1.20 study univariate linear regression

1.20 study

One 、 introduction (Introduction)

1.1 welcome

But in fact, I only know the algorithm 、 Math doesn't solve the practical problems you care about . therefore , We will spend a lot of time doing exercises , So you can implement each of these algorithms yourself , So as to understand the internal mechanism .

that , Why is machine learning so popular ？ as a result of , Machine learning is not just used in the field of artificial intelligence .

Finally, learning algorithms are used to understand human learning and understand the brain .

In the next video , We will begin to give a more formal definition , What is machine learning . Then we will start to learn the main problems and algorithms of machine learning. You will know some main machine learning terms , And start to understand different algorithms , Which algorithm is more appropriate .

1.2 What is machine learning ？

The first definition of machine learning comes from Arthur Samuel. He defined machine learning as , In the case of specific programming , Areas that give computer learning ability .Samuel The definition of can be traced back to 50 years , He wrote a chess program . The magic of this program is , The programmer himself is not a chess player . But because he's too good , So through programming , Let the chess program play tens of thousands of chess with itself . By observing which layout （ Chessboard position ） It will win , Which layout will lose , in the course of time , This chess program understands what a good layout is , What is a bad layout . And then it's crazy , After learning the program , The level of playing chess exceeds Samuel. This is definitely a remarkable achievement .

Another more recent definition , from Tom Mitchell Put forward , From Carnegie Mellon University ,Tom The definition of machine learning is , A good learning problem is defined as follows , He said ,== A program is thought to be able to learn from experience E Middle school learning , Solve the task T, Performance metrics reached P, If and only if , Experience gained E after , after P judge , The program is processing T Improved performance when .== I think experience E It is the experience and task of tens of thousands of self-practice procedures T Is playing chess . Performance metrics P Well , It's when it plays against some new opponents , Probability of winning the game .

In this lesson , I want to teach you about different types of learning algorithms . There are several different types of learning algorithms . The two main types are called supervised learning and unsupervised learning . In the next few videos , I will give the definitions of these terms . Here are two simple sentences , The idea of supervised learning means , We will teach computers how to complete tasks , And in unsupervised learning , We intend to let it learn by itself . If you are still confused about these two terms , Please don't worry , In the next two videos , I will introduce these two learning algorithms in detail .

This is machine learning , These are the topics I want to teach . In the next video , I will define what supervised learning is , What is unsupervised learning . Besides , Explore when to use both .

1.3 Supervised learning

Reference video : 1 - 3 - Supervised Learning (12 min).mkv

In this video , I want to define what is probably the most common machine learning problem ： That is supervised learning . I will formally define supervised learning later .

Let's use an example to introduce what is supervised learning. The formal definition is introduced later . If you want to predict house prices .

A while ago , A student collected data on house prices from a research institute in Portland, Oregon . You draw these data , It looks like this ： The horizontal axis represents the area of the house , The unit is square feet , The vertical axis represents house prices , The unit is thousands of dollars . Based on this set of data , If you have a friend , He has a set of 750 Square foot house , Now he wants to sell his house , He wants to know how much the house can sell .

So about this question , How will machine learning algorithms help you ？

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We apply learning algorithms , You can draw a straight line in this set of data , Or to put it another way , Fit a straight line , From this line, we can infer , The house may sell $$150,000$,WhenhoweverthisNoyesonlyOneOfcountLaw.cancanalsoYesmoregoodOf,ThanSuch\; asIPeopleNousestraightLinequasiclosethissomeCountAccording\; to\; the,useTwoTimeFangchengGo\; toquasiclosecancaneffectfruitMeetingmoregood.rootAccording\; to\; theTwoTimeFangchengOfsongLine,IPeoplecanWithfromthisindividualspotPUSHmeasuringOut,thissetroomSoncansellPick\; upnear$$200,000$. Later we will discuss how to choose a learning algorithm , How to decide whether to use a straight line or a quadratic equation to fit . One of the two options can make your friend's house sell more reasonably . These are good examples of learning algorithms . The above is an example of supervised learning .

It can be seen that , Supervised learning means that we give the learning algorithm a data set . By this data set “ right key ” form . In the case of house prices , We gave a series of house data , We give the correct price for each sample in a given dataset , That's how much they actually sell, and then using learning algorithms , Work out more correct answers . For example, the price of your friend's new house . In terms of terms , It's called The return question . We Try to guess the result of a continuous value , The price of the house .

