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zzO7yQ0zI9RDW+8#QZ6Bfz^je`Gt=W&9d)k$90C*&;Fy*IcwABr0VWXOn3f6DxiUEf zC?LQwEd}tnq#OcFAiyy#6R2}#atKgBfMZ$;;BiSg1eieJe_Z7pWp{g-VgLXD07*qo IM6N<$f;cz6CIA2c literal 0 HcmV?d00001 diff --git a/notes/week1.md b/notes/week1.md index 206bd29..93d3bd6 100644 --- a/notes/week1.md +++ b/notes/week1.md @@ -33,11 +33,11 @@ - Arthur Samuel (1959). Machine Learning: Field of study that gives computers the ability to learn without being explicitly programmed. - 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。 + 这个定义有点不正式但提出的时间最早,来自于一个懂得计算机编程的下棋菜鸟,编程使得计算机通过不断的对弈,不断地计算布局的好坏来“学习”,从而积累经验,这样,这个计算机程序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**. - 此定义是**第一个正式的机器学习定义**,有点拗口,视频中介绍了一个例子,即垃圾邮件分类。对于垃圾邮件分类,文中的三个字母分别代表: + Tom Mitchell 的定义更现代,也有点拗口,视频中介绍了一个例子,即垃圾邮件分类。对于垃圾邮件分类,文中的三个字母分别代表: - T(task): 对垃圾邮件分类这个任务。 - P(Performance): 垃圾邮件分类的准确程度。 @@ -69,7 +69,7 @@ 回归问题即为预测一系列的**连续值**。 - 在房屋价格预测的例子中,给出了一系列的房屋面基数据,根据这些数据来预测任意面积的房屋价格。 + 在房屋价格预测的例子中,给出了一系列的房屋面基数据,根据这些数据来预测任意面积的房屋价格。给出照片-年龄数据集,预测给定照片的年龄。 ![](image\20180105_194712.png) @@ -77,7 +77,9 @@ 分类问题即为预测一系列的**离散值**。 - 即根据数据预测被预测对象属于哪个分类。视频中举了癌症肿瘤这个例子,针对诊断结果,分别分类为良性或恶性。上个视频中的垃圾邮件分类问题,也同样属于监督学习中的分类问题。 + 即根据数据预测被预测对象属于哪个分类。 + + 视频中举了癌症肿瘤这个例子,针对诊断结果,分别分类为良性或恶性。还例如垃圾邮件分类问题,也同样属于监督学习中的分类问题。 ![](image\20180105_194839.png) @@ -86,21 +88,24 @@ ## 1.4 无监督学习(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),而是由计算机通过无监督学习算法来自行分析,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。 +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。 -无监督学习一般由两种: +无监督学习一般分为两种: 1. 聚类(Clustering) -2. 关联(Associative) + - 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 +2. 非聚类(Non-clustering) + - 鸡尾酒问题 + +**新闻聚合** -这里列举一些无监督学习的例子: +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。 -- 新闻聚合分类 -- DNA 个体聚类 -- 社交网络 -- 市场细分 -- 天文数据分析 +**鸡尾酒问题** -**例子,鸡尾酒问题** ![](image/20180105_201639.png) 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\~~。 @@ -110,7 +115,9 @@ 神奇的一行代码: `[W,s,v] = svd((repmat(sum(x.*x,1),size(x,1),1).*x)*x');` -在机器学习刚开始时,**推荐使用 Octave 类的工程计算软件**,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。**** +**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。 @@ -136,9 +143,9 @@ 2. **问题解决模型** -:![](image/20180105_212048.png) +![](image/20180105_212048.png) -其中 $h$ 代表结果函数,也称为**假设(hypothesis)** 。这个结果函数根据输入(房屋的面积),给出预测结果输出(房屋的价格)。 +其中 $h$ 代表结果函数,也称为**假设(hypothesis)** 。这个函数 $h$ 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。 $h_\theta(x)=\theta_0+\theta_1x$,为其中一种可行的表达式。 @@ -150,7 +157,7 @@ $h_\theta(x)=\theta_0+\theta_1x$,为其中一种可行的表达式。 ## 2.2 损失函数(Cost Function) -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,则问题可表达为**求解 $\sum\limits_{i=0}^{m}(h_\theta(x^{(i)})-y^{(i)})$ 的最小值**。 +我们的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 > @@ -166,9 +173,9 @@ $h_\theta(x)=\theta_0+\theta_1x$,为其中一种可行的表达式。 为了求解最小值,引入损失函数(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): -$$ J\left( \theta \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}^{2}}} $$ +$$J(\theta_0, \theta_1) = \dfrac {1}{2m} \displaystyle \sum _{i=1}^m \left ( \hat{y}_{i}- y_{i} \right)^2 = \dfrac {1}{2m} \displaystyle \sum _{i=1}^m \left (h_\theta (x_{i}) - y_{i} \right)^2$$ -> 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。 +> 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**。 @@ -183,13 +190,11 @@ $$ J\left( \theta \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( {{h}_{\the - 损失函数(Cost Function): $ J\left( \theta_0, \theta_1 \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}^{2}}} $ - 目标(Goal): $\underset{\theta_0, \theta_1}{\text{minimize}} J \left(\theta_0, \theta_1 \right)$ -为了直观理解损失函数到底是在做什么,先假设 $\theta_1 = 0$,并假设训练集有三个数据,分别为$\left(1, 1\right), \left(2, 2\right), \left(3, 3\right)$。 - -TODO: 可更换为动图<--> +为了直观理解损失函数到底是在做什么,先假设 $\theta_1 = 0$,并假设训练集有三个数据,分别为$\left(1, 1\right), \left(2, 2\right), \left(3, 3\right)$,这样在平面坐标系中绘制出 $h_\theta\left(x\right)$ ,并分析 $J\left(\theta_0, \theta_1\right)$ 的变化。 ![](image/20180106_085915.png) -上图中 $J\left(\theta_0, \theta_1\right)$ 随着 $\theta_1$ 的变化而变化,**当 $\theta_1 = 1$ 时,$J\left(\theta_0, \theta_1 \right) = 0$,取得最小值。