Chapter 1 - Introduction

Arthur Samuel

The field of study that gives computers the ability to learn without being explicitly programmed.
Tom Mitchell

A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P, if its performance at tasks in T, as measured by P, improves with experience E.

1.2.1 Classification of Machine Learning

Supervised Learning 监督学习：given a labeled data set; already know what a correct output/result should look like
- Regression 回归：continuous output
- Classification 分类：discrete output
Unsupervised Learning 无监督学习：given an unlabeled data set or an data set with the same labels; group the data by ourselves
- Clustering 聚类：group the data into different clusters
- Non-Clustering 非聚类
Others: Reinforcement Learning, Recommender Systems…

1.2.2 Model Representation

Training Set 训练集

\[\begin{matrix}
x^{(1)}_1&x^{(1)}_2&\cdots&x^{(1)}_n&&y^{(1)}\\
x^{(2)}_1&x^{(2)}_2&\cdots&x^{(2)}_n&&y^{(2)}\\
\vdots&\vdots&\ddots&\vdots&&\vdots\\
x^{(m)}_1&x^{(m)}_2&\cdots&x^{(m)}_n&&y^{(m)}
\end{matrix}\]
符号说明

$m=$ the number of training examples 训练样本的数量 - 行数

$n=$ the number of features 特征数量 - 列数

$x=$ input variable/feature 输入变量/特征

$y=$ output variable/target variable 输出变量/目标变量

$(x^{(i)}_j,y^{(i)})$ ：第$j$个特征的第 $i$ 个训练样本，其中 $i=1, …, m$，$j=1, …, n$

1.2.3 Cost Function 代价函数

1.2.4 Gradient Descent 梯度下降

Chapter 2 - Linear Regression 线性回归

\[\begin{matrix}
x_0&x^{(1)}_1&x^{(1)}_2&\cdots&x^{(1)}_n&&y^{(1)}\\
x_0&x^{(2)}_1&x^{(2)}_2&\cdots&x^{(2)}_n&&y^{(2)}\\
\vdots&\vdots&\vdots&\ddots&\vdots&&\vdots\\
x_0&x^{(m)}_1&x^{(m)}_2&\cdots&x^{(m)}_n&&y^{(m)}\\
\\
\theta_0&\theta_1&\theta_2&\cdots&\theta_n&&
\end{matrix}\]

Hypothesis Function

\[h_{\theta}(x)=\theta_0+\theta_1x
\]
Cost Function - Square Error Cost Function 平方误差代价函数

\[J(\theta_0,\theta_1)=\frac{1}{2m}\displaystyle\sum_{i=1}^m(h_{\theta}(x^{(i)})-y^{(i)})^2
\]

Goal

\[\min_{(\theta_0,\theta_1)}J(\theta_0,\theta_1)
\]
Hypothesis Function

\[\theta=
\left[
\begin{matrix}
\theta_0\\
\theta_1\\
\vdots\\
\theta_n
\end{matrix}
\right],\
x=
\left[
\begin{matrix}
x_0\\
x_1\\
\vdots\\
x_n
\end{matrix}
\right]\]

\[\begin{aligned}h_\theta(x)&=\theta_0+\theta_1x_1+\theta_2x_2+\cdots+\theta_nx_n\\
&=\theta^Tx
\end{aligned}\]
Cost Function

\[J(\theta^T)=\frac{1}{2m}\displaystyle\sum_{i=1}^m(h_{\theta}(x^{(i)})-y^{(i)})^2
\]
Goal

\[\min_{\theta^T}J(\theta^T)
\]

2.3.1 Gradient Descent 梯度下降法

算法过程

Repeat until convergence(simultaneous update for each $j=1, …, n$)

\[\begin{aligned}
\theta_j
&:=\theta_j-\alpha{\partial\over\partial\theta_j}J(\theta^T)\\
&:=\theta_j-\alpha{1\over{m}}\displaystyle\sum_{i=1}^m(h_{\theta}(x^{(i)})-y^{(i)})x^{(i)}_j
\end{aligned}\]

Feature Scaling 特征缩放

对每个特征 $x_j$ 有$$x_j={{x_j-\mu_j}\over{s_j}}$$

其中 $\mu_j$ 为 $m$ 个特征 $x_j$ 的平均值，$s_j$ 为 $m$ 个特征 $x_j$ 的范围（最大值与最小值之差）或标准差。
Learning Rate 学习率

2.3.2 Normal Equation(s) 正规方程（组）

令

\[X=\left[
\begin{matrix}
x_0&x^{(1)}_1&x^{(1)}_2&\cdots&x^{(1)}_n\\
x_0&x^{(2)}_1&x^{(2)}_2&\cdots&x^{(2)}_n\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
x_0&x^{(m)}_1&x^{(m)}_2&\cdots&x^{(m)}_n\\
\end{matrix}
\right],\
y=\left[
\begin{matrix}
y^{(1)}\\
y^{(2)}\\
\vdots\\
y^{(m)}\\
\end{matrix}
\right]\]

其中 $X$ 为 $m\times(n+1)$ 维矩阵，$y$ 为 $m$ 维的列向量。则

\[\theta=(X^TX)^{-1}X^Ty
\]

如果 $X^TX$ 不可逆（noninvertible），可能是因为：

Redundant features 冗余特征：存在线性相关的两个特征，需要删除其中一个；
特征过多，如 $m\leq n$：需要删除一些特征，或对其进行正规化（regularization）处理。

If a linear $h_\theta(x)$ can't fit the data well, we can change the behavior or curve of $h_\theta(x)$ by making it a quadratic, cubic or square root function(or any other form).

e.g.

$h_{\theta}(x)=\theta_0+\theta_1x_1+\theta_2x_1^2,\ x_2=x_1^2$
$h_{\theta}(x)=\theta_0+\theta_1x_1+\theta_2x_1^2+\theta_3x_1^3,\ x_2=x_1^2,\ x_3=x_1^3$
$h_{\theta}(x)=\theta_0+\theta_1x_1+\theta_2\sqrt{x_1},\ x_2=\sqrt{x_1}$

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