Generally, the price of a house will be recorded in cents , So house prices are actually a series of discrete values , But we usually regard house prices as real numbers , As a scalar , So we think of it as a continuous number .

The word return means , We are trying to infer this series of continuous value attributes .

Let me give another example of supervised learning . I have studied this before with some friends . Suppose you want to guess whether breast cancer is benign or not by looking at medical records. , If someone detects a breast tumor , Malignant tumors are harmful and dangerous , Benign tumors are less harmful , So people obviously care about this problem .

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Let's look at a set of data ： This dataset , The horizontal axis shows the size of the tumor , On the longitudinal axis , I marked 1 and 0 Indicates whether it is a malignant tumor or not . The tumors we've seen before , If it is malignant, it is recorded as 1, Not malignant , Or benign as 0.

I have a 5 Benign tumor samples , stay 1 The location of this is 5 A malignant tumor sample . Now we have a friend who unfortunately has a breast tumor . Suppose her tumor is about this big , So the problem with machine learning is , Can you estimate the probability that the tumor is malignant or benign . In terms of terms , This is a question of classification .

Classification refers to , We're trying to figure out the discrete output ：0 or 1 Benign or malignant , In fact, in the problem of classification , The output may have more than two values . For example, there may be three types of breast cancer , So you want to predict discrete output 0、1、2、3.0 Represents good ,1 It means the first one 1 Breast cancer ,2 It means the first one 2 Carcinoid ,3 It means the first one 3 class , But this is also a classification problem .

Because these discrete outputs correspond to benign signals respectively , Type I, type II or type III cancer , In the classification problem, we can draw these data points in another way .

Now I use different symbols to represent these data . Since we regard the size of the tumor as a feature to distinguish malignant from benign , Then I can draw like this , I use different symbols to represent benign and malignant tumors . Or negative samples and positive samples. Now we don't draw all of them X, Benign tumors are replaced with O Express , Malignant continue to use X Express . To predict whether the tumor is malignant or not .

In some other machine learning problems , More than one feature may be encountered . for instance , We don't just know the size of the tumor , Also know the age of the corresponding patient . In other machine learning problems , We usually have more characteristics , When my friend studied this problem , These features are usually used , Such as mass density , Consistency of tumor cell size and shape, etc , There are other features . This is one of the most interesting learning algorithms we are about to learn .

That algorithm can not only handle 2 Kind of 3 To plant or 5 Species characteristics , Even an infinite number of features can be processed .

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Above picture , I listed a total of 5 Different characteristics , Two on the axis and one on the right 3 Kind of , But in some learning problems , You wish you didn't just 3 To plant or 5 Species characteristics . contrary , You want to use an infinite variety of features , So that your algorithm can take advantage of a large number of features , Or clues to speculate . How do you deal with infinite features , Even how to store these features is problematic , Your computer certainly doesn't have enough memory .== We'll talk about an algorithm later , It's called support vector machine , There is a clever mathematical skill , It allows the computer to process an infinite number of features .== Imagine , I didn't write down these two and the three features on the right , But in an infinitely long list , Keep writing, keep writing , Write down an infinite number of features , in fact , We can use algorithms to deal with them .

Now let's review , In this lesson, we introduced supervised learning . The basic idea is this ,== Each sample in our dataset has a corresponding “ right key ”. And then make predictions based on these samples ,== As in the case of houses and tumors . We also introduce the regression problem , That is, a continuous output is derived by returning , Then we introduce the classification problem , The goal is to derive a set of discrete results .

1.4 Unsupervised learning

Reference video : 1 - 4 - Unsupervised Learning (14 min).mkv

In this video , We will introduce the second major machine learning problem . It's called unsupervised learning .

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In the last video , Supervised learning has been introduced . Think back to the data set at that time , As the chart shows , Each data in this data set has been marked as negative or positive , That is, benign or malignant tumors . therefore , For each data in supervised learning , We already know , The correct answer corresponding to the training set , Is it benign or malignant .

In unsupervised learning , The data we know . It looks a little different , Different from the data of supervised learning , That is, there is no label in unsupervised learning, or there is the same label or no label . So we know the data set , But I don't know how to deal with , Nor does it tell what each data point is . I don't know anything else , It's just a data set . Can you find some structure in the data ？ For datasets , Unsupervised learning can judge that the data has two different clusters . This is a , That's another one , They are different . Yes , Unsupervised learning algorithms may divide these data into two different clusters . So it is called clustering algorithm . The fact proved that , It can be used in many places .

