.inf):n, d = X.shapew = np.zeros(d)t = 0i = 0last = 0while not (Judge(X, y, w)) and t<max_step:if np.sign(X[i, :].dot(w) * y[i]) <= 0:t += 1w += eta * y[i] * X[i, :]last = ii += 1if i == n:i = 0return t, last, w#运行g算法n次，1代表训练集，2代表测试集
def average_of_n(g, X1, y1, X2, y2, n, eta=1, max_step=np.inf):result = []data = np.c_[X1, y1]m = X2.shape[0]for i in range(n):np.random.shuffle(data)X = data[:, :-1]y = data[:, -1]w = g(X, y, eta=eta, max_step=max_step)[-1]result.append(count(X2, y2, w) / m)plt.hist(result)plt.xlabel("错误率")plt.title("平均错误率为"+str(np.mean(result)))plt.show()

主函数：

import matplotlib.pyplot as plt
import numpy as np
import utils as util
plt.rcParams['font.sans-serif']=['SimHei'] #用来正常显示中文标签
plt.rcParams['axes.unicode_minus']=False #用来正常显示负号data_train = np.genfromtxt("hw1_18_train.dat")
data_test = np.genfromtxt("hw1_18_test.dat")X_train, y_train = util.preprocess(data_train)
X_test, y_test = util.preprocess(data_test)util.average_of_n(util.PLA, X_train, y_train, X_test, y_test, 2000, max_step=50)

Question20-修改Pocket_PLA迭代次数

utils函数：

import matplotlib.pyplot as plt
import numpy as npdef count(X, y, w):#判断是否同号num = np.sum(X.dot(w) * y <= 0)return np.sum(num)def Judge(X, y, w):n = X.shape[0]#判断是否同号num = np.sum(X.dot(w) * y > 0)return num == ndef preprocess(data):"""数据预处理"""# 获取维度n, d = data.shape# 分离XX = data[:, :-1]# 添加偏置项1X = np.c_[np.ones(n), X]# 分离yy = data[:, -1]return X, ydef Pocket_PLA(X, y, eta=1, max_step=np.inf):#max_step 限制迭代次数#获得数据维度n, d = X.shape#初始化w = np.zeros(d)#记录最优向量w0 = np.zeros(d)#记录次数t = 0#记录最少错误数量error = count(X, y, w0)#记录元素的下标i = 0while (error != 0 and t < max_step):if np.sign(X[i, :].dot(w) * y[i]) <= 0:w += eta * y[i] * X[i, :]#迭代次数增加t += 1#记录当前错误error_now = count(X, y, w)if error_now < error:error = error_noww0 = np.copy(w)#移动到下一个元素i += 1#如果达到n，则重置为0if i == n:i = 0return error, w0#运行g算法n次，1代表训练集，2代表测试集
def average_of_n(g, X1, y1, X2, y2, n, eta=1, max_step=np.inf):result = []data = np.c_[X1, y1]m = X2.shape[0]for i in range(n):np.random.shuffle(data)X = data[:, :-1]y = data[:, -1]w = g(X, y, eta=eta, max_step=max_step)[-1]result.append(count(X2, y2, w) / m)plt.hist(result)plt.xlabel("错误率")plt.title("平均错误率为"+str(np.mean(result)))plt.show()

主函数：

import matplotlib.pyplot as plt
import numpy as np
import utils as util
plt.rcParams['font.sans-serif']=['SimHei'] #用来正常显示中文标签
plt.rcParams['axes.unicode_minus']=False #用来正常显示负号data_train = np.genfromtxt("hw1_18_train.dat")
data_test = np.genfromtxt("hw1_18_test.dat")X_train, y_train = util.preprocess(data_train)
X_test, y_test = util.preprocess(data_test)util.average_of_n(util.Pocket_PLA, X_train, y_train, X_test, y_test, 2000, max_step=100)