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<!DOCTYPE html>
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<meta name="description" content="机器学习深度思考学习笔记。从最简单的机器学习基本概念，逐层剖析，挖掘机器学习最本质的问题，展示机器学习背后的数学之美">
<meta name="keywords" content="机器学习，统计学习，逻辑回归，决策树，支持向量机，条件随机场，聚类">
<meta name="author" content="xjtu-zhongyingLi">
<title>深度思考之机器学习系列</title>
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<body class="book toc2 toc-left">
<div id="header">
<h1>深度思考之机器学习系列</h1>
<div class="details">
<span id="author" class="author">xjtu-zhongyingLi</span><br>
<span id="revdate">2018-07-22</span>
</div>
<div id="toc" class="toc2">
<div id="toctitle">目录</div>
<ul class="sectlevel1">
<li><a href="#_机器学习基本概念">1. 机器学习基本概念</a>
<ul class="sectlevel2">
<li><a href="#_统计学习">1.1. 统计学习</a></li>
<li><a href="#_统计学习三要素">1.2. 统计学习三要素</a>
<ul class="sectlevel3">
<li><a href="#_模型">1.2.1. 模型</a></li>
<li><a href="#_策略">1.2.2. 策略</a>
<ul class="sectlevel4">
<li><a href="#_损失函数">1.2.2.1. 损失函数</a></li>
<li><a href="#_经验风险最小化">1.2.2.2. 经验风险最小化</a></li>
<li><a href="#_结构风险最小化">1.2.2.3. 结构风险最小化</a></li>
</ul>
</li>
<li><a href="#_算法">1.2.3. 算法</a></li>
</ul>
</li>
<li><a href="#_模型评估">1.3. 模型评估</a>
<ul class="sectlevel3">
<li><a href="#_正则化">1.3.1. 正则化</a></li>
<li><a href="#_交叉验证">1.3.2. 交叉验证</a></li>
<li><a href="#_泛化能力">1.3.3. 泛化能力</a></li>
<li><a href="#_生成模型和判别模型">1.3.4. 生成模型和判别模型</a>
<ul class="sectlevel4">
<li><a href="#_判别模型">1.3.4.1. 判别模型</a></li>
<li><a href="#_生成模型">1.3.4.2. 生成模型</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#_深度思考">1.4. 深度思考</a>
<ul class="sectlevel3">
<li><a href="#_贝叶斯理论">1.4.1. 贝叶斯理论</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#_线性回归与逻辑回归">2. 线性回归与逻辑回归</a>
<ul class="sectlevel2">
<li><a href="#_线性回归">2.1. 线性回归</a>
<ul class="sectlevel3">
<li><a href="#_概念">2.1.1. 概念</a></li>
<li><a href="#_梯度下降">2.1.2. 梯度下降</a></li>
<li><a href="#_梯度下降的局限性">2.1.3. 梯度下降的局限性</a></li>
<li><a href="#_深度思考_2">2.1.4. 深度思考</a></li>
</ul>
</li>
<li><a href="#_逻辑回归">2.2. 逻辑回归</a>
<ul class="sectlevel3">
<li><a href="#_揭开面纱">2.2.1. 揭开面纱</a></li>
<li><a href="#_sigmoid函数">2.2.2. Sigmoid函数</a></li>
<li><a href="#_参数更新">2.2.3. 参数更新</a></li>
<li><a href="#_深度思考_3">2.2.4. 深度思考</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#_特征工程">3. 特征工程</a>
<ul class="sectlevel2">
<li><a href="#_引子">3.1. 引子</a></li>
<li><a href="#_特征工程_2">3.2. 特征工程</a>
<ul class="sectlevel3">