An example of clustering application is in Google News . If you've never seen it before , You can go to this URL website news.google.com Go to see . Google news is online every day , Collect a lot , A lot of online news content . It then groups these news , Make up relevant news . So what Google news does is search a lot of news events , Automatically cluster them together . therefore , These news events are all on the same theme , So show together .

The fact proved that , Clustering algorithm and unsupervised learning algorithm are also used in many other problems .

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Among them is the understanding and application of genetics . One DNA Examples of microdata . The basic idea is to input a group of different individuals , For each of them , You have to analyze whether they have a specific gene . technical , How many specific genes do you want to analyze . So these colors , red , green , Grey and so on , These colors show the corresponding degree , Whether different individuals have a specific gene . All you can do is run a clustering algorithm , Clustering individuals into different classes or groups of different types （ people ）……

So this is unsupervised learning , Because we didn't tell the algorithm some information in advance , such as , This is the first group of people , Those are the second kind of people , There is a third category , wait . We just said , Yes , This is a pile of data . I don't know what's in the data . I don't know who is what type . I don't even know what different types of people there are , What are these types . But can you automatically find the structure in the data ？ That means you have to Automatically cluster those individuals into categories , I can't know in advance what is what . Because we did not give the correct answer to the algorithm to respond to the data in the data set , So this is unsupervised learning .

Two 、 Univariate linear regression (Linear Regression with One Variable)

2.1 Model to represent

Reference video : 2 - 1 - Model Representation (8 min).mkv

Our first learning algorithm is the linear regression algorithm . In this video , You will see an overview of the algorithm , More importantly, you will understand the complete process of supervising the learning process .

Let's start with an example ： This is an example of forecasting housing prices , We're going to use a dataset , The data set contains housing prices in Portland, Oregon . ad locum , I want to sell the price according to different house sizes , Draw my dataset . For example , If your friend's house is 1250 Square foot , You have to tell them how much the house will sell for . that , One thing you can do is build a model , Maybe it's a straight line , From this data model , Maybe you can tell your friends , He can approximate 220000( dollar ) Sell the house for about the price . This is an example of a supervised learning algorithm .

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It is called supervised learning because for each data , We have given “ The right answer ”, Tell us ： According to our data , What is the actual price of the house , and , More specifically , This is a question of return . The term regression refers to , We predict an accurate output value based on the previous data , For this example, the price , meanwhile , There's another most common form of supervised learning , It's called the classification problem , When we want to predict discrete output values , for example , We're looking for cancer , And want to determine whether the tumor is benign or malignant , This is it. 0/1 The problem of discrete output . To go a step further , In supervised learning, we have a data set , This data set is called the training set .

I will use lowercase throughout the course $m$ To represent the number of training samples .

Take the previous housing transaction as an example , Let's go back to the training set of the problem （Training Set） As shown in the following table ：

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The markup we will use to describe this regression problem is as follows :

$m$ Represents the number of instances in the training set

$x$ On behalf of the characteristic / The input variable

$y$ Represents the target variable / Output variables

$\left(x,y\right)$ Represents an instance of a training set

$(x^{(i)},y^{(i)})$ On behalf of the $i$ Two observation examples

$h$ Solutions or functions that represent learning algorithms are also called assumptions （hypothesis）

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This is how a supervised learning algorithm works , We can see here the house prices in our training set
We feed it to our learning algorithm , The work of learning algorithms , Then output a function , Usually expressed in lowercase $h$ Express .$h$ representative hypothesis( hypothesis ),$h$ Represents a function , The input is the house size , Like the house your friend wants to sell , therefore $h$ According to the input $x$ Value derived $y$ value ,$y$ The value corresponds to the price of the house therefore ,$h$ It is a slave. $x$ To $y$ Function mapping of .

I will choose the initial usage rules $h$ representative hypothesis, thus , To solve the problem of house price prediction , We're actually going to put the training set “ feed ” Give us the learning algorithm , Then learn to get a hypothesis $h$, Then we input the size of the house we want to predict as an input variable to $h$, Predict the transaction price of the house as the output variable and output as the result . that , For our house price forecast , How should we express $h$？

One possible expression is ：$h_{\theta }\left(x\right)={\theta }_0+{\theta }_{1}x$, Because it contains only one feature / The input variable , So this kind of problem is called univariate linear regression problem .

