table of Contents
- 1. Introduction (Introduction)
- 2 Univariate linear regression (Linear Regression with One Variable)
- 2.1 model representation (Model Representation)
- 2.2 cost function (Cost Function)
- 2.3 cost function - intuitive understanding 1 (Cost Function - Intuition I)
- 2.4 cost function - intuitive understanding 2 (Cost Function - Intuition II)
- 2.5 gradient descent (Gradient Descent)
- 2.6 gradient descent intuitive understanding (Gradient Descent Intuition)
- Gradient descent (Gradient Descent For Linear Regression) 2.7 linear regression
- 3 Linear Algebra Review
@
1. Introduction (Introduction)
1.1 Welcome
As the Internet continues to accumulate data, escalating hardware iteration, in this era of information explosion, machine learning has been applied in all walks of life, it is everywhere.
Some common applications of machine learning, such as:
- Handwriting recognition
- Spam Classification
- search engine
- Image Processing
- …
Use some cases Machine Learning:
- Data Mining
- Web clickstream data analysis
- Manual work can not handle (large)
- Handwriting recognition
- Computer Vision
- Personal customization
- Recommended system
- Brain research
- ……
1.2 What is machine learning (What is Machine Learning)
- Machine learning definitions
There are two main definitions:
Arthur Samuel (1959). Machine Learning: Field of study that gives computers the ability to learn without being explicitly programmed.
This definition is a bit formal, but the first time proposed, from a computer programming know how to play chess rookie. He wrote a program, but did not explicitly programmed every step of how to get there, but to himself and his chess computer, and continue to calculate the layout is good or bad, to determine high probability of winning under what circumstances, in order to gain experience, like learning, and finally, the computer program has become a powerful player than his own.
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 of more modern and formal definitions. In this example, filter spam, the e-mail system based on user mark (yes / no spam) e-mail constantly learning, so as to enhance the accuracy of spam filtering, the definition of three letters represent:
- T (Task): filter spam task.
- P (Performance): e-mail system spam filter accuracy.
- E (Experience): mark a user's e-mail.
Machine learning algorithms
There are two kinds of machine learning algorithm classification
- Supervised learning
- Unsupervised Learning
The difference is the need for human intervention annotation data results . A large proportion of the contents of the two parts, and very important, the master can save a lot a lot of time in the later application -
Some algorithms also belong to the field of machine learning, such as:
- Semi-supervised learning: supervised learning ranged between unsupervised learning
- Recommendation algorithm: Yes, that is after those who were buying a commodity is also recommended that the same section of the algorithm used by a shopping site.
- Reinforcement Learning: by observing to learn how to make a movement, every action will have an impact on the environment, and the environment but also the feedback to guide the learning algorithm.
- Transfer learning
1.3 Supervised Learning (Supervised Learning)
Supervised learning, that is, to teach the computer how to complete the prediction task (feedback), to advance a certain amount of input data and corresponding results of that is training set, model fitting, and let the computer predictions of unknown data.
Supervised learning generally two types:
Regression (Regression)
Regression is the predicted number of consecutive values .
In the case of housing price forecast, we are given a series of data groups housing surface, to predict housing prices to any area based on these data. Given photo - age data sets to predict given the age of the photo.
Classification (Classification)
Classification is the prediction of a series of discrete values .
I.e. the prediction data according to which category the object belongs to the prediction.
Video cites the example of cancerous tumors, for diagnosis were classified as benign or malignant. Also, for example spam classification issues, also belong to the classification of supervised learning.
Video mentioned in support vector machine algorithm, to address the situation when the characteristic large amount of time (ie, features a variety of features such as tumor size, color, odor and other cancers in the example), computer memory will not enough. SVM computer processing allows an unlimited number of features.