<li><a href="#_数据清洗">3.2.1. 数据清洗</a></li>
<li><a href="#_数据采样">3.2.2. 数据采样</a></li>
<li><a href="#_特征处理">3.2.3. 特征处理</a>
<ul class="sectlevel4">
<li><a href="#_数值型特征">3.2.3.1. 数值型特征</a></li>
<li><a href="#_类别型特征">3.2.3.2. 类别型特征</a></li>
<li><a href="#_时间型特征">3.2.3.3. 时间型特征</a></li>
<li><a href="#_文本型特征">3.2.3.4. 文本型特征</a></li>
<li><a href="#_统计特征">3.2.3.5. 统计特征</a></li>
<li><a href="#_组合特征">3.2.3.6. 组合特征</a></li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
<li><a href="#_支持向量机">4. 支持向量机</a>
<ul class="sectlevel2">
<li><a href="#_开门见山">4.1. 开门见山</a></li>
<li><a href="#_高维空间">4.2. 高维空间</a>
<ul class="sectlevel3">
<li><a href="#_超平面的表示">4.2.1. 超平面的表示</a></li>
<li><a href="#_点到平面距离">4.2.2. 点到平面距离</a></li>
</ul>
</li>
<li><a href="#_问题优化">4.3. 问题优化</a>
<ul class="sectlevel3">
<li><a href="#_理解对偶问题">4.3.1. 理解对偶问题</a></li>
<li><a href="#_等价性证明">4.3.2. 等价性证明</a></li>
</ul>
</li>
<li><a href="#_对偶问题求解">4.4. 对偶问题求解</a>
<ul class="sectlevel3">
<li><a href="#_推导和结论">4.4.1. 推导和结论</a></li>
<li><a href="#_smo算法">4.4.2. SMO算法</a>
<ul class="sectlevel4">
<li><a href="#_smo算法思想">4.4.2.1. SMO算法思想</a></li>
<li><a href="#_二次规划求解方法">4.4.2.2. 二次规划求解方法</a></li>
<li><a href="#_公式证明">4.4.2.3. 公式证明</a></li>
<li><a href="#_变量选择方法">4.4.2.4. 变量选择方法</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#_软间隔分类器">4.5. 软间隔分类器</a>
<ul class="sectlevel3">
<li><a href="#_优化问题">4.5.1. 优化问题</a></li>
<li><a href="#_合页损失">4.5.2. 合页损失</a></li>
</ul>
</li>
<li><a href="#_核技巧">4.6. 核技巧</a></li>
<li><a href="#_深度思考_4">4.7. 深度思考</a>
<ul class="sectlevel3">
<li><a href="#_深入理解kkt条件">4.7.1. 深入理解KKT条件</a>
<ul class="sectlevel4">
<li><a href="#_什么是kkt条件">4.7.1.1. 什么是KKT条件</a></li>
<li><a href="#_等式约束优化问题">4.7.1.2. 等式约束优化问题</a></li>
<li><a href="#_不等式约束优化问题">4.7.1.3. 不等式约束优化问题</a></li>
</ul>
</li>
<li><a href="#_如何理解高斯核映射至无穷维">4.7.2. 如何理解高斯核映射至无穷维</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#_主题模型">5. 主题模型</a>
<ul class="sectlevel2">
<li><a href="#_解决什么问题">5.1. 解决什么问题</a></li>
<li><a href="#_给问题建模">5.2. 给问题建模</a>
<ul class="sectlevel3">
<li><a href="#_建模">5.2.1. 建模</a></li>
<li><a href="#_第一次思考">5.2.2. 第一次思考</a></li>
<li><a href="#_第二次思考">5.2.3. 第二次思考</a>
<ul class="sectlevel4">
<li><a href="#_一个重大的发现">5.2.3.1. 一个重大的发现</a></li>
<li><a href="#_metropolis_hastings算法">5.2.3.2. Metropolis Hastings算法</a></li>
<li><a href="#_gibbs_sampling_算法">5.2.3.3. Gibbs Sampling 算法</a></li>
</ul>
</li>
<li><a href="#_第三次思考">5.2.4. 第三次思考</a></li>
