From a3b6c32a4462dae64cde1b4009039ebb8d5d8600 Mon Sep 17 00:00:00 2001 From: Yujie Lu Date: Thu, 19 Mar 2020 18:42:43 +0800 Subject: [PATCH] update week3 notes (latex) (#12) --- latex_version/notes/note.pdf | Bin 268067 -> 332359 bytes latex_version/notes/note.tex | 248 ++++++++++++++++++---------------- latex_version/notes/week2.tex | 171 +++++++++++++++++++++++ 3 files changed, 303 insertions(+), 116 deletions(-) create mode 100644 latex_version/notes/week2.tex diff --git a/latex_version/notes/note.pdf b/latex_version/notes/note.pdf index cd36dcfec0a160ee54b2c0d390c52865e4b0a228..361f291743c73793951af3f736bc67335b7782e2 100644 GIT binary patch delta 315626 zcmZsCV{k4^vu$kKc6Myr=8m&t+k9f%Hg;^=wr$%^?t9*Q>(qDZ+mvaQKXR@Zx}+p&^vV(wVt9S0>NJ>AjIlUMqE+)_IrY0JGu8!HsguAu3kLhCsn_qS-H5W_rcV_)g&w&BWE!4H`(c>yBa~K>=NBn!UD8@1JW9_Orb$pNntQT_ zvBV=p-G`MOg{?DLf9Xzr{!{_I-0PO(hci|NV~qi-8dE1#p-0%S1ci%=*a!Q=@qF58 z2;;^!_>>o}t(D0KqY&E07xJFaUp!5`usRr1J|vXevf|Qt$6+`r`71bR#@^ zjNMk=o+t_*#-m*lYD$ZO>rI`P__y|!miadTFz$bG?B-I*)}9ED>S!!wdZx@ob0F)#sapB zJ1#*)c

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zY>I6$6uV*=*2i${hTX9ZHpdp&0NY`E?0_Ay8Fs>k*aSOc7i@{Gur)Tq#@GYL-~jB6 zJ+T+|$3EB>`{8IDgadIf4#y!l6eDmLj>HidiBUKTXJa&u!znlkr(+z>z?pwIPbECf zp8tH6o&~lSs0_}wy-+3MyX{3P!>`$1tP-7Ldx=WSAKOb+*6y~wOl8|T+sjqr%e#MO zLZp=yDhDdKlJv{UN|oejS5og^JdX<;HuDpNpx3WQ{Ak6kgmDh>x=$l+Cn^X$@U3u$kWwXk=j;?%& zv$93y<1tr0EwQpy0PByOE0-P;1sn}BHa9UjISM5uMNdWwyz-F2 diff --git a/latex_version/notes/note.tex b/latex_version/notes/note.tex index b3af72f..c31ec8d 100644 --- a/latex_version/notes/note.tex +++ b/latex_version/notes/note.tex @@ -6,161 +6,177 @@ \tableofcontents \input{week1} -\chapter{March 9} -\section{判别模型和生成模型} -\begin{itemize} - \item 判别模型 - \begin{itemize} - \item 确定性判别:$y=f_{\theta}\left( x \right) $ - \item 随机判别:$p_{\theta}\left( y \mid x \right) $ - \end{itemize} - \item 生成模型:建立联合概率分布(一般不用) - \[ - p_{\theta}\left( y \mid x \right) = \frac{p_{\theta}\left( x,y \right)}{p_{\theta}\left( x \right) } - .\] -\end{itemize} - -\subsection{生成模型} -\begin{itemize} - \item 频率派,$\theta$ 是具体的一个点,易于计算 - \item 贝叶斯派,$\theta$ 是一个分布 -\end{itemize} - - -\section{线性回归} -一维的线性回归和二次回归都是线性模型:比如 +\input{week2} +\chapter{March 16} +\section{逻辑回归} +概率判别模型在分类时往往更加可靠,因为如果用函数的话就无法求导了。 +\subsection{二分类} +\noindent \textbf{损失函数} +用交叉熵来衡量 +\[ + \mathcal{L}\left( y,x \right) = -\sum_{k} \delta\left( y=c_{k} \right) \log p_{\theta}\left( y=c_k \mid x \right) +.