1.4 unsupervised learning (Unsupervised Learning)
相对于监督学习，训练集不会有人为标注的结果（无反馈），我们不会给出结果或无法得知训练集的结果是什么样，而是单纯由计算机通过无监督学习算法自行分析，从而“得出结果”。计算机可能会把特定的数据集归为几个不同的簇，故叫做聚类算法。
无监督学习一般分为两种：
- 聚类(Clustering)
- 新闻聚合
- DNA 个体聚类
- 天文数据分析
- 市场细分
- 社交网络分析
- 非聚类(Non-clustering)
- 鸡尾酒问题
新闻聚合
在例如谷歌新闻这样的网站中，每天后台都会收集成千上万的新闻，然后将这些新闻分组成一个个的新闻专题，这样一个又一个聚类，就是应用了无监督学习的结果。
鸡尾酒问题
在鸡尾酒会上，大家说话声音彼此重叠，几乎很难分辨出面前的人说了什么。我们很难对于这个问题进行数据标注，而这里的通过机器学习的无监督学习算法，就可以将说话者的声音同背景音乐分离出来，看视频，效果还不错呢~~。
嗯，这块是打打鸡血的，只需要一行代码就解决了问题，就是这么简单！当然，我没复现过 ^_^……
神奇的一行代码：
[W,s,v] = svd((repmat(sum(x.*x,1),size(x,1),1).*x)*x');
编程语言建议
在机器学习刚开始时，推荐使用 Octave 类的工程计算编程软件，因为在 C++ 或 Java 等编程语言中，编写对应的代码需要用到复杂的库以及要写大量的冗余代码，比较耗费时间，建议可以在学习过后再考虑使用其他语言来构建系统。
另外，在做原型搭建的时候也应该先考虑使用类似于 Octave 这种便于计算的编程软件，当其已经可以工作后，才将模型移植到其他的高级编程语言中。
注：Octave 与 MATLAB 语法相近，由于 MATLAB 为商业软件，课程中使用开源且免费的 Octave。
机器学习领域发展迅速，现在也可使用 Tensorflow 等开源机器学习框架编写机器学习代码，这些框架十分友好，易于编写及应用。
2 单变量线性回归(Linear Regression with One Variable)
2.1 模型表示(Model Representation)
- 房价预测训练集
| Size in $feet^2$ ($x$) | Price ($) in 1000's($y$) |
|---|---|
| 2104 | 460 |
| 1416 | 232 |
| 1534 | 315 |
| 852 | 178 |
| ... | ... |
房价预测训练集中，同时给出了输入 $x$ 和输出结果 $y$，即给出了人为标注的”正确结果“，且预测的量是连续的，属于监督学习中的回归问题。
- 问题解决模型
[外链图片转存失败(img-kjB3RLia-1568369880643)(images/20180105_212048.png)]
其中 $h$ 代表结果函数，也称为假设(hypothesis) 。假设函数根据输入(房屋的面积)，给出预测结果输出(房屋的价格)，即是一个 $X\to Y$ 的映射。
$h_\theta(x)=\theta_0+\theta_1x$，为解决房价问题的一种可行表达式。
$x$: 特征/输入变量。
上式中，$\theta$ 为参数，$\theta$ 的变化才决定了输出结果，不同以往，这里的 $x$ 被我们视作已知(不论是数据集还是预测时的输入)，所以怎样解得 $\theta$ 以更好地拟合数据，成了求解该问题的最终问题。
单变量，即只有一个特征(如例子中房屋的面积这个特征)。
2.2 代价函数(Cost Function)
李航《统计学习方法》一书中，损失函数与代价函数两者为同一概念，未作细分区别，全书没有和《深度学习》一书一样混用，而是统一使用损失函数来指代这类类似概念。
吴恩达(Andrew Ng)老师在其公开课中对两者做了细分。如果要听他的课做作业，不细分这两个概念是会被打小手扣分的！这也可能是因为老师发现了业内混用的乱象，想要治一治吧。
损失函数(Loss/Error Function): 计算单个样本的误差。
代价函数(Cost Function): 计算整个训练集所有损失函数之和的平均值
我们的目的在于求解预测结果 $h$ 最接近于实际结果 $y$ 时 $\theta$ 的取值，则问题可表达为求解 $\sum\limits_{i=0}^{m}(h_\theta(x^{(i)})-y^{(i)})$ 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
[外链图片转存失败(img-VyozXzkj-1568369880643)(images/20180105_224648.png)]
上图展示了当 $\theta$ 取不同值时，$h_\theta\left(x\right)$ 对数据集的拟合情况，蓝色虚线部分代表建模误差（预测结果与实际结果之间的误差），我们的目标就是最小化所有误差之和。
为了求解最小值，引入代价函数(Cost Function)概念，用于度量建模误差。考虑到要计算最小值，应用二次函数对求和式建模，即应用统计学中的平方损失函数（最小二乘法）：
$$J(\theta_0,\theta_1)=\dfrac{1}{2m}\displaystyle\sum_{i=1}^m\left(\hat{y}{i}-y{i} \right)^2=\dfrac{1}{2m}\displaystyle\sum_{i=1}^m\left(h_\theta(x_{i})-y_{i}\right)^2$$
$\hat{y}$: $y$ 的预测值
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，平方的导数会抵消掉 $\frac{1}{2}$ 。
讨论到这里，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 代价函数 - 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): $\theta_0, \theta_1$
- 代价函数(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): $\underset{\theta_0, \theta_1}{\text{minimize}} J \left(\theta_0, \theta_1 \right)$
为了直观理解代价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$\left(1, 1\right), \left(2, 2\right), \left(3, 3\right)$，这样在平面坐标系中绘制出 $h_\theta\left(x\right)$ ，并分析 $J\left(\theta_0, \theta_1\right)$ 的变化。
右图 $J\left(\theta_0, \theta_1\right)$ 随着 $\theta_1$ 的变化而变化，可见当 $\theta_1 = 1$ 时，$J\left(\theta_0, \theta_1 \right) = 0$，取得最小值，对应于左图青色直线，即函数 $h$ 拟合程度最好的情况。
2.4 代价函数 - 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可跳过。
给定数据集：
参数在 $\theta_0$ 不恒为 $0$ 时代价函数 $J\left(\theta\right)$ 关于 $\theta_0, \theta_1$ 的3-D图像，图像中的高度为代价函数的值。