2.2 Cost function

Reference video : 2 - 2 - Cost Function (8 min).mkv

In this video, we will define the concept of cost function , This helps us figure out how to fit the most likely straight line to our data . Pictured ：

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In linear regression, we have a training set like this ,$m$ Represents the number of training samples , such as $m=47$. And our hypothetical function , That's the function used to predict , It's a linear function like this ：$h_{\theta }\left(x\right)={\theta }_0+{\theta }_{1}x$.

Next, we will introduce some terms. What we need to do now is to choose the right one for our model Parameters （parameters）${\theta }_0$ and ${\theta }_1$, In the case of house price problem, it is the slope of the straight line and $y$ The intercept of the axis .

The parameters we choose determine the accuracy of our line relative to our training set , The gap between the predicted value of the model and the actual value of the training set （ The blue line is indicated in the figure below ） Namely Modeling error （modeling error）.

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Our goal is to select the model parameters that minimize the sum of squares of modeling errors . Even if you get the cost function $J\left({\theta }_0,{\theta }_1\right)=\frac{1}{2m}\sum _{i=1}^m{\left(h_{\theta }(x^{(i)})-y^{(i)}\right)}^2$ Minimum .

Let's draw a contour map , The three coordinates are ${\theta }_0$ and ${\theta }_1$ and $J({\theta }_0,{\theta }_1)$：

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Then we can see that there is a problem in three-dimensional space that makes $J({\theta }_0,{\theta }_1)$ The smallest point .

The cost function is also called the square error function , Sometimes called the square error cost function . The reason why we ask for the sum of squares of errors , Because the square cost function of the error , For most problems , Especially the problem of return , Is a reasonable choice . There are other cost functions that work well , But the square error cost function is probably the most common way to solve regression problems .

Maybe this function $J({\theta }_0,{\theta }_1)$ A bit abstract , Maybe you still don't know what it means , In the next few videos , Let's go a step further and explain the cost function J How it works , And try to explain more intuitively what it is calculating , And why we use it .

2.3 An intuitive understanding of the cost function I

Reference video : 2 - 3 - Cost Function - Intuition I (11 min).mkv

In the previous video , We give a mathematical definition of the cost function . In this video , Let's get some intuitive feelings through some examples , See what the cost function is doing .

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2.4 An intuitive understanding of the cost function II

Reference video : 2 - 4 - Cost Function - Intuition II (9 min).mkv

In this class , We will learn more about the role of the cost function , The content of this video assumes that you already know the contour map , If you are not familiar with contour maps , You may not understand some of the content in this video , But it doesn't matter. , If you skip this video , It doesn't matter , Not listening to this class will have little impact on the understanding of subsequent courses .

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What the cost function looks like , Contour map , Then we can see that there is a problem in three-dimensional space that makes $J({\theta }_0,{\theta }_1)$ The smallest point .

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Through these graphics , I hope you can better understand these cost functions $ J$thesurfacereachOfvalueyesWhatWellsampleOf,itPeopleYesShould\; beOffalseset\; upyesWhatWellsampleOf,WithAndWhatWellsampleOffalseset\; upYesShould\; beOfspot,morePick\; upnearOngenerationpriceLetterCount$J$ The minimum value of .

Of course , What we really need is an effective algorithm , Can automatically find out these cost functions $J$ The parameter with the minimum value ${\theta }_0$ and ${\theta }_1$ Come on .

We don't want to write a program to draw these dots , Then read the values of these points manually , This is obviously not a good idea . We will encounter more complicated 、 Higher dimensions 、 More parameters , And these situations are difficult to draw , Therefore, it is even more impossible to visualize , So what we really need to do is write a program to find out how to minimize these cost functions ${\theta }_0$ and ${\theta }_1$ Value , In the next video , We will introduce an algorithm , Can automatically find out the cost function $J$ Minimized parameter ${\theta }_0$ and ${\theta }_1$ Value .

2.5 gradient descent

Reference video : 2 - 5 - Gradient Descent (11 min).mkv

Gradient descent is an algorithm used to find the minimum value of a function , We will use the gradient descent algorithm to find the cost function $J({\theta }_0,{\theta }_1)$ The minimum value of .