</ul>
</li>
<li><a href="#_文本建模">5.3. 文本建模</a>
<ul class="sectlevel3">
<li><a href="#_文档是如何产生的">5.3.1. 文档是如何产生的</a></li>
<li><a href="#_plsa模型">5.3.2. PLSA模型</a></li>
<li><a href="#_lda模型">5.3.3. LDA模型</a>
<ul class="sectlevel4">
<li><a href="#_二项分布">5.3.3.1. 二项分布</a></li>
<li><a href="#_神奇的_gamma_函数">5.3.3.2. 神奇的 <code>Gamma</code> 函数</a></li>
<li><a href="#_beta_函数">5.3.3.3. <code>Beta</code> 函数</a></li>
<li><a href="#_第四次思考">5.3.3.4. 第四次思考</a></li>
<li><a href="#_dirichlet_分布">5.3.3.5. <code>Dirichlet</code> 分布</a></li>
<li><a href="#_lda_模型">5.3.3.6. <code>LDA</code> 模型</a></li>
<li><a href="#_模型训练和推演">5.3.3.7. 模型训练和推演</a></li>
</ul>
</li>
<li><a href="#_变分em算法求解plsa模型">5.3.4. 变分EM算法求解pLSA模型</a>
<ul class="sectlevel4">
<li><a href="#_em算法">5.3.4.1. EM算法</a></li>
<li><a href="#_求解plsa算法">5.3.4.2. 求解pLSA算法</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#_终篇">5.4. 终篇</a></li>
</ul>
</li>
<li><a href="#_最大熵理论和多分类器">6. 最大熵理论和多分类器</a>
<ul class="sectlevel2">
<li><a href="#_最大熵原理">6.1. 最大熵原理</a>
<ul class="sectlevel3">
<li><a href="#_最大熵模型的定义">6.1.1. 最大熵模型的定义</a></li>
<li><a href="#_最大熵模型的推导">6.1.2. 最大熵模型的推导</a></li>
<li><a href="#_多分类器">6.1.3. 多分类器</a></li>
</ul>
</li>
<li><a href="#_指数分布簇">6.2. 指数分布簇</a></li>
<li><a href="#_广义线形模型">6.3. 广义线形模型</a></li>
<li><a href="#_广义线形模型应用_多项式分布">6.4. 广义线形模型应用-多项式分布</a></li>
</ul>
</li>
<li><a href="#_em_算法">7. <code>EM</code> 算法</a>
<ul class="sectlevel2">
<li><a href="#_前置知识点">7.1. 前置知识点</a>
<ul class="sectlevel3">
<li><a href="#_jensen不等式">7.1.1. Jensen不等式</a></li>
<li><a href="#_最大似然估计">7.1.2. 最大似然估计</a></li>
</ul>
</li>
<li><a href="#_em算法_2">7.2. EM算法</a>
<ul class="sectlevel3">
<li><a href="#_直通结论">7.2.1. 直通结论</a></li>
<li><a href="#_em推导">7.2.2. EM推导</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#_决策树和集成学习">8. 决策树和集成学习</a>
<ul class="sectlevel2">
<li><a href="#_决策树模型">8.1. 决策树模型</a>
<ul class="sectlevel3">
<li><a href="#_决策树思想">8.1.1. 决策树思想</a></li>
<li><a href="#_特征选择">8.1.2. 特征选择</a>
<ul class="sectlevel4">
<li><a href="#_信息熵">8.1.2.1. 信息熵</a></li>
<li><a href="#_信息增益">8.1.2.2. 信息增益</a></li>
<li><a href="#_信息增益比">8.1.2.3. 信息增益比</a></li>
<li><a href="#_基尼指数">8.1.2.4. 基尼指数</a></li>
</ul>
</li>
<li><a href="#_决策树生成">8.1.3. 决策树生成</a>
<ul class="sectlevel4">
<li><a href="#_cart生成">8.1.3.1. CART生成</a></li>
</ul>
</li>
<li><a href="#_决策树剪枝">8.1.4. 决策树剪枝</a></li>
</ul>
</li>
<li><a href="#_集成学习">8.2. 集成学习</a>