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（下图中的右图）来作直观理解，其中相同颜色的一个圈代表着同一高度（同一 $J\left(\theta\right)$ 值）。
$\theta_0 = 360, \theta_1 =0$ 时：
大概在 $\theta_0 = 0.12, \theta_1 =250$ 时：
上图中最中心的点（红点），近乎为图像中的最低点，也即代价函数的最小值，此时对应 $h_\theta\left(x\right)$ 对数据的拟合情况如左图所示，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)
在特征量很大的情况下，即便是借用计算机来生成图像，人工的方法也很难读出 $J\left(\theta\right)$ 的最小值，并且大多数情况无法进行可视化，故引入梯度下降(Gradient Descent)方法，让计算机自动找出最小化代价函数时对应的 $\theta$ 值。
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，计算代价函数，然后寻找下一个能使得代价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。
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，直到我们到达山下的某一处陆地。
梯度下降公式：
$\begin{align} & \text{repeat until convergence:} ; \lbrace \newline ; &{{\theta }{j}}:={{\theta }{j}}-\alpha \frac{\partial }{\partial {{\theta }{j}}}J\left( {\theta{0}},{\theta_{1}} \right) \newline \rbrace \end{align}$
${\theta }_{j}$: 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
公式中，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“。
注意，在计算时要批量更新 $\theta$ 值，即如上图中的左图所示，否则结果上会有所出入，原因不做细究。
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 $\theta_1 := \theta_1 - \alpha\frac{d}{d\theta_1}J\left(\theta_1\right)$，此处为了数学定义上的精确性，用的是 $\frac{d}{d\theta_1}J\left(\theta_1\right)$，如果不熟悉微积分学，就把它视作之前的 $\frac{\partial}{\partial\theta}$ 即可。
把红点定为初始点，切于初始点的红色直线的斜率，表示了函数 $J\left(\theta\right)$ 在初始点处有正斜率，也就是说它有正导数，则根据梯度下降公式 ，${{\theta }{j}}:={{\theta }{j}}-\alpha \frac{\partial }{\partial {{\theta }_{j}}}J\left( \theta_0, \theta_1 \right)$ 右边的结果是一个正值，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）。
初始 $\theta$ 值（初始点）是任意的，若初始点恰好就在极小值点处，梯度下降算法将什么也不做（$\theta_1 := \theta_1 - \alpha*0$）。
不熟悉斜率的话，就当斜率的值等于图中三角形的高度除以水平长度好啦，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能使得梯度下降算法运行良好。
- 学习速率过小图示：
收敛的太慢，需要更多次的迭代。
- 学习速率过大图示：
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，随着斜率越来越接近于0，代价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，代价函数随着迭代的进行，变化的幅度越来越小。
最后，梯度下降不止可以用于线性回归中的代价函数，还通用于最小化其他的代价函数。
2.7 线性回归中的梯度下降(Gradient Descent For Linear Regression)
线性回归模型
- $h_\theta(x)=\theta_0+\theta_1x$
- $J\left( \theta_0, \theta_1 \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}^{2}}}$
梯度下降算法
- $\begin{align} & \text{repeat until convergence:} ; \lbrace \newline ; &{{\theta }{j}}:={{\theta }{j}}-\alpha \frac{\partial }{\partial {{\theta }{j}}}J\left( {\theta{0}},{\theta_{1}} \right) \newline \rbrace \end{align}$
直接将线性回归模型公式代入梯度下降公式可得出公式
当 $j = 0, j = 1$ 时，线性回归中代价函数求导的推导过程：
$\frac{\partial}{\partial\theta_j} J(\theta_1, \theta_2)=\frac{\partial}{\partial\theta_j} \left(\frac{1}{2m}\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}^{2}} \right)=$
$\left(\frac{1}{2m}*2\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}} \right)*\frac{\partial}{\partial\theta_j}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}} =$
$\left(\frac{1}{m}\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}} \right)*\frac{\partial}{\partial\theta_j}{{\left(\theta_0{x_0^{(i)}} + \theta_1{x_1^{(i)}}-{{y}^{(i)}} \right)}}$
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：
$\frac{\partial}{\partial\theta_1} J(\theta)=\frac{1}{m}\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}} *x_1^{(i)}$
Gradient descent mentioned above, are batch gradient descent (Batch Gradient Descent), i.e. each time calculations are used for all data sets $ \ left (\ sum \ limits_ {i = 1} ^ {m} \ right) $ updates.
