From 73e9df836d009d54f30053fce2b8ad1b482a60ed Mon Sep 17 00:00:00 2001 From: scruel Date: Sat, 6 Jan 2018 20:56:05 +0800 Subject: [PATCH] finish "week1 - 2 " MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 单变量线性回归(Linear Regression with One Variable) --- notes/image/20180106_085915.png | Bin 0 -> 300710 bytes notes/image/20180106_090904.png | Bin 0 -> 163134 bytes notes/image/20180106_091307.png | Bin 0 -> 153417 bytes notes/image/20180106_092119.png | Bin 0 -> 121601 bytes notes/image/20180106_101659.png | Bin 0 -> 79361 bytes notes/image/20180106_184926.png | Bin 0 -> 45624 bytes notes/image/20180106_190944.png | Bin 0 -> 11963 bytes notes/image/20180106_191023.png | Bin 0 -> 16747 bytes notes/image/20180106_191956.png | Bin 0 -> 51545 bytes notes/image/20180106_203726.png | Bin 0 -> 70972 bytes .../db48c81304317847870d486ba5bb2015.jpg | Bin 0 -> 61973 bytes notes/week1.md | 152 ++++++++++++++++-- 12 files changed, 142 insertions(+), 10 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--- a/notes/week1.md +++ b/notes/week1.md @@ -34,7 +34,7 @@ 这个定义是非正式但是是最早的,来自于一个会计算机编程的下棋菜鸟,计算机通过不断的对弈,通过对弈计算布局的好坏,通过不断“学习”,积累经验,成为了一个厉害的棋手。 - - Tom Mitchell (1998) Well-posed Learning Problem: A computer program is said to learn from experience E with respect to some task T and some performance measure P, if its performance on T, as measured by P, improves with experience E. + - Tom Mitchell (1998) Well-posed Learning Problem: A computer program is said to learn from experience E with respect to some **task T** and some **performance measure P**, if its performance on T, as measured by P, improves with **experience E**. 这个定义是第一个正式的机器学习定义,有点拗口,视频介绍了一个例子,即垃圾邮件分类的例子,对于垃圾邮件分类,三个字母分别代表: @@ -137,42 +137,174 @@ $h_\theta(x)=\theta_0+\theta_1x$,为其中一种可行的表达式。 +> $x$: 特征/输入变量。 + 上式中,$\theta$ 为参数,$\theta$ 的变化决定了输出结果,不同以往,这里的 $x$ 被我们**视作已知**(即不论是数据集还是预测前的输入),所以怎样解得 $\theta$ 以拟合数据,成了求解该问题的最终问题。 -单变量即只有一个特征(房屋的面积) +单变量,即只有一个特征(如例子中房屋的面积这个特征)。 ## 2.2 损失函数(Cost Function) 为了求解上个视频中提到的 $\theta$,本视频引入了损失函数(Cost Function)这个概念,损失函数,用于度量预测结果于实际结果的**建模误差**。 -目的在于求解预测结果 $h_\theta(x)$ 最接近于实际结果 $y$ 时 $\theta$ 的取值,则可以把问题转化为**求解 $\sum\limits_{i=0}^{n}(h_\theta(x^{(i)})-y^{(i)})$ 的最小值**。 +目的在于求解预测结果 $h_\theta(x)$ 最接近于实际结果 $y$ 时 $\theta$ 的取值,则可以把问题转化为**求解 $\sum\limits_{i=0}^{m}(h_\theta(x^{(i)})-y^{(i)})$ 的最小值**。 + +> m: 训练集中的样本总数 +> +> y: 目标变量/输出变量 +> +> $\left(x, y\right)$: 训练集中的实例 +> +> $\left(x^\left(i\right), y^\left(i\right)\right)$: 训练集中的第 $i$ 个样本实例 -![](image\20180105_224648.png) +![](image/20180105_224648.png) 上图中,为在 $\theta$ 取不同值时的拟合情况,蓝色虚线代表的即为误差,要最小化的就是每个预测点的误差值。 这里则要引入损失函数的概念,我们对求最小值的式子进行数学建模,让其变得易于求最小值,而其中一种办法就是建模为二次函数,即统计学上的平方损失函数(最小二乘法): -$$ J\left( \theta \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}^{2}}} $$ ,其中 $m$ 为训练集中的样本数量。 +$$ J\left( \theta \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}^{2}}} $$ + +> 系数 $\frac{1}{2}$ 存在与否都不会影响结果,这里是为了在应用梯度下降时便于求解。 对于之前的房价问题来说,我们的问题就转化成了**求解 $J\left( \theta_0, \theta_1 \right)$ 的最小值**。 -## 2.3 Cost Function - Intuition I +## 2.3 损失函数 - 直观理解1(Cost Function - Intuition I) + +根据上节视频,列出如下定义: + +- 假设函数(Hypothesis): $h_\theta(x)=\theta_0+\theta_1x$ +- 参数(Parameters): $\theta_0, \theta_1$ +- 损失函数(Cost Function): $ J\left( \theta_0, \theta_1 \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}^{2}}} $ +- 目标(Goal): $\underset{\theta_0, \theta_1}{\text{minimize}} J \left(\theta_0, \theta_1 \right)$ + +为了直观的理解损失函数到底是在做什么,先假设 $\theta_1 = 0$,并假设训练集有三个数据,分别为$\left(1, 1\right), \left(2, 2\right), \left(3, 3\right)$。 + +TODO: 可更换为动图<--> + +![](image/20180106_085915.png) + +上图显示了 $J\left(\theta_0, \theta_1\right)$ 随着 $\theta_1$ 的变化而变化,**当 $\theta_1 = 1$ 时,$J\left(\theta_0, \theta_1 \right) = 0$,取得最小值。** + +## 2.4 损失函数 - 直观理解2(Cost Function - Intuition II) + +> 注:该部分由于涉及到了多变量成像,可能较难理解,要求只需要理解上节内容即可,该节如果不能较好理解也可跳过。 + +数据集: + +![](image\20180106_091307.png) + +下图是在该数据集中,平方损失函数在 $\theta_0 \neq 0$ 时绘制出的3-D图像,空间的高度则为 $J\left(\theta\right)$ 的值。 + +![](image/20180106_090904.png) + +由于3-D图形不便于标注,所以将3-D图形转换为**轮廓图(contour plot)**,下面用轮廓图来作直观理解,即下图中的右图。 + +![](image/20180106_092119.png) + +轮廓图中,相同颜色的一圈代表同一高度(同一 $J\left(\theta\right)$ 值),最中心的部分,即红点的部分,就是这个损失函数的最小值,对应的 $h_\theta\left(x\right)$ 以及对数据的拟合情况就是左图啦,嗯,一看就拟合的很不错,预测应该比较精准啦 + +## 2.5 梯度下降(Gradient Descent) + +由于数据量、特征量会随着数据集的增长而越来越大,即便是借用计算机来生成图像,人工的方法也无法读出 $J\left(\theta\right)$ 的最小值,并且大多数情况无法进行可视化,所以引入**梯度下降(Gradient Descent),让计算机自动找出 $\theta$ 值,最小化损失函数。** + +梯度下降背后的思想是:开始时,我们随机选择一个参数组合$\left( {\theta_{0}},{\theta_{1}},......,{\theta_{n}} \right)$即起始点,计算损失函数,然后寻找下一个能使得损失函数下降最多的参数组合。不断迭代,直到找到一个**局部最小值(local minimum)**,由于下降的情况只考虑当前参数组合周围的情况,所以无法确定当前的局部最小值是否就是**全局最小值(global minimum)**,不同的初始参数组合,可能会产生不同的局部最小值。 + +下图根据不同的起始点,产生了两个不同的局部最小值。 + +![](image/db48c81304317847870d486ba5bb2015.jpg) + +视频中举了下山的例子,即我们在山顶上的某个位置,为了下山,就不断地看一下周围**下一步往哪走**下山比较快,然后就**迈出那一步**,一直重复,直到我们到达某一块山下的**陆地**。 + +先给出梯度下降的公式: +$$ +{{\theta }_{j}}:={{\theta }_{j}}-\alpha \frac{\partial }{\partial {{\theta }_{j}}}J\left( \theta_0, \theta_1 \right) +$$ + +> ${\theta }_{j}$: 第 $j$ 个特征参数 +> +> ”:=“: 赋值操作符 +> +> $\alpha$: 学习速率(learning rate), $\alpha > 0$ +> +> $\frac{\partial }{\partial {{\theta }_{j}}}J\left( \theta_0, \theta_1 \right)$: $J\left( \theta_0, \theta_1 \right)$ 的偏导 + +学习速率决定了参数值变化的速率即”**走多少距离**“,而偏导这部分决定了下降的方向即”**下一步往哪里**“走(当然实际上的走多少距离是由偏导值给出的,学习速率起到调整后决定的作用),收敛处的局部最小值又叫做极小值,即”**陆地**“。 + +![](image/20180106_101659.png) -## 2.4 Cost Function - Intuition II +这里非常重要的一个点是,在计算时**要批量更新 $\theta$ 值**,否则结果上会有所出入,原因不做细究。 -## 2.5 Gradient Descent +## 2.6 梯度下降直观理解(Gradient Descent Intuition) +该节探讨 $\theta_1$ 的梯度下降更新过程,即 $\theta_1 := \theta_1 - \alpha\frac{d}{d\theta_1}J\left(\theta_1\right)$,注意到这里为了数学定义上的精确性,是 $\frac{d}{d\theta_1}J\left(\theta_1\right)$,如果不熟悉微积分学,就把它视作之前的 $\frac{\partial}{\partial\theta}$ 即可。 +![](image/20180106_184926.png) -## 2.6 Gradient Descent Intuition +把红点定为初始点,切于初始点的红色直线的斜率,代表 $J\left(\theta\right)$ 在初始点处有**正斜率**,也就是说它有**正导数**,则 $\theta_1$ 等于 初始值 - 学习速率 * 导数,即 $\theta_1$ 会**向左边移动**。这样不断重复,直到收敛(达到局部最小值,即斜率为0,所以如果 $\theta$ 值已经在极小值点处时,梯度下降算法将什么也不做)。 -## 2.7 Gradient Descent For Linear Regression +> 不熟悉斜率的话,可以理解为斜率的值等于图中三角形的高度除以水平长度,精确地求斜率的方法是求导。 + +对于学习速率 $\alpha$ ,需要选取一个合适的值才可以收敛。 + +- 学习速率过小: + + ![](image/20180106_190944.png) + + 收敛的太慢,需要更多次的迭代。 + +- 学习速率过大: + + ![](image/20180106_191023.png) + + 可能越过最低点,甚至导致无法收敛。 + +学习速率只需选定即可,不需要在运行梯度下降算法的时候进行动态改变,因为由于斜率的变化(越来越接近于0),损失函数的变化幅度会越来越小,直到收敛到局部极小值。 + +如图,品红色点为初始点,损失函数随着迭代的进行,变化的幅度越来越小。 + +![](image/20180106_191956.png) + +梯度下降不止可以用于线性回归中的损失函数,还通用于最小化其他的损失函数。 + +## 2.7 线性回归中的梯度下降(Gradient Descent For Linear Regression) + +将线性回归模型和梯度下降算法结合,则可得出单变量线性问题的解。 + +- $h_\theta(x)=\theta_0+\theta_1x$ +- $ J\left( \theta_0, \theta_1 \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}^{2}}} $ +- ${{\theta }_{j}}:={{\theta }_{j}}-\alpha \frac{\partial }{\partial {{\theta }_{j}}}J\left( \theta_0, \theta_1 \right)$ + +这里将线性回归模型公式直接带入梯度下降公式即可得出公式: + +![](image/20180106_203726.png) + +其实上文中提到的梯度下降,都为批量梯度下降(Batch Gradient Descent),即每次计算都使用所有的数据集 $\left(\sum\limits_{i=1}^{m}\right)$ 更新。 + +视频中直接给出了 $j = 0, j = 1$ 时求解偏导的计算方法,这里给出推导过程如下: + +$\frac{\partial}{\partial\theta_j} J(\theta_1, \theta_2)=\frac{\partial}{\partial\theta_j} \left(\frac{1}{2m}\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}^{2}} \right)=$ + +$\left(\frac{1}{2m}*2\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}} \right)*\frac{\partial}{\partial\theta_j}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}} =$ + +$\left(\frac{1}{m}\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}} \right)*\frac{\partial}{\partial\theta_j}{{\left(\theta_0{x_0^{(i)}} + \theta_1{x_1^{(i)}}-{{y}^{(i)}} \right)}}$ + +所以当 $j = 0$ 时: + +$\frac{\partial}{\partial\theta_0} J(\theta)=\frac{1}{m}\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}} *x_0^{(i)}$ + +所以当 $j = 1$ 时: + +$\frac{\partial}{\partial\theta_1} J(\theta)=\frac{1}{m}\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}} *x_1^{(i)}$ + +使用循环代码较为冗余,后面会讲到如何使用**向量化(Vectorization)**来简化代码并优化计算,使得梯度下降运行的更快。 # 3 Linear Algebra Review +这部分,学过线性代数的可以复习一下,比较基础。笔记整理暂留。 + ## 3.1 Matrices and Vectors ## 3.2 Addition and Scalar Multiplication

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