机器学习笔记

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机器学习 Machine Learning简介

属人工智能(AI). 赋予电脑新能力:

反垃圾邮件数据挖掘: 网络, 医疗, 生物无法手工编程:自动驾驶 (Autonomous)手写识别自然语言处理 (NLP)机器视觉 (Computer Vision)自助客服: 各种网络服务的推荐系统帮助理解人类学习和真正智能定义无需明确编程, 使电脑能自己学习

随着经验的积累,完成任务的成绩越来越好

任务: 计算结果 Hypothesis function 成绩: 误差反馈 Cost function 经验: 训练样本 Training set, $X \to y$

通过反馈误差, 不断拟合训练样本, 改进内部模型, 提高任务成绩. 学习已知, 预测未知. 博古而知今

分类

监督学习 (Supervised learning): 读书考试, 有正确答案来纠正训练

回归 (Regression): 连续值, 房价预估分类 (Classification): 离散值, 肿瘤疹断

非监督学习 (Unsupervised learning): 探索发现, 只有实事没有对错

聚类分析 (Clustering)

新闻分组基因测序机器集群部署社交网络分析市场分段天文分析

鸡尾酒会问题: 杂音消除, 声音分离 (如何做?)

步骤

关键是如何得到好模型 (类比人类心智模型)

计算主义: 宇宙就是一场计算, 它的模型就是自身. 所有理论只是宇宙原模型某角度某层面的简化.

所谓神者, 就是某种拥有自指结构的模型. 通神, 就是通过自指达到内部的无限, 以贴近外部宇宙的无限. 自指可再分等级和结构.

先选模型类别 骨架

线性回归 (Linear regression): 用于简单模型, 预测连续值逻辑回归 (Logistic Regression): 用于分类, 得到离散值多项式回归 (Polynomial regression): 用于复杂模型, 可归约到线性回归非线性神经网络 (Neural network)

再定模型架构 肌肉

回归与分类

特性数量多项式特性结果通过 Sigmoid 函数逻辑化非线性特性?

非线形神经网络

网络层数每层单元数

用训练数据确定模型权重 纹理

定义假设函数 (Hypothesis): $h_\theta(X)$找出代价函数 (Cost) 及其偏导: $J(\theta)$, $\frac{\partial }{\partial \theta} J(\theta)$ 计算模型参数 $\theta$ 值, 使得训练结果总代价最小叠代逼近最值:梯度下降 (Gradient descent) 等算法Conjugate gradientBFGSL-BFGS反向传播 (Backpropagation)分析计算最值正规方程 (Normal equation): 利用偏导函数求极值只适用小数据量简单回归模型X 不可逆: 特性太多, 大于样本数 特性不正交独立, 互有影响从天文观测资料中机器学习得到牛顿定律

骨架是基础, 纹理是皮毛. 骨架错则余皆不问.非线性网络通用但复杂

过拟合

模型与学习样本高度拟合, 但面对新样本性能明显下降. 高分低能, 知识不能指导实践就不是真知.

原因: 样本有错误, 误差, 特例和噪音. 过度贴近样本会使模型变形, 连错误和噪音也一并学习.事物发展有偶然性与必然性, 模型要体现决定事物发展的必然性, 排除偶然性.

如何减少:

减少特性人工筛选特性模型选择算法正规化 (Regularization)保留所有特性, 但通过对模型参数过大进行惩罚来减少参数过大, 避免过度匹配如果众多特性平均影响结果, 则不能减少特性. 这种情况正规化效果很好

过拟合时图形曲折, 通常是 $|\theta|$ 过大的结果. 通过把 $\theta^2$ 加入代价函数,惩罚过大的 $\theta$ 来避免.

$$ J_{R}(\theta) = J(\theta) + \lambda \sum \theta^2$$

凡事过犹不及, 如果惩罚过重, 则会产生欠拟合, 低分低能. 这里的火候全在 $\lambda$

神经网络

脑神经回路特性:

通用算法: 听说读写… 轻松学习新技能大量神经元互联而成: 容错性, 冗余性多个输入, 多个输出阈值限制 激活态/非激活态: Sigmoid 函数

人工神经网络算法:

通过矩阵进行输入输出关联中间多个隐藏层是特性层, 从原始数据中抽取抽象特征正向传播 (Forward propagation): $h(\theta)$反向传播 (Backpropagation): $\frac{\partial }{\partial \theta} J(\theta)$参数展开: 将二维矩阵参数展开为一维向量参数梯度验证: 数值近似计算偏分值, 验证反向传播随机初始化 $\theta$:局域最值反对称技巧

过犹不及, 不偏不倚, 术也.