Since the linear regression function showing the bowl , and only a global optimal value, the function must always converge to the global minimum (learning rate is not too large). Meanwhile, the function is called $ $ J convex quadratic function , the minimum of the linear regression functions belonging Solving Convex Optimization Function .
Further, use of recycled solving code, code redundancy, will be mentioned later how to quantization (the Vectorization) to simplify the code optimization and the gradient descent run faster and better.
3 Linear Algebra Review
This part, studied linear algebra can review the basis of comparison. Note finishing persistence.
3.1 Matrices and Vectors
Octave / Matlab code for:
% The ; denotes we are going back to a new row.
A = [1, 2, 3; 4, 5, 6; 7, 8, 9; 10, 11, 12]
% Initialize a vector
v = [1;2;3]
% Get the dimension of the matrix A where m = rows and n = columns
[m,n] = size(A)
% You could also store it this way
dim_A = size(A)
% Get the dimension of the vector v
dim_v = size(v)
% Now let's index into the 2nd row 3rd column of matrix A
A_23 = A(2,3)Results of the:
A =
1 2 3
4 5 6
7 8 9
10 11 12
v =
1
2
3
m = 4
n = 3
dim_A =
4 3
dim_v =
3 1
A_23 = 63.2 Addition and Scalar Multiplication
Octave / Matlab code for:
% Initialize matrix A and B
A = [1, 2, 4; 5, 3, 2]
B = [1, 3, 4; 1, 1, 1]
% Initialize constant s
s = 2
% See how element-wise addition works
add_AB = A + B
% See how element-wise subtraction works
sub_AB = A - B
% See how scalar multiplication works
mult_As = A * s
% Divide A by s
div_As = A / s
% What happens if we have a Matrix + scalar?
add_As = A + sResults of the:
A =
1 2 4
5 3 2
B =
1 3 4
1 1 1
s = 2
add_AB =
2 5 8
6 4 3
sub_AB =
0 -1 0
4 2 1
mult_As =
2 4 8
10 6 4
div_As =
0.50000 1.00000 2.00000
2.50000 1.50000 1.00000
add_As =
3 4 6
7 5 43.3 Matrix Vector Multiplication
Octave / Matlab code for:
% Initialize matrix A
A = [1, 2, 3; 4, 5, 6;7, 8, 9]
% Initialize vector v
v = [1; 1; 1]
% Multiply A * v
Av = A * v
Results of the:
A =
1 2 3
4 5 6
7 8 9
v =
1
1
1
Av =
6
15
24
3.4 Matrix Matrix Multiplication
Octave / Matlab code for:
% Initialize a 3 by 2 matrix
A = [1, 2; 3, 4;5, 6]
% Initialize a 2 by 1 matrix
B = [1; 2]
% We expect a resulting matrix of (3 by 2)*(2 by 1) = (3 by 1)
mult_AB = A*B
% Make sure you understand why we got that resultResults of the:
A =
1 2
3 4
5 6
B =
1
2
mult_AB =
5
11
17
3.5 Matrix Multiplication Properties
Octave / Matlab code for:
% Initialize random matrices A and B
A = [1,2;4,5]
B = [1,1;0,2]
% Initialize a 2 by 2 identity matrix
I = eye(2)
% The above notation is the same as I = [1,0;0,1]
% What happens when we multiply I*A ?
IA = I*A
% How about A*I ?
AI = A*I
% Compute A*B
AB = A*B
% Is it equal to B*A?
BA = B*A
% Note that IA = AI but AB != BAResults of the:
A =
1 2
4 5
B =
1 1
0 2
I =
Diagonal Matrix
1 0
0 1
IA =
1 2
4 5
AI =
1 2
4 5
AB =
1 5
4 14
BA =
5 7
8 103.6 Inverse and Transpose
Octave / Matlab code for:
% Initialize matrix A
A = [1,2,0;0,5,6;7,0,9]
% Transpose A
A_trans = A'
% Take the inverse of A
A_inv = inv(A)
% What is A^(-1)*A?
A_invA = inv(A)*A
Results of the:
A =
1 2 0
0 5 6
7 0 9
A_trans =
1 0 7
2 5 0
0 6 9
A_inv =
0.348837 -0.139535 0.093023
0.325581 0.069767 -0.046512
-0.271318 0.108527 0.038760
A_invA =
1.00000 -0.00000 0.00000
0.00000 1.00000 -0.00000
-0.00000 0.00000 1.00000