\] +其中 \[ - f\left( x \right) =\theta_0+\theta_1x+\theta_2 x^2 = \theta^{T} \phi\left( x \right) + \delta\left( z \right) = \begin{cases} + 1, \quad z \text{ is true} \\ + 0, \quad z \text{ else} + \end{cases} .\] -$\phi$ 实际上是一种feature engineering。 - -\noindent \textbf{学习目标} +为了输出一个$\left( 0,1 \right) $ 之间的值,我们会给$z=\theta^{T}x$套上一个sigmoid函数 \[ - \mathcal{L}\left( \theta \right) = - \frac{1}{n} \sum_{i=1}^{n} \mathcal{L}\left( y_{i},f_{\theta}\left( x_{i} \right) \right) + \sigma\left( z \right) = \frac{1}{1+\exp\left( -z \right) } .\] -损失函数选择:min square loss - -\noindent \textbf{学习方法} - -梯度下降 +即 \[ - \theta_{new} \leftarrow \theta_{old} - \eta \frac{\partial \mathcal{L}\left( \theta \right) }{\partial \theta} + p\left( y=1 \mid x \right) =\sigma\left( z \right), \quad p\left( y=0 \mid x \right) = 1-\sigma\left( z \right) .\] - -\section{梯度下降} -\subsection{批量梯度下降} -根据整个批量数据的梯度更新数据 +带入损失函数有 \[ - \frac{\partial J\left( \theta \right) }{\partial \theta} = - -\frac{1}{N} \sum_{i=1}^{N} \left( y_{i}-f_{\theta}\left( x_{i} \right) \right) x_{i} + \mathcal{L}\left( y,x \right) = -y\log \sigma\left( z \right) - \left( 1-y \right) \log\left( 1-\sigma\left( z \right) \right) .\] - -\subsection{随机梯度下降} -\href{https://en.wikipedia.org/wiki/Stochastic_gradient_descent}{Stochastic Gradient Descent} -在数据量较大时比批量梯度下降更为优秀。但是学习中存在\textbf{震荡}或不确定性。 - -\noindent \textbf{优化目标} +有趣的是,$\sigma$ 求导后 +\[ + \frac{\partial \sigma\left( z \right) }{\partial z} = \sigma\left( z \right) \left( 1-\sigma\left( z \right) \right) +.\] +从而梯度为 +\[ + \frac{\partial \mathcal{L}\left( y,x \right) }{\partial \theta} = \left( \sigma\left( z \right) -y \right) x +.\] +可以看到,梯度形式与线性回归十分相似。 +\subsection{多分类} +与二分类同样的,只是引入一个类别集 +\[ +C = \{c_1,c_2,\ldots,c_{n}\} +.\] +使用softmax \[ - J^{(i)} \left( \theta \right) = \frac{1}{2} \left( y_{i}-f_{\theta}\left( x_{i} \right) \right) ^2 + p_{\theta}\left( x=c_{j} \mid x \right) = \frac{\exp\left( \theta_{j}^{T}x \right) }{\sum_{i=1}^{m} \exp\left( \theta_{i^{T}x} \right) } .\] -\subsection{小批量梯度下降} -前面两种方式的结合,将训练集分为$K$ 个mini-batch。对每一个小批量更新参数 +参数$\theta = \{\theta_{1},\ldots,\theta_{m}\} $, 可以同时除掉$\theta_{1}$ 来减少一个参数。 + +其实可以看成$m$ 个二分类,目标 \[ - J^{(k)}\left( \theta \right) = \frac{1}{2N_{k}} \sum_{i=1}^{N_k} \left( y_i-f_{\theta}\left( x_{i} \right) \right) ^2 + \mathrm{max} \log p_{\theta}\left( y=c_{j} \mid x \right) .\] -\noindent \textbf{优点} +求梯度有 +\begin{align*} + \frac{\partial \log p_{\theta}}{\partial \theta_{j}} + &= x - p_{\theta}x \\ + &= \left( 1-p_{\theta}\left( y=c_j \mid x \right) \right) x +.