The idea behind the gradient drop is ： At first, we randomly choose a combination of parameters $\left({\theta }_0,{\theta }_1,......,{\theta }_n\right)$, Computational cost function , Then we look for the next parameter combination that can reduce the cost function value the most . We keep doing this until we find a local minimum （local minimum）, Because we haven't tried all the parameter combinations , So it's not sure if the local minimum we get is the global minimum （global minimum）, Choose different combinations of initial parameters , Different local minima may be found .

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Imagine that you are standing at this point in the mountain , Stand on the red mountain in your imaginary Park , In gradient descent algorithm , All we have to do is rotate 360 degree , Look around us , And ask yourself to be in a certain direction , Go down the mountain in small steps as soon as possible . Where do these little steps need to go ？ If we stand at this point on the hillside , Look around , You will find the best way to go down the mountain , Look around again , And then think about it again , In what direction should I take a small step down the mountain ？ Then you take another step in your own judgment , Repeat the above steps , From this new point , You look around , And decide which direction will go down the mountain as soon as possible , And then a small step forward , And so on , Until you get close to the local lowest point .

Batch gradient descent （batch gradient descent） The formula of the algorithm is ：

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among $a$ It's the learning rate （learning rate）, It determines how far down we go in the direction where the cost function can go down the most , In the decline of batch gradients , Each time we subtract all the parameters from the learning rate times the derivative of the cost function .

In gradient descent algorithm , There's a more subtle question , The gradient is falling , We need to update ${\theta }_0$ and ${\theta }_1$ , When $j=0$ and $j=1$ when , There will be updates , So you're going to update $J\left({\theta }_0\right)$ and $J\left({\theta }_1\right)$. The subtlety of implementing gradient descent algorithm is , In this expression , If you want to update this equation , You need to update it at the same time ${\theta }_0$ and ${\theta }_1$, I mean in this equation , We need to update it like this ：

${\theta }_0$:= ${\theta }_0$ , And update the ${\theta }_1$:= ${\theta }_1$.

The way to achieve this is ： You should calculate the right part of the formula , Calculate from that part ${\theta }_0$ and ${\theta }_1$ Value , Then update at the same time ${\theta }_0$ and ${\theta }_1$.

Let me elaborate further on this process ：

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In gradient descent algorithm , This is the right way to implement simultaneous updates . I'm not going to explain why you need to update at the same time , At the same time, updating is a common method in gradient descent . We'll talk about , Synchronous update is a more natural way to achieve . When people talk about gradient decline , They mean sync update .

$\alpha \frac{\partial }{\partial {\theta }_0}J({\theta }_0,{\theta }_1)$,$\alpha \frac{\partial }{\partial {\theta }_1}J({\theta }_0,{\theta }_1)$.

If you're not familiar with calculus , Never mind , Even if you haven't seen calculus before , Or have not been exposed to partial derivatives , In the next video , You'll get everything you need to know , How to calculate this differential term .

In the next video , I hope we can give all the knowledge to realize the gradient descent algorithm .

2.6 An intuitive understanding of gradient descent

Reference video : 2 - 6 - Gradient Descent Intuition (12 min).mkv

In the previous video , We give a mathematical definition of gradient descent , Let's take a closer look at this video , Feel more intuitively what this algorithm does , And what's the significance of the update process of gradient descent algorithm . The gradient descent algorithm is as follows ：

${\theta }_j:={\theta }_j-\alpha \frac{\partial }{\partial {\theta }_j}J\left(\theta \right)$

describe ： Yes $\theta $Fuvalue,sendhave\; to$J\left( \theta \right)$PressladderdegreeNextdropmostfastFangtowardsInto\; theThat\text{'}s\; ok,OnestraightOverlappinggenerationNextGo\; to,mostendhave\; toTogameMinistrymostSmallvalue.Itsin$a$ It's the learning rate （learning rate）, It determines how far down we go in the direction where the cost function can go down the most .

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For this question , The purpose of derivation is , Basically, let's take the tangent line of the red dot , It's such a red line , Just tangent to the function at this point , Let's look at the slope of this red line , This is the line that is tangent to the function curve , The slope of the line is exactly the height of the triangle divided by the horizontal length , Now? , This line has a positive slope , That is to say, it has a positive derivative , therefore , I got a new ${\theta }_1$,${\theta }_1$ Updated equals ${\theta }_1$ Subtract a positive number times $a$.

