Coursera 学习笔记｜Machine Learning by Standford University - 吴恩达

/ 20220404 Week 1 - 2 /

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Chapter 1 - Introduction

1.1 Definition

• 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 Concepts

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

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

2.1 Linear Regression with One Variable 单元线性回归

• 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

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

2.2 Multivariate Linear Regression 多元线性回归

• 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 Algorithm Optimization

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：

1. 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；
2. 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。

2.4 Polynomial Regression 多项式回归

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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