\end{align*} +\subsection{逻辑回归应用} +\noindent \textbf{在线广告点击率(CTR)估算} +Feature engineering十分重要 \begin{itemize} - \item 结合map-reduce可以比较容易实现并行 - \item 更新速度快,不确定性低 + \item One-Hot二进制编码 + \item 例如\verb|weekday=friday| 转化为一个七维向量。容易发现转化的高维向量极为稀疏。 \end{itemize} -\subsection{基本搜索步骤} -随机选择初始化参数,走到局部最优值。 +\section{支持向量机} +线性分类器本质上是划一个决策边界,但是考虑到数据噪声后,分类器很容易出错。 +我们发现边界与数据距离最大时是最 +robust的,也称之为最大间隔边界。 -\begin{defi} - $f:\R^{n}\to \R$ 是凸函数:$\mathrm{dom} f$\footnote{dom 指函数定义域} 是一个凸集,并且 - \[ - f\left( tx_1+\left( 1-t \right) x_2 \right) \le tf\left( x_1 \right) - + \left( 1-t \right) f\left( x_2 \right) - .\] -\end{defi} -而凸函数是一定有最值的。 +我们的优化目标便是最短距离最大化。逻辑回归的打分函数$s=\theta^{T} x$,容易 +发现如果打分越高的样例越远离决策边界,分类器越可靠。 -\section{从代数角度看线性回归} -\noindent \textbf{目标函数} +\subsection{SVM最优化问题} +假设 +\begin{itemize} + \item 标签$y \in \{-1,1\} $ + \item $h\left( x \right) = g\left( w^{T}x + b \right) $ \footnote{g = sgn} +\end{itemize} +由高中知识我们知道,点到直线的距离即几何间隔为 \[ - J\left( \bm{\mu} \right) = \frac{1}{2} \left( \bm{y} - \bm{X\mu} \right)^{T} - \left( \bm{y} - \bm{X\mu} \right) + \gamma^{(i)} = \frac{ \lvert w^{T}x^{(i)} + b \rvert }{\|w\|^2} .\] -而梯度为 +不妨设$\|w\|^2 = 1$,且由于$y^{(i)}$ 的取值只为$\{-1,1\} $,上式可简化为 +\begin{equation} + \gamma^{(i)} = y^{(i)} \left( w^{T}x^{(i)} + b \right) \label{eqn:1} +.\end{equation} + +除此之外还可以通过简单的代数手段进行推导: +\[ + w^{T} \left( x^{(i)} - \gamma^{(i)} y^{(i)} \frac{w}{\|w\|} \right) + b= 0 +.\] +解得 +\begin{equation} + \gamma^{(i)} = y^{(i)} \left[ \left( \frac{w}{\|w\|} \right) ^{T} x^{(i)} + \frac{b}{\|w\|} \right] \label{eqn:2} +.\end{equation} +容易发现\eqref{eqn:1}和\eqref{eqn:2}本质相同,从而最小几何间隔为 \[ - \frac{\partial J(\bm{\mu} )}{\partial \mu} - = - \bm{X}^{T} \left( \bm{y}-\bm{X\mu} \right) + \hat{ \gamma} = \mathrm{min}_{i} \left( \gamma^{(1)},\ldots, \gamma^{(m)} \right) .\] -甚至可以直接求出最优参数 +等价于在 \[ - \frac{\partial J(\bm{\mu} )}{\partial \mu} = 0 \implies \hat{\mu} = \left( \bm{X^{T}X} \right) ^{-1} \bm{X}^{T} \bm{y} + y^{(i)} \left( w^{T}x^{(i)} + b \right) \ge \hat{\gamma} .\] -但是当$\bm{X}$ 很大时,这是很难计算的。 - -\noindent \textbf{当$\bm{X^{T}X}$ 为奇异矩阵 } - -其逆矩阵无法计算,解决方法 -\begin{itemize} - \item Regulariazation - \item $J_1\left( \mu \right) = J\left( \mu \right) + \frac{\lambda}{2} \|\mu\|^2_{2} $ -\end{itemize} -从而 +时求$\max \gamma$,但是非凸目标函数不易求最值,所以需要进行转换。 +将函数间隔固定为1,即$\hat{\gamma} = 1$,目标函数转化为 \[ - \hat{\mu}= \left( \bm{X^{T}X } + \lambda \bm{I} \right) ^{-1} \bm{X^{T} y} +\min \frac{1}{2} \|w \|^2 .