<ul class="sectlevel3">
<li><a href="#_bagging和随机森林">8.2.1. Bagging和随机森林</a></li>
<li><a href="#_boosting">8.2.2. Boosting</a>
<ul class="sectlevel4">
<li><a href="#_adboost详解">8.2.2.1. Adboost详解</a></li>
<li><a href="#_gbdt详解">8.2.2.2. GBDT详解</a></li>
<li><a href="#_xgboost详解">8.2.2.3. XGBoost详解</a></li>
<li><a href="#_总结">8.2.2.4. 总结</a></li>
<li><a href="#_深度思考_5">8.2.2.5. 深度思考</a></li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
<li><a href="#_推荐系统">9. 推荐系统</a></li>
<li><a href="#_聚类和近邻算法">10. 聚类和近邻算法</a>
<ul class="sectlevel2">
<li><a href="#_k_means算法">10.1. K-means算法</a></li>
<li><a href="#_k_means算法_2">10.2. k-means++算法</a>
<ul class="sectlevel3">
<li><a href="#_基本理论">10.2.1. 基本理论</a></li>
<li><a href="#_算法核心">10.2.2. 算法核心</a></li>
</ul>
</li>
<li><a href="#_基于层次的聚类">10.3. 基于层次的聚类</a></li>
<li><a href="#_knn算法">10.4. KNN算法</a></li>
<li><a href="#_kd树">10.5. kd树</a>
<ul class="sectlevel3">
<li><a href="#_构造kd树">10.5.1. 构造\(kd\)树</a></li>
<li><a href="#_搜索kd树">10.5.2. 搜索\(kd\)树</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#_贝叶斯理论_2">11. 贝叶斯理论</a>
<ul class="sectlevel2">
<li><a href="#_一个公式的故事">11.1. 一个公式的故事</a></li>
<li><a href="#_一些常见题目">11.2. 一些常见题目</a>
<ul class="sectlevel3">
<li><a href="#_两信封和美元的问题">11.2.1. 两信封和美元的问题</a></li>
<li><a href="#_三扇门和开奖的问题">11.2.2. 三扇门和开奖的问题</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#_隐马尔可夫模型">12. 隐马尔可夫模型</a></li>
<li><a href="#_条件随机场">13. 条件随机场</a></li>
<li><a href="#_深度学习概论">14. 深度学习概论</a></li>
<li><a href="#_深度学习中的优化算法">15. 深度学习中的优化算法</a></li>
<li><a href="#_深度学习中的正则化">16. 深度学习中的正则化</a></li>
<li><a href="#_深度神经网络">17. 深度神经网络</a></li>
<li><a href="#_卷积神经网络">18. 卷积神经网络</a></li>
<li><a href="#_循环神经网络">19. 循环神经网络</a></li>
<li><a href="#_神经网络番外篇">20. 神经网络番外篇</a></li>
<li><a href="#_生成对抗网络">21. 生成对抗网络</a></li>
<li><a href="#_迁移学习">22. 迁移学习</a></li>
<li><a href="#_强化学习">23. 强化学习</a></li>
<li><a href="#_数学之美">24. 数学之美</a>
<ul class="sectlevel2">
<li><a href="#_数学中的空间">24.1. 数学中的空间</a>
<ul class="sectlevel3">
<li><a href="#_空间关系">24.1.1. 空间关系</a></li>
<li><a href="#_希尔伯特空间">24.1.2. 希尔伯特空间</a></li>
</ul>
</li>
<li><a href="#_似然函数">24.2. 似然函数</a></li>
</ul>
</li>
<li><a href="#_深度学习框架">25. 深度学习框架</a></li>
<li><a href="#_走进互联网">26. 走进互联网</a></li>
</ul>
</div>
</div>
<div id="content">
<div id="preamble">
<div class="sectionbody">
<div class="quoteblock">
<blockquote>
数学是人类智慧的结晶，统计学习是数学领域最璀璨的明珠，算法可以让这个明珠照亮整个世界!