This is the update rule of my gradient descent method ：${\theta }_j:={\theta }_j-\alpha \frac{\partial }{\partial {\theta }_j}J\left(\theta \right)$

Let's see if $a$ Too small or $a$ What happens in the General Assembly ：

If $a$ Is too small , That is, my learning rate is too small , As a result, I can only move like a little baby , To try to get to the bottom , It will take many steps to reach the lowest point , So if $a$ Too small , It could be very slow , Because it moves a little bit , It will take many steps to reach the global lowest point .

If $a$ Too big , Then the gradient descent method might go past the lowest point , It may not even converge , The next iteration takes another big step , Over and over again , Over again , Over and over again , Until you find that it's actually getting further and further from the bottom , therefore , If $a$ Too big , It leads to non convergence , Even divergence .

Now? , I have another question , When I first studied this place , It took me a long time to understand the problem , If we put ${\theta }_1$ Put it at a local lowest point , How do you think the gradient descent method will work next ？

Suppose you will ${\theta }_1$ Initialization at local minimum , Here it is , It's already at a local optimum or local minimum . The result is that the derivative of the local best will be equal to zero , Because it's the slope of that tangent . This means that you've got the local best , It makes the ${\theta }_1$ No more changes , That's new ${\theta }_1$ Equal to the original ${\theta }_1$, therefore , If your parameter is already at the local lowest point , So the gradient descent method actually doesn't do anything , It doesn't change the value of the parameter . This also explains why even the learning rate $a$ When it stays the same , Gradient descent can also converge to the local lowest point .

Let's take an example , This is the cost function $J\left(\theta \right)$.

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I want to find the minimum value , First, initialize my gradient descent algorithm , Initialize at that magenta point , If I update a step gradient descent , Maybe it will take me to this point , Because the derivative of this point is quite steep . Now? , At this green point , If I update one more step , You will find my derivative , That's slope , It's not that steep . As I get close to the bottom , My derivative is getting closer to zero , therefore , One step down the gradient , The new derivative will be a little smaller . Then I want to take another step down the gradient , At this green dot , Naturally, I will use one to compare it with the magenta point just now , A little bit smaller , To the new red dot , Closer to the global minimum , So the derivative at this point will be smaller than at the green point . therefore , When I take another step down the gradient , My derivative term is smaller ,${\theta }_1$ The update will be smaller . So as the gradient descent method works , You'll automatically move less and less , Until the final movement is very small , You'll find that , Has converged to the local minimum .

Take a look back. , In gradient descent method , When we approach the local minimum , Gradient descent will automatically take a smaller range , This is because when we approach the local minimum , Obviously, at the local minimum, the derivative equals zero , So when we get close to the local minimum , The derivative value will automatically get smaller and smaller , So the gradient drop will automatically take a smaller magnitude , This is how the gradient falls . So there's really no need to reduce it any more $a$.

This is the gradient descent algorithm , You can use it to minimize any cost function $J$, It's not just a cost function in linear regression $J$.

In the next video , We're going to use the cost function $J$, Go back to its essence , Cost function in linear regression . That's the square error function we got earlier , Combined with gradient descent method , And the square cost function , We'll get the first machine learning algorithm , Linear regression algorithm .

2.7 Linear regression of gradient descent

Reference video : 2 - 7 - GradientDescentForLinearRegression (6 min).mkv

In the previous video, we talked about gradient descent algorithm , Gradient descent is a very common algorithm , It's not only used in linear regression and linear regression models 、 Squared error cost function . In this video , We want to combine gradient descent with cost function . We will use this algorithm , And apply it to the specific linear regression algorithm of fitting straight line .

The comparison between gradient descent algorithm and linear regression algorithm is shown in the figure ：

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For our previous linear regression problem, we use the gradient descent method , The key is to find the derivative of the cost function , namely ：

$\frac{\partial }{\partial {\theta }_j}J({\theta }_0,{\theta }_1)=\frac{\partial }{\partial {\theta }_j}\frac{1}{2m}{\sum _{i=1}^m\left(h_{\theta }(x^{(i)})-y^{(i)}\right)}^2$

$j=0$ when ：$\frac{\partial }{\partial {\theta }_0}J({\theta }_0,{\theta }_1)=\frac{1}{m}\sum _{i=1}^m\left(h_{\theta }(x^{(i)})-y^{(i)}\right)$