\] -\subsection{泛线性模型} -本质上是做一个替换$ \bm x\to \phi\left( \bm x \right) $, -$\phi\left( x \right) $ 是$\R^{d}\to\R^{h}$ -的向量函数。加入了$\phi$ 的线性回归也叫核线性回归。 +使得$y^{(i)} \left( w^{T}x^{(i)} + b \right) \ge 1$ -\subsection{核线性回归的矩阵形式} -使用\href{https://zhuanlan.zhihu.com/p/45223109}{线性代数技巧}得到: +\section{支持向量机优化} +\subsection{拉格朗日对偶问题} +对于凸优化问题,可以使用拉格朗日乘数法。但是只能处理限制条件是等式的 +情况,为了处理不等式,我们需要引入松弛变量将其转化为等式,例如 \[ - \hat{y} = \Phi \hat{\theta} = \Phi\Phi^{T} \left( \Phi\Phi^{T} + \lambda I_{n} \right) ^{-1} y -.\] -只需关心核矩阵 + g_{i}\left( w \right) = 1 - y^{(i)} \left( w^{T}x^{(i)} + b \right) \le 0 +.\] +$g_{i}\left( w \right) $ 即支持向量。 +这等价于存在$a_{i}$使得 \[ - K = \Phi\Phi^{T} = \{ k\left( x^{(i)}, x^{(j)} \right) \} + g_{i}\left( w \right) + a_{i}^2 = 0 .\] - -\section{最大似然估计} - -带高斯白噪声的线性拟合: +得到拉格朗日函数 \[ - y = f_{\theta} \left( x \right) + \ep + \mathcal{L} \left( w,\lambda,a \right) = + f\left( w \right) + \sum_{i=1}^{n} \lambda_{i} \left( g_{i}\left( w \right) + + a_{i}^2 \right) +.\] +其中$f$ 即优化目标$\frac{1}{2} \|w\|^2 $由拉格朗日乘数法知 +\[ +\begin{cases} + \frac{\partial \mathcal{L}}{\partial w} = 0 \\ + \lambda_{i} a_{i} = 0 \\ + g_{i} \left( w \right) + a_{i} ^2 = 0 \\ + \lambda_{i} \ge 0 +\end{cases} .\] -其中$\ep \sim \mathcal{N} \left( 0, \sigma^2 \right) $. +对$\lambda_{i}a_{i} = 0$讨论,若$a_{i}=0$则$g_{i}\left( w \right) =0$, +若$a_{i}\neq 0$ 则必有$\lambda_{i} = 0$。 -\noindent \textbf{优化目标} +知$a_{i}$ 这个变量对我们求最值没有任何约束,我们可以用$\lambda_{i}g_{i}\left( w \right) +=0$ 来代替上式的条件。删去$a_{i}$ 即可得到\href{https://zhuanlan.zhihu.com/p/77750026}{KKT条件}, -最大似然(Likelyhood). +从而拉格朗日函数可以重写为 \[ - p\left( y \mid x \right) =\frac{1}{\sqrt{2\pi \sigma^2} } e^{- \frac{\ep^2}{2\sigma^2} }= \frac{1}{\sqrt{2\pi \sigma^2} } e^{- \frac{y-\theta^{T}x }{2\sigma^2} } + \mathcal{L}\left( w,\lambda \right) = f(w) + \sum_{i=1}^{n} \lambda_{i}g_{i}\left( w \right) .\] -最大化这个似然 +注意到$\mathcal{L} \le f\left( w \right) $ ,当调整$\lambda$使得 +$\sum_{i=1}^{n} \lambda_{i}g_{i}\left( w \right) $大的时候, +$f\left( w \right) $一定会更小,故我们的最优化问题等价于 \[ - \mathrm{max}_{\theta} \prod_{i=1}^{N} p\left( y_{i} \mid x_{i} \right) + \min_{w} \max_{\lambda} \mathcal{L} \left( w, \lambda \right) .\] -等价于最小均方误差的学习(取对数即可证明) - -\section{分类指标} -分类器有以下几个指标: -\begin{align*} - \mathrm{Accuracy} &= \frac{TP + TN}{TP + FP + TN + FN} \\ - \mathrm{Precision} &= \frac{TP}{TP+FP} \\ - \mathrm{Recall} &= \frac{TP}{TP+FN} -.\end{align*} -为了判别分类器好不好,我们有F1 score +\subsection{支持向量机优化求解} +可以证明KKT条件下有强对偶性 \[ -F_1 = \frac{2 \times \mathrm{Precision} \times \mathrm{Recall} }{ \mathrm{Precision} + \mathrm{Recall} } +\min \max f = \max \min f +.