</blockquote>
<div class="attribution">
— 李中英<br>
<cite>世界知名互联网公司高级算法研究猿👍👍👍</cite>
</div>
</div>
</div>
</div>
<div class="sect1">
<h2 id="_机器学习基本概念">1. 机器学习基本概念</h2>
<div class="sectionbody">
<div class="sect2">
<h3 id="_统计学习">1.1. 统计学习</h3>
<div class="paragraph">
<p>统计学习是关于计算机基于数据构建概率统计模型并运用模型对数据进行预测与分析的一门学科，也成为统计机器学习。
统计学习的对象是数据，它从数据出发，提取数据的特征，抽象出数据的模型，发现数据中的知识，又回到对数据的分析与预测中去，
统计学习关于数据的基本假设是：同类数据具有一定的统计规律性.<br></p>
</div>
<div class="paragraph">
<p>统计学习的目的是对数据进行预测与分析，是通过构建概率统计模型实现的，统计学习总的目的就是考虑学习什么样的模型和如何学习模型，
以使模型能对数据进行准确的预测和分析，同时考虑尽可能的提高学习效率.<br></p>
</div>
<div class="paragraph">
<p>统计学习的方法包括：监督学习、非监督学习、半监督学习和强化学习，我们重点讨论监督学习.<br></p>
</div>
</div>
<div class="sect2">
<h3 id="_统计学习三要素">1.2. 统计学习三要素</h3>
<div class="sect3">
<h4 id="_模型">1.2.1. 模型</h4>
<div class="paragraph">
<p>在监督学习中，模型就是指要学习的条件概率分布或决策函数。假设空间中的模型一般有无穷多个，假设空间可定义为：<br></p>
</div>
<div class="paragraph">
<p>\begin{equation}
F=\left \{ f|Y=f(X) \right \}
\end{equation}</p>
</div>
<div class="paragraph">
<p>假设空间通常是由参数向量决定的函数簇，其中参数向量取值于n维欧氏空间，称为参数空间。</p>
</div>
<div class="paragraph">
<p>\begin{equation}
F=\left \{ f|Y=f_{\theta }(X),\theta \in R^{n} \right \}
\end{equation}</p>
</div>
<div class="paragraph">
<p>假设空间也可以定义为条件概率的集合:</p>
</div>
<div class="paragraph">
<p>\begin{equation}
F=\left \{ P|P_{\theta }(Y|X),\theta \in R^{n} \right \}
\end{equation}</p>
</div>
</div>
<div class="sect3">
<h4 id="_策略">1.2.2. 策略</h4>
<div class="paragraph">
<p>有了模型的假设空间，统计学习接着要考虑的是按照什么样的准则学习或选择最优的模型，这就是策略。<br></p>
</div>
<div class="sect4">
<h5 id="_损失函数">1.2.2.1. 损失函数</h5>
<div class="paragraph">
<p>策略用来解决最优模型的选择问题，那么如何评价模型优劣，这就是损失函数或代价函数，下面是一些常见的损失函数：<br>
（1）0-1损失<br></p>
</div>
<div class="paragraph">
<p>\begin{equation}
L(Y|f(X))=\left\{\begin{matrix}
1,Y\neq f(X) & \\
0,Y=f(X)&
\end{matrix}\right.