** +右图 $J\left(\theta_0, \theta_1\right)$ 随着 $\theta_1$ 的变化而变化,可见**当 $\theta_1 = 1$ 时,$J\left(\theta_0, \theta_1 \right) = 0$,取得最小值,**对应于左图青色直线,即函数 $h$ 拟合程度最好的情况。 ## 2.4 损失函数 - 直观理解2(Cost Function - Intuition II) @@ -203,11 +208,17 @@ $$ J\left( \theta \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( {{h}_{\the ![](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。 +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: + +![](image/0f38a99c8ceb8aa5b90a5f12136fdf43.png) + +大概在 $\theta_0 = 0.12, \theta_1 =250$ 时: ![](image/20180106_092119.png) -右图轮廓图中,相同颜色的一个圈代表同一高度(同一 $J\left(\theta\right)$ 值),最中心的点(红点),是图像中的最低点,也即损失函数的最小值,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。 +上图中最中心的点(红点),近乎为图像中的最低点,也即损失函数的最小值,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) @@ -222,11 +233,11 @@ $$ J\left( \theta \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( {{h}_{\the 视频中举了下山的例子,即我们在山顶上的某个位置,为了下山,就不断地看一下周围**下一步往哪走**下山比较快,然后就**迈出那一步**,一直重复,直到我们到达山下的某一处**陆地**。 给出梯度下降的公式: -$$ -{{\theta }_{j}}:={{\theta }_{j}}-\alpha \frac{\partial }{\partial {{\theta }_{j}}}J\left( \theta_0, \theta_1 \right) -$$ -> ${\theta }_{j}​$: 第 $j​$ 个特征参数 +​ repeat until convergence: +​ ${{\theta }_{j}}:={{\theta }_{j}}-\alpha \frac{\partial }{\partial {{\theta }_{j}}}J\left( \theta_0, \theta_1 \right)$ + +> ${\theta }_{j}$: 第 $j$ 个特征参数 > > ”:=“: 赋值操作符 > @@ -246,7 +257,9 @@ $$ ![](image/20180106_184926.png) -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 ,${{\theta }_{j}}:={{\theta }_{j}}-\alpha \frac{\partial }{\partial {{\theta }_{j}}}J\left( \theta_0, \theta_1 \right)$ 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,当然如果 $\theta$ 值开始就在极小值点处时,梯度下降算法将什么也不做)。 +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 ,${{\theta }_{j}}:={{\theta }_{j}}-\alpha \frac{\partial }{\partial {{\theta }_{j}}}J\left( \theta_0, \theta_1 \right)$ 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)。 + +当然如果 $\theta$ 值开始就在极小值点处时,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。 @@ -286,7 +299,7 @@ $$ ![](image/20180106_203726.png) -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: +对于 $j = 0, j = 1$ 时,给出偏导计算公式的推导过程如下: $\frac{\partial}{\partial\theta_j} J(\theta_1, \theta_2)=\frac{\partial}{\partial\theta_j} \left(\frac{1}{2m}\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}^{2}} \right)=$ @@ -306,7 +319,11 @@ $\frac{\partial}{\partial\theta_1} J(\theta)=\frac{1}{m}\sum\limits_{i=1}^{m}{{\ 上文中所提到的梯度下降,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。 -使用循环求解,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,使梯度下降运行的更快更好。 +由于线性回归函数呈现**碗状**,且**只有一个**全局的最优值,所以函数**一定总会**收敛到全局最小值(学习速率不可过大)。同时,函数 $J$ 被称为凸二次函数,而线性回归函数求解最小值问题属于**凸函数优化问题**。 + +![](image/24e9420f16fdd758ccb7097788f879e7.png) + +另外,使用循环求解,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,使梯度下降运行的更快更好。 # 3 Linear Algebra Review @@ -314,13 +331,315 @@ $\frac{\partial}{\partial\theta_1} J(\theta)=\frac{1}{m}\sum\limits_{i=1}^{m}{{\ ## 3.1 Matrices and Vectors +Octave/Matlab 代码: + +```matlab +% 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) +``` + +执行结果: + +``` +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 = 6 +``` + ## 3.2 Addition and Scalar Multiplication +Octave/Matlab 代码: + +```matlab +% 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 + s +``` + +执行结果: + +``` +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 4 +``` ## 3.3 Matrix Vector Multiplication +Octave/Matlab 代码: + +```matlab +% 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 + + +``` + +执行结果: + +``` +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 代码: + +```matlab +% 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 result +``` + +执行结果: + +``` +A = + + 1 2 + 3 4 + 5 6 + +B = + + 1 + 2 + +mult_AB = + + 5 + 11 + 17 + +``` ## 3.5 Matrix Multiplication Properties +Octave/Matlab 代码: + +```matlab +% 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 != BA +``` + +执行结果: + +``` +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 10 +``` ## 3.6 Inverse and Transpose +Octave/Matlab 代码: + +```matlab +% 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 + + +``` + +执行结果: + +``` +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 + +``` \ No newline at end of file