\] +从而我们可以很容易求解之前的问题,先对$w$求偏导求出$\min_{w}f$, +计算后可以知道问题转化为 +$ +\sum_{i=1}^{m} \lambda_{i} y^{i} = 0 +$ +时求解 +\[ +\max_{\lambda} \left[ \sum_{i=1}^{m} \lambda_{i} + +\frac{1}{2}\lambda_{i}\lambda_{j} y^{(i)} y^{(j)} \left( (x^{(i)})^{T} x^{(j)} \right) \right] .\] \end{document} diff --git a/latex_version/notes/week2.tex b/latex_version/notes/week2.tex new file mode 100644 index 0000000..0941ac8 --- /dev/null +++ b/latex_version/notes/week2.tex @@ -0,0 +1,171 @@ + +\chapter{March 9} +\section{判别模型和生成模型} +\begin{itemize} + \item 判别模型 + \begin{itemize} + \item 确定性判别:$y=f_{\theta}\left( x \right) $ + \item 随机判别:$p_{\theta}\left( y \mid x \right) $ + \end{itemize} + \item 生成模型:建立联合概率分布(一般不用) + \[ + p_{\theta}\left( y \mid x \right) = \frac{p_{\theta}\left( x,y \right)}{p_{\theta}\left( x \right) } + .\] +\end{itemize} + +\subsection{生成模型} +\begin{itemize} + \item 频率派,$\theta$ 是具体的一个点,易于计算 + \item 贝叶斯派,$\theta$ 是一个分布 +\end{itemize} + + +\section{线性回归} +一维的线性回归和二次回归都是线性模型:比如 +\[ + f\left( x \right) =\theta_0+\theta_1x+\theta_2 x^2 = \theta^{T} \phi\left( x \right) +.\] +$\phi$ 实际上是一种feature engineering。 + + +\noindent \textbf{学习目标} +\[ + \mathcal{L}\left( \theta \right) = + \frac{1}{n} \sum_{i=1}^{n} \mathcal{L}\left( y_{i},f_{\theta}\left( x_{i} \right) \right) +.\] +损失函数选择:min square loss + +\noindent \textbf{学习方法} + +梯度下降 +\[ + \theta_{new} \leftarrow \theta_{old} - \eta \frac{\partial \mathcal{L}\left( \theta \right) }{\partial \theta} +.\] + +\section{梯度下降} +\subsection{批量梯度下降} +根据整个批量数据的梯度更新数据 +\[ + \frac{\partial J\left( \theta \right) }{\partial \theta} = + -\frac{1}{N} \sum_{i=1}^{N} \left( y_{i}-f_{\theta}\left( x_{i} \right) \right) x_{i} +.\] + +\subsection{随机梯度下降} +\href{https://en.wikipedia.org/wiki/Stochastic_gradient_descent}{Stochastic Gradient Descent} +在数据量较大时比批量梯度下降更为优秀。但是学习中存在\textbf{震荡}或不确定性。 + +\noindent \textbf{优化目标} +\[ + J^{(i)} \left( \theta \right) = \frac{1}{2} \left( y_{i}-f_{\theta}\left( x_{i} \right) \right) ^2 +.\] +\subsection{小批量梯度下降} +前面两种方式的结合,将训练集分为$K$ 个mini-batch。对每一个小批量更新参数 +\[ + J^{(k)}\left( \theta \right) = \frac{1}{2N_{k}} \sum_{i=1}^{N_k} \left( y_i-f_{\theta}\left( x_{i} \right) \right) ^2 +.\] +\noindent \textbf{优点} +\begin{itemize} + \item 结合map-reduce可以比较容易实现并行 + \item 更新速度快,不确定性低 +\end{itemize} + +\subsection{基本搜索步骤} +随机选择初始化参数,走到局部最优值。 + +\begin{defi} + $f:\R^{n}\to \R$ 是凸函数:$\mathrm{dom} f$\footnote{dom 指函数定义域} 是一个凸集,并且 + \[ + f\left( tx_1+\left( 1-t \right) x_2 \right) \le tf\left( x_1 \right) + + \left( 1-t \right) f\left( x_2 \right) + .