[notes of wuenda] fundamentals of machine learning

Fundamentals of machine learning

What is machine learning ？

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 .

Supervised Learning Supervised learning

Supervised learning ： There are more than features in a dataset feature-X, And labels target-Y

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 .

Unsupervised Learning Unsupervised learning

Unsupervised learning : There are only features in the dataset feature

clustering algorithm : Separate audio at different distances , Distinguish whether the mailbox is a junk mailbox, etc

[W,s,v] = svd((repmat(sum(x.*x,1),size(x,1),1).*x)*x');

Self supervised learning

Explain a : Self supervised learning enables us to obtain high-quality representation without large-scale annotation data , Instead, we can use a large number of unlabeled data and optimize predefined pretext Mission . We can then use these features to learn about new tasks that lack data .

Explain two : self-supervised learning It's a kind of unsupervised learning , The main purpose is to learn a common feature expression for downstream tasks . The main way is to supervise yourself , For example, remove a few words from a paragraph , Use his context to predict missing words , Or remove some parts of the picture , Rely on the information around it to predict the missing patch.

effect :

Learn useful information from unlabeled data , For subsequent tasks .

Self monitoring tasks （ Also known as pretext Mission ） We are asked to consider the supervisory loss function . However , We usually don't care about the final performance of the task . actually , We are only interested in the intermediate representations we have learned , We expect these representations to cover good semantic or structural meaning , And it can be beneficial to various downstream practical tasks .

Specific understanding

Linear Regression with One Variable Linear regression of single variable

Univariate linear regression ：

• 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 .

Selling a house : Already know the price of the previous sale , Based on the previous data set, predict the price at which your friend's house can be sold .

Training Set（ Training set ） 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）

Cost Function Cost function

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 .

The cost function makes us $h_{\theta }\left(x\right)={\theta }_0+{\theta }_{1}x$ Better choice of parameters **parameters ** ${\theta }_0{\theta }_1$, So that the most possible lines and data fit each other .

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 .

${\theta }_0$ and ${\theta }_1$ and $J({\theta }_0,{\theta }_1)$ Visualization of relationships

At present, the global minimum cost function is obtained , Simplify ${\theta }_0=0$

Yes ${\theta }_1$ Continuously assign values to get the corresponding $J({\theta }_1)$, obtain $J({\theta }_1)$ and ${\theta }_1$ Relationship

Contour map ： Corresponding ${\theta }_0=360$,${\theta }_1=0$, Corresponding to the position in the contour map

Gradient Descent gradient descent

gradient descent ： To solve the cost function $J({\theta }_0,{\theta }_1)$ Minimum value ${\theta }_0$,${\theta }_1$

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 .

Gradient descent algorithm

• $a$ 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 .

• The right is right , Assign values after all values are calculated , The left side is wrong

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 .

If only consider ${\theta }_1$,${\theta }_0=0$ when ,$\alpha \frac{\partial }{\partial {\theta }_1}J\left(\theta \right)$, The front is the learning rate , Then comes the cost function $J\left(\theta \right)$ About ${\theta }_1$ The derivative of .

Cost function $J\left({\theta }_1\right)$ and ${\theta }_1$ Image ,

If the learning rate is too small , More iterations

If the learning rate is too high , May cross the local minimum , Back and forth oscillation deviates from the local minimum value point .

take Gradient descent and cost function combination , And apply it to the specific linear regression algorithm of fitting straight line .

Gradient descent algorithm and linear regression algorithm are shown in the figure below :

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 {

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

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

​ **}

Refer to https://zhuanlan.zhihu.com/p/328261042

Batch gradient descent , All training sets are used for each gradient descent

summary

1. Hypothesis function （Hypothesis）

A linear function is used to fit the sample data set , It can be simply defined as ：

among

and

Is the parameter .

2. Cost function （Cost Function）

Measure the... Of a hypothetical function “ Loss ”, Also known as “ Square sum error function ”（Square Error Function）, The following definitions are given ：

amount to , Sum up the square of the difference between the assumed value and the true value of all samples , Divide by the number of samples m, Get an average of “ Loss ”. Our task is to find out

and

Make this “ Loss ” Minimum .

3. gradient descent （Gradient Descent）

gradient ： The directional derivative of a function at that point is maximized in that direction , That is, the rate of change at this point （ Slope ） Maximum .

gradient descent ： Make the argument

Along make

Move in the direction of the fastest descent , Get... As soon as possible

The minimum value of , The following definitions are given ：

In wuenda's course, I learned , Gradient descent requires all independent variables to be simultaneously “ falling ” Of , therefore , We can translate it into separate pairs

and

Find the partial derivative , It's fixed

take

Take the derivative as a variable , The opposite is true of

equally .

We know that the cost function is

, among

, that , According to the derivation principle of composite function ,dx/dy=（*du/dy）*∗（dx/du）, That is to translate it into ：

Finally, the results of the course ：