\end{equation}</p>
</div>
<div class="paragraph">
<p>（2）平方损失<br></p>
</div>
<div class="paragraph">
<p>\begin{equation}
L(Y|f(X))=(Y-f(X))^{2}
\end{equation}</p>
</div>
<div class="paragraph">
<p>（3）绝对损失<br></p>
</div>
<div class="paragraph">
<p>\begin{equation}
L(Y|f(X))=\left |Y-f(X) \right |
\end{equation}</p>
</div>
<div class="paragraph">
<p>（4）对数损失<br></p>
</div>
<div class="paragraph">
<p>\begin{equation}
L(Y|f(X))=-log(P(Y|X)
\end{equation}</p>
</div>
<div class="paragraph">
<p>损失函数越小，模型就越好，理论上的最优模型应该是损失函数的期望值最小，而理论模型是关于联合分布下的平均损失，称为期望损失或风险损失，然而现实的问题是联合分布是未知的，如果已知也就不需要学习了。<br></p>
</div>
<div class="paragraph">
<p>所以机器学习采用的方法时是通过用<strong>训练数据集上的平均损失近似期望损失</strong>，训练集上的风险我们称为经验风险，根据大数定理，当训练样本数量趋近于无穷时，经验风险趋近于期望风险。<br></p>
</div>
<div class="paragraph">
<p>但是现实中的训练样本数量是有限的，甚至很小，所以直接使用经验风险估计期望风险常常不理想，需要对经验风险进行一定的矫正，这就关系到监督学习的两个基本策略：<strong>经验风险最小化和结构风险最小化</strong>。<br></p>
</div>
</div>
<div class="sect4">
<h5 id="_经验风险最小化">1.2.2.2. 经验风险最小化</h5>
<div class="paragraph">
<p>经验风险最小化的策略认为：经验风险最小化的模型是最优模型，根据这一策略，
按照经验风险最小化策略求最优模型就是求解最优化问题：<br></p>
</div>
<div class="paragraph">
<p>\begin{equation}
min\frac{1}{m}\sum_{i=1}^{m}L(y_{i},f(x_{i}))
\end{equation}</p>
</div>
<div class="paragraph">
<p>极大似然估计就是经验风险最小化的典型例子，当模型是条件概率，损失函数是对数损失函数时，经验风险最小化就等价于极大似然估计。<br></p>
</div>
<div class="paragraph">
<p>但是，当样本量较少时，经验风险最小化学习的效果未必很好，会产生过拟合的问题，结构风险最小化是为了防止过拟合而提出的策略。<br></p>
</div>
</div>
<div class="sect4">
<h5 id="_结构风险最小化">1.2.2.3. 结构风险最小化</h5>
<div class="paragraph">
<p>结构风险最小化等价于正则化，结构风险在经验风险上加上表示模型复杂度的正则化项或惩罚项，结构风险求解的最优化问题是：<br></p>
</div>
<div class="paragraph">
<p>\begin{equation}
min\frac{1}{m}\sum_{i=1}^{m}L(y_{i},f(x_{i}))+\lambda J(f)
\end{equation}</p>
</div>
<div class="paragraph">
<p>其中 \(J(f)\) 为模型复杂度，\(\lambda \geqslant 0\)是系数，用以权衡经验风险和模型复杂度。结构风险小需要经验风险和模型复杂度同时小，结构风险小的模型往往对未知的测试数据和已知的训练数据都有较好的预测.<br></p>
</div>
<div class="paragraph">
<p>比如，贝叶斯估计中的最大后验概率估计就是结构风险最小化的一个例子。<strong>当模型是条件概率分布、损失函数是对数损失函数、模型复杂度由模型的先验概率表示时，结构风险最小化就等价于最大后验概率估计</strong>。<br></p>
</div>
<div class="admonitionblock important">
<table>
<tr>
<td class="icon">
<i class="fa icon-important" title="Important"></i>
</td>
<td class="content">
<div class="title">经验风险最小化深度理解</div>
<div class="paragraph">
<p>模型是条件概率分布，优化目标可以表示为概率连乘的形势，如果损失函数是对数损失，即可以写成连加的形势，由于最大后验概率可以写成似然函数和先验概率分布的乘积，对数损失后，先验概率就变成似然函数连加后的一项，对比上面公式，先验概率就刚好等价于模型复杂度，因此这种情况下，结构风险最小化就等价于最大后验概率估计。<br></p>
</div>
<div class="ulist">
<ul>
<li>
<p>我们将会在一个独立的小节阐述如何用贝叶斯理论理解本章的概念</p>
</li>
</ul>
</div>
</td>
</tr>