$j=1$ when ：$\frac{\partial }{\partial {\theta }_1}J({\theta }_0,{\theta }_1)=\frac{1}{m}\sum _{i=1}^m\left(\left(h_{\theta }(x^{(i)})-y^{(i)}\right)\cdot x^{(i)}\right)$

Then the algorithm is rewritten to ：

Repeat {

​ ${\theta }_0:={\theta }_0-a\frac{1}{m}\sum _{i=1}^m\left(h_{\theta }(x^{(i)})-y^{(i)}\right)$

​ ${\theta }_1:={\theta }_1-a\frac{1}{m}\sum _{i=1}^m\left(\left(h_{\theta }(x^{(i)})-y^{(i)}\right)\cdot x^{(i)}\right)$

​ }

The algorithm we just used , Sometimes called batch gradient descent . actually , In machine learning , You don't usually name algorithms , But the name ” Batch gradient descent ”, In every step of the gradient descent , We all used all the training samples , In gradient descent , In computing the derivative of a differential , We need to sum , therefore , In each individual gradient descent , We all end up calculating something like this , This item needs to be applied to all $m$ Sum the training samples . therefore , The name batch gradient descent shows that we need to consider all of this " batch " The training sample , As a matter of fact , Sometimes there are other types of gradient descent , It's not like this " Batch " Type , Regardless of the entire training set , It's about focusing on a small subset of the training set at a time . In a later lesson , We will also introduce these methods .

But for now , Apply the algorithm just learned , You should have mastered the batch gradient algorithm , And it can be applied to linear regression , This is the gradient descent method for linear regression .

If you've studied linear algebra before , Some students may have studied advanced linear algebra before , You should know that there is a cost function $J$ Numerical solution of minimum value , There is no need for gradient descent iterative algorithm . In a later lesson , We also talked about this method , It can be used without multi-step gradient descent , Can also solve the cost function $J$ The minimum value of , This is another kind of equation called normal equation (normal equations) Methods . In fact, when the amount of data is large , The gradient descent method is more suitable than the normal equation .

Now we have the gradient descent , We can use the gradient descent method in different environments , We will also make extensive use of it in different machine learning problems . therefore , Congratulations on learning your first machine learning algorithm .

In the next video , Tell you the generalized gradient descent algorithm , This will make the gradient descent more powerful .

2.8 What's next

Reference video : 2 - 8 - What_'s Next (6 min).mkv

In the next set of videos , I will do a quick review of linear algebra . If you've never been in touch with vectors and matrices , So everything on this courseware is new to you , Or you know something about linear algebra , But because of the long separation , Forget about it , Then learn the next set of videos , I will quickly review your knowledge of linear algebra .

Through them , You can implement and use more powerful linear regression models . in fact , Linear algebra is not only widely used in linear regression , Its matrices and vectors will help us to implement more machine learning models in the future , And more efficient in calculation . It is precisely because these matrices and vectors provide an effective way to organize large amounts of data , Especially when we deal with huge training sets , If you are not familiar with Linear Algebra , If you think linear algebra looks complicated 、 Terrible concept , Especially for people who have never touched it before , Don't worry about , in fact , To implement machine learning algorithms , We only need some very, very basic knowledge of linear algebra . Through the next few videos , You can quickly learn all you need to know about linear algebra . say concretely , To help you decide if you need to learn the next set of videos , I'll talk about what matrices and vectors are , Talk about how to add 、 reduce 、 Multiply matrices and vectors , The concepts of inverse matrix and transpose matrix are discussed .

If you are very familiar with these concepts , So you can skip this elective video on linear algebra , But if you're still a little uncertain about these concepts , Not sure what these numbers or these matrices mean , So please take a look at the next group of videos , It will quickly teach you something you need to know about linear algebra , It is convenient to write machine learning algorithms and process large amounts of data .

3、 ... and 、 Review of Linear Algebra (Linear Algebra Review)

3.1 Matrices and vectors

Reference video : 3 - 1 - Matrices and Vectors (9 min).mkv

Pictured ： This is 4×2 matrix , namely 4 That's ok 2 Column , Such as $m$ For the line ,$n$ Column , that $m\times n$ namely 4×2

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The dimension of the matrix is the number of rows × Number of columns

Matrix elements （ Matrix term ）：$A=\left[\begin{array}{cc}1402& 191\\ 1371& 821\\ 949& 1437\\ 147& 1448\end{array}\right]$

$A_{ij}$ Refers to the first $i$ That's ok , The first $j$ The elements of the column .