学习速率 $\alpha$ 太大易发散太小难收敛画图直观确认, 从小到大试验 $\alpha$ 的值: …, 0.01, 0.1, 1, …拟合程度过拟合 (Overfitting) 不好, 学习样本有噪音和错误, 好的坏的一起学欠拟合 (Underfitting) 不好, 连已知都解释不好, 怎么拿来预测特性归一化处理:Feature scaling: $ -1 \leq X \leq 1$Mean normalization: $ -0.5 \leq X \leq 0.5$, $\bar{X} = 0$少用循环. 批量化, 向量化 (Vectorization) 提高性能. 使用 NumPy 等向量矩阵库只有凸函数 (Convex function) 才能确保找到全局最优. 用 $\log$ 函数改造非凸函数非线性模型一般只能找到局部最优用 Sigmoid 函数将线性回归扩展到逻辑回归, 用于分类将 $X^n$ 加入 X 中, 使线性回归扩展为多项式回归常数项要恰好, 以减少复杂度. 例如 $\frac{1}{2}m\sum{(h(X)-y)^2}$ 这里 $\frac{1}{2}$ 求导后恰好消除为何要加入 $X_0 = 1$ 偏见?通过一对多 (One-vs-all) 策略将二分法扩展到处理多分类: $\underset{i}{max}\ h_\theta^{(i)}(x)$数值验证

练习

使用 NumPy, matplotlib, SymPy 等 Python 科学库实现

In[1]:

from __future__ import unicode_literalsfrom __future__ import division%matplotlib inlinefrom IPython.display import *import numpy as npimport scipy.optimize as opimport matplotlib as mplimport matplotlib.pyplot as pltfrom mpl_toolkits.mplot3d import Axes3Dimport sympy as spsp.init_printing(use_latex=True, use_unicode=True)plt.rcParams['font.sans-serif'] = ['Arial Unicode MS']plt.rcParams['font.size'] = 14plt.rcParams['axes.unicode_minus'] = Falseplt.rcParams['figure.figsize'] = 8, 6

线性回归

用线性函数拟合训练集 $X \to y$,得到线性模型 $h(\theta^T x)$ 以预测新样本.代价函数 $J(\theta_n) = \frac{1}{2m}\sum_m (h_m(\theta) – y_m)^2$ 做为性能指标, 最低值来确定权重 $\theta_n$用梯度下降等迭代法得到最优值, 可作图直观调试收敛情况用正规函数直接计算得到最优值

In[2]:

def ex1(data_file_path):    data = np.loadtxt(data_file_path, delimiter=',')    x, y = data[:,0:-1], data[:, -1:]    bias = np.ones((x.shape[0], 1))    x_ = np.hstack((bias, x))    theta = np.zeros(x_.shape[1])    # 算法    def h(x, theta):        '''hypothesis'''        return np.dot(x, theta).reshape(-1, 1)    def J(x, y, theta):        '''cost'''        return np.power(h(x, theta) - y, 2).mean()/2    def gradient_descent(x, y, theta, alpha, iterations):        thetas = [theta]        costs = []        for i in range(iterations):            delta = alpha * np.multiply(h(x, theta)-y, x).mean(0)            theta = theta - delta            thetas.append(theta)            cost = J(x, y, theta)            costs.append(cost)        return thetas, costs    def normal_equation(x, y):        x = np.matrix(x)        y = np.matrix(y)        return np.linalg.pinv(x.T * x) * x.T * y    # 数据结果图    thetas, costs = gradient_descent(x_, y, theta, 0.01, 1500)    theta_gradient_descent = thetas[-1]    theta_normal_equation = normal_equation(x_, y)    theta_minimize = op.minimize(lambda theta: J(x_, y, theta), [0, 0])['x']    plt.plot(costs)    plt.title('代价迭代收敛')    plt.xlabel('迭代次数')    plt.ylabel('代价')    plt.show()    display(Math(r'$\theta_{梯度下降} = \begin{bmatrix}%.2f\\%.2f\end{bmatrix}\                 ,\ \theta_{正规函数} = \begin{bmatrix}%.2f\\%.2f\end{bmatrix}\                 ,\ \theta_{最小化BFGS} = \begin{bmatrix}%.2f\\%.2f\end{bmatrix}$' %                  (theta_gradient_descent[0], theta_gradient_descent[1],                   theta_normal_equation[0], theta_normal_equation[1],                  theta_minimize[0], theta_minimize[1])))    y_35 = h([1, 3.5], theta_gradient_descent)[0, 0]    y_7 = h([1, 7], theta_gradient_descent)[0, 0]    plt.plot(x, y, 'x', label='样本')    px = range(2, 30, 2)    py = [h(np.array([1, i]), theta_gradient_descent)[0, 0] for i in px]    plt.plot(px, py, label='线性回归模型')    plt.plot([3.5, 7], [y_35, y_7], 'o', label='预测')    plt.title('线性回归')    plt.xlabel('食品车利润(万)')    plt.ylabel('城市人口(万)')    plt.legend(loc=4)    plt.show()    display(Math(r'h(%s) = %.2f' % (3.5, y_35)))    display(Math(r'h(%s) = %.2f' % (7, y_7)))    # 代价函数图    theta0s = np.linspace(-10, 10, 100)    theta1s = np.linspace(-1, 4, 100)    theta0s, theta1s = np.meshgrid(theta0s, theta1s)    m, n = theta0s.shape    hs = np.array([J(x_, y, [theta0s[i, j], theta1s[i, j]])                    for i in range(m) for j in range(n)]).reshape(m, n)    fig = plt.figure()    ax = fig.add_subplot(111, projection='3d')    ax.plot_surface(theta0s, theta1s, hs,                    rstride=3, cstride=3, cmap=mpl.cm.jet, linewidth=0.3, antialiased=True)    plt.xlabel(r'$\theta_0$')    plt.ylabel(r'$\theta_1$')    plt.title(r'代价函数 $J(\theta)$')    plt.show()    plt.contour(theta0s, theta1s, hs, np.logspace(-2, 3, 30))    plt.xlabel(r'$\theta_0$')    plt.ylabel(r'$\theta_1$')    plt.title(r'代价函数等高线')    plt.plot([i[0] for i in thetas], [i[1] for i in thetas], 'r-')    plt.plot(theta_gradient_descent[0], theta_gradient_descent[1], 'rx')    plt.plot(theta_normal_equation[0, 0], theta_normal_equation[1, 0], 'k+')    plt.plot(theta_minimize[0], theta_minimize[1], 'yx')    plt.show()ex1('ex1data1.txt')