\] +\end{defi} +而凸函数是一定有最值的。 + +\section{从代数角度看线性回归} +\noindent \textbf{目标函数} +\[ + J\left( \bm{\mu} \right) = \frac{1}{2} \left( \bm{y} - \bm{X\mu} \right)^{T} + \left( \bm{y} - \bm{X\mu} \right) +.\] +而梯度为 +\[ + \frac{\partial J(\bm{\mu} )}{\partial \mu} + = - \bm{X}^{T} \left( \bm{y}-\bm{X\mu} \right) +.\] +甚至可以直接求出最优参数 +\[ + \frac{\partial J(\bm{\mu} )}{\partial \mu} = 0 \implies \hat{\mu} = \left( \bm{X^{T}X} \right) ^{-1} \bm{X}^{T} \bm{y} +.\] +但是当$\bm{X}$ 很大时,这是很难计算的。 + +\noindent \textbf{当$\bm{X^{T}X}$ 为奇异矩阵 } + +其逆矩阵无法计算,解决方法 +\begin{itemize} + \item Regulariazation + \item $J_1\left( \mu \right) = J\left( \mu \right) + \frac{\lambda}{2} \|\mu\|^2_{2} $ +\end{itemize} +从而 +\[ + \hat{\mu}= \left( \bm{X^{T}X } + \lambda \bm{I} \right) ^{-1} \bm{X^{T} y} +.\] +\subsection{泛线性模型} +本质上是做一个替换$ \bm x\to \phi\left( \bm x \right) $, +$\phi\left( x \right) $ 是$\R^{d}\to\R^{h}$ +的向量函数。加入了$\phi$ 的线性回归也叫核线性回归。 + +\subsection{核线性回归的矩阵形式} +使用\href{https://zhuanlan.zhihu.com/p/45223109}{线性代数技巧}得到: +\[ + \hat{y} = \Phi \hat{\theta} = \Phi\Phi^{T} \left( \Phi\Phi^{T} + \lambda I_{n} \right) ^{-1} y +.\] +只需关心核矩阵 +\[ + K = \Phi\Phi^{T} = \{ k\left( x^{(i)}, x^{(j)} \right) \} +.\] + +\section{最大似然估计} + +带高斯白噪声的线性拟合: +\[ + y = f_{\theta} \left( x \right) + \ep +.\] +其中$\ep \sim \mathcal{N} \left( 0, \sigma^2 \right) $. + +\noindent \textbf{优化目标} + +最大似然(Likelyhood). +\[ + p\left( y \mid x \right) =\frac{1}{\sqrt{2\pi \sigma^2} } e^{- \frac{\ep^2}{2\sigma^2} }= \frac{1}{\sqrt{2\pi \sigma^2} } e^{- \frac{y-\theta^{T}x }{2\sigma^2} } +.\] +最大化这个似然 +\[ + \mathrm{max}_{\theta} \prod_{i=1}^{N} p\left( y_{i} \mid x_{i} \right) +.\] +等价于最小均方误差的学习(取对数即可证明) + +\section{分类指标} +分类器有以下几个指标: +\begin{align*} + \mathrm{Accuracy} &= \frac{TP + TN}{TP + FP + TN + FN} \\ + \mathrm{Precision} &= \frac{TP}{TP+FP} \\ + \mathrm{Recall} &= \frac{TP}{TP+FN} +.\end{align*} +为了权衡准确率和召回率,可以调整阀值$h$: +\[ +\hat{y} = \begin{cases} + 1, \quad p_{\theta}\left( y=1 \mid x \right) > h \\ + 0, \quad \text{else} +\end{cases} +.\] + +为了判别分类器好不好,我们有F1 score +\[ +F_1 = \frac{2 \times \mathrm{Precision} \times \mathrm{Recall} }{ \mathrm{Precision} + \mathrm{Recall} } +.\] + +\subsection{ROC 曲线} +当$h$ 不断提升时,可以做出一条$\frac{TP}{P} \sim \frac{FP}{N}$曲线( +Receiver operating characteristic). +AUC\footnote{一般处于$\left( 0.5,1 \right) $} (Area under ROC curve) 可以用来衡量准确率 +,处于$0.75$以上时可以认为分类模型比较可靠。