Vector is a special kind of matrix , The vectors in the handout are usually column vectors , Such as ：
$y=\left[\begin{array}{c}460\\ 232\\ 315\\ 178\end{array}\right]$

Is a four-dimensional column vector （4×1）.

The picture below shows 1 Index vectors and 0 Index vector , Left for 1 Index vector , The right to 0 Index vector , Usually we use 1 Index vector .

$y=\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\\ y_4\end{array}\right]$,$y=\left[\begin{array}{c}{y}_0\\ y_1\\ y_2\\ y_3\end{array}\right]$

3.2 Addition and scalar multiplication

Reference video : 3 - 2 - Addition and Scalar Multiplication (7 min).mkv

Addition of matrix ： If the number of rows and columns is equal, you can add .

example ：

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Multiplication of matrices ： Every element has to be multiplied by

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Combinatorial algorithms are similar .

3.3 Matrix vector multiplication

Reference video : 3 - 3 - Matrix Vector Multiplication (14 min).mkv

The multiplication of matrix and vector is shown in the figure ：$m\times n$ Matrix times $n\times 1$ Vector , Get is $m\times 1$ Vector

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Examples of algorithms ：

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3.4 Matrix multiplication

Reference video : 3 - 4 - Matrix Matrix Multiplication (11 min).mkv

Matrix multiplication ：

$m\times n$ Matrix times $n\times o$ matrix , become $m\times o$ matrix .

If this is not easy to understand, give an example to illustrate , For example, there are now two matrices $A$ and $B$, Then their product can be expressed in the form shown in the figure .

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3.5 The properties of matrix multiplication

Reference video : 3 - 5 - Matrix Multiplication Properties (9 min).mkv

The properties of matrix multiplication ：

The multiplication of matrices does not satisfy the commutative law ：$A\times B\ne B\times A$

The multiplication of matrices satisfies the associative law . namely ：$A\times (B\times C)=(A\times B)\times C$

Unit matrix ： In matrix multiplication , There is a matrix that plays a special role , As in the multiplication of numbers 1, We call this kind of matrix identity matrix ． It's a square matrix , It's usually used $I$ perhaps $E$ Express , This handout is used $I$ Identity matrix , The diagonal from the top left to the bottom right （ Called the main diagonal ） The elements on are 1 All other than 0. Such as ：

$A{A}^{-1}=A^{-1}A=I$

For the identity matrix , Yes $AI=IA=A$

3.6 The inverse 、 Transposition

Reference video : 3 - 6 - Inverse and Transpose (11 min).mkv

The matrix of the inverse ： Like matrix $A$ It's a $m\times m$ matrix （ Matrix ）, If there is an inverse matrix , be ：$A{A}^{-1}=A^{-1}A=I$

We are usually in OCTAVE perhaps MATLAB To calculate the inverse matrix of the matrix .

The transpose of the matrix ： set up $A$ by $m\times n$ Order matrix （ namely $m$ That's ok $n$ Column ）, The first $i $That\text{'}s\; ok$j $ColumnOfelementplainyes$a(i,j)$,namely：$A=a(i,j)$

Definition $A$ Transpose to such a $n\times m$ Order matrix $B$, Satisfy $B=a(j,i)$, namely $b(i,j)=a(j,i)$（$B$ Of the $i$ Xing di $j$ The column elements are $A$ Of the $j$ Xing di $i$ Column elements ）, remember $A^T=B$.( Some secretaries are A’=B）

Intuitive to see , take $A$ All of the elements around a from 1 Xing di 1 The bottom right of the column elements 45 The rays of degree are mirror reversed , Or get $A$ The transpose .

example ：

${\left|\begin{array}{cc}a& b\\ c& d\\ e& f\end{array}\right|}^T=\left|\begin{array}{ccc}a& c& e\\ b& d& f\end{array}\right|$

The basic properties of transposition of matrix :

$ { {\left( A\pm B \right)}^{{T}}={ {A}}{T}}\pm { {B}^{T}} $
${\left(A\times B\right)}^T=B^T\times A^T$
${ {\left( { {A}^{T}} \right)}^{T}}=A $
${ {\left( KA \right)}^{{T}}=K{ {A}}{T}} $

matlab Matrix transpose in ： Just write it ,x=y'.