$$\theta_{梯度下降} = \begin{bmatrix}-3.63\\1.17\end{bmatrix} ,\ \theta_{正规函数} = \begin{bmatrix}-3.90\\1.19\end{bmatrix} ,\ \theta_{最小化BFGS} = \begin{bmatrix}-3.90\\1.19\end{bmatrix}$$

$$h(3.5) = 0.45$$

$$h(7) = 4.53$$

逻辑回归用 Sigmoid 函数扩展线性回归, 得到简单两分法分类模型. 使用一对多策略扩展到多于两个类别的分类模型

In[3]:

z = sp.symbols('z')sigmoid = 1/(1+sp.exp(-z))sp.plotting.plot(sigmoid, (z, -8, 8))display(sigmoid)

$$\frac{1}{1 + e^{- z}}$$

这个函数值在 0 与 1 之间, 用来表示预测的分类概率. 该函数需要用 $log$ 函数来使代价函数为凸起

In[4]:

def ex2(data_file_path):    data = np.loadtxt(data_file_path, delimiter=',')    x, y = data[:,0:-1], data[:, -1:]    bias = np.ones((x.shape[0], 1))    x_ = np.hstack((bias, x))    theta = np.zeros(x_.shape[1])    # 算法    def sigmoid(z):        return 1 / (1+np.exp(-z))    def h(x, theta):        return sigmoid(np.dot(x, theta).reshape(-1, 1))    def J(x, y, theta):        h_val = h(x, theta)        return (-y * np.log(h_val) - (1-y)*np.log(1-h_val)).mean();    def gradient(x, y, theta):        return np.multiply(h(x, theta)-y, x).mean(0)    print('初始代价 %.3f' % J(x_, y, theta))    cost = lambda t: J(x_, y, t)    gradient_ = lambda t: gradient(x_, y, t)    # TODO: make it work with other minimize algorithm    #op.minimize(lambda t: J(x_, y, t),     #            np.zeros(x_.shape[1]), method = 'TNC', jac=lambda t: gradient(x_, y, t))    #print gradient_(init)    #print 'fmin_cg()\n', op.fmin_cg(cost, init, fprime=gradient_)    #op.check_grad(cost, gradient_, init)    theta_fmin = op.fmin(cost, theta, disp=False)    display(Math(r'\theta_{fmin} = \begin{bmatrix}%.3f\\%.3f\\%.3f\end{bmatrix}' % tuple(theta_fmin)))    pos = np.where(y == 0)[0]    neg = np.where(y != 0)[0]    plt.plot(x[pos, 0], x[pos, 1], 'yo', label='拒绝')    plt.plot(x[neg, 0], x[neg, 1], 'k+', label='录取')    plt.xlabel('甲科考分')    plt.ylabel('乙科考分')    plot_x = np.array([x[:,0].min()-2, x[:,0].max()+2])    plot_y =  (theta_fmin[0] + theta_fmin[1]*plot_x) / -theta_fmin[2]    plt.plot(plot_x, plot_y, label='决策边界') # TODO how to plot decision bondary generally?    plt.legend(loc=3)    plt.show()    print('两门成绩 45 分和 85 分的考生录取率为 {:.1%}'.format(h([1, 45, 85], theta_fmin)[0][0]))    print('总拟合准确率 {:.1%}'.format(((h(x_, theta_fmin) >= 0.5) == y).mean()))ex2('ex2data1.txt')

初始代价 0.693

$$\theta_{fmin} = \begin{bmatrix}-25.161\\0.206\\0.201\end{bmatrix}$$

两门成绩 45 分和 85 分的考生录取率为 77.6%总拟合准确率 89.0%

原文地址：机器学习笔记, 感谢原作者分享。 爱上一个人的时候，总会有点害怕，怕得到他;怕失掉他。