Programming Exercise 5: Regularized Linear Regression and Bias v.s. Variance Solution

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Description

Introduction

 

In this exercise, you will implement regularized linear regression and use it to study models with di erent bias-variance properties. Before starting on the programming exercise, we strongly recommend watching the video lectures and completing the review questions for the associated topics.

 

To get started with the exercise, you will need to download the starter code and unzip its contents to the directory where you wish to complete the exercise. If needed, use the cd command in Octave to change to this directory before starting this exercise.

 

Files included in this exercise

 

ex5.m – Octave script that will help step you through the exercise ex5data1.mat – Dataset

 

submit.m – Submission script that sends your solutions to our servers submitWeb.m – Alternative submission script featureNormalize.m – Feature normalization function fmincg.m – Function minimization routine (similar to fminunc)

 

plotFit.m – Plot a polynomial t

 

trainLinearReg.m – Trains linear regression using your cost function

 

[?] linearRegCostFunction.m – Regularized linear regression cost func-tion

 

[?] learningCurve.m – Generates a learning curve

 

[?] polyFeatures.m – Maps data into polynomial feature space

 

[?] validationCurve.m – Generates a cross validation curve

 

  • indicates  les you will need to complete

 

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Throughout the exercise, you will be using the script ex5.m. These scripts set up the dataset for the problems and make calls to functions that you will write. You are only required to modify functions in other les, by following the instructions in this assignment.

 

Where to get help

 

We also strongly encourage using the online Q&A Forum to discuss exer-cises with other students. However, do not look at any source code written by others or share your source code with others.

 

If you run into network errors using the submit script, you can also use an online form for submitting your solutions. To use this alternative submis-sion interface, run the submitWeb script to generate a submission le (e.g., submit ex5 part2.txt). You can then submit this le through the web submission form in the programming exercises page (go to the programming exercises page, then select the exercise you are submitting for). If you are having no problems submitting through the standard submission system us-ing the submit script, you do not need to use this alternative submission interface.

 

 

 

 

 

  • Regularized Linear Regression

 

In the rst half of the exercise, you will implement regularized linear regres-sion to predict the amount of water owing out of a dam using the change of water level in a reservoir. In the next half, you will go through some diag-nostics of debugging learning algorithms and examine the e ects of bias v.s. variance.

 

The provided script, ex5.m, will help you step through this exercise.

 

1.1 Visualizing the dataset

 

We will begin by visualizing the dataset containing historical records on the change in the water level, x, and the amount of water owing out of the dam,

 

  • .

 

This dataset is divided into three parts:

 

ˆ A training set that your model will learn on: X, y

 

 

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  • A cross validation set for determining the regularization parameter: Xval, yval

 

  • A test set for evaluating performance. These are \unseen” examples which your model did not see during training: Xtest, ytest

 

The next step of ex5.m will plot the training data (Figure 1). In the following parts, you will implement linear regression and use that to t a straight line to the data and plot learning curves. Following that, you will implement polynomial regression to nd a better t to the data.

 

 

 

 

 

 

Water flowing out of the dam (y)

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−50 −40 −30 −20 −10 0 10 20 30 40

 

Change in water level (x)

 

Figure 1: Data

 

 

1.2 Regularized linear regression cost function

 

Recall that regularized linear regression has the following cost function:

J( ) = 2m   =1 (h (x(i))  y(i))2! + 2m  j=1  j2! ;
1 m     n  
Xi   X  
         
         

where is a regularization parameter which controls the degree of regu-larization (thus, help preventing over tting). The regularization term puts a penalty on the overal cost J. As the magnitudes of the model parameters j increase, the penalty increases as well. Note that you should not regular-ize the 0 term. (In Octave, the 0 term is represented as theta(1) since indexing in Octave starts from 1).

 

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You should now complete the code in the le linearRegCostFunction.m. Your task is to write a function to calculate the regularized linear regression cost function. If possible, try to vectorize your code and avoid writing loops. When you are nished, the next part of ex5.m will run your cost function using theta initialized at [1; 1]. You should expect to see an output of 303.993.

 

You should now submit your regularized linear regression cost function.

 

1.3 Regularized linear regression gradient

 

Correspondingly, the partial derivative of regularized linear regression’s cost for j is de ned as

 

@J( ) = 1     m (h (x(i)) y(i))xj(i)       for j = 0
               
@ 0           Xi          
      m =1          
                y(i))xj ! + m j   for j   1
  @ j =   m =1 (h (x(i))
@J( )     1     m (i)      
                  Xi          
                           

In linearRegCostFunction.m, add code to calculate the gradient, re-turning it in the variable grad. When you are nished, the next part of ex5.m will run your gradient function using theta initialized at [1; 1]. You should expect to see a gradient of [-15.30; 598.250].

 

You should now submit your regularized linear regression gradient func-tion.

 

1.4 Fitting linear regression

 

Once your cost function and gradient are working correctly, the next part of ex5.m will run the code in trainLinearReg.m to compute the optimal values of . This training function uses fmincg to optimize the cost function.

 

In this part, we set regularization parameter to zero. Because our current implementation of linear regression is trying to t a 2-dimensional , regularization will not be incredibly helpful for a of such low dimension. In the later parts of the exercise, you will be using polynomial regression with regularization.

 

Finally, the ex5.m script should also plot the best t line, resulting in an image similar to Figure 2. The best t line tells us that the model is

 

 

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not a good t to the data because the data has a non-linear pattern. While visualizing the best t as shown is one possible way to debug your learning algorithm, it is not always easy to visualize the data and model. In the next section, you will implement a function to generate learning curves that can help you debug your learning algorithm even if it is not easy to visualize the data.

 

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(y)                    
the dam 25                  
20                  
out of                  
                   
flowing 15                  
                   
Water 10                  
                   
  5                  
  0                  
  −5 −40 −30 −20 −10 0 10 20 30 40
  −50

 

Change in water level (x)

 

Figure 2: Linear Fit

 

 

 

  • Bias-variance

 

An important concept in machine learning is the bias-variance tradeo . Mod-els with high bias are not complex enough for the data and tend to under t, while models with high variance over t to the training data.

 

In this part of the exercise, you will plot training and test errors on a learning curve to diagnose bias-variance problems.

 

2.1 Learning curves

 

You will now implement code to generate the learning curves that will be useful in debugging learning algorithms. Recall that a learning curve plots

 

 

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training and cross validation error as a function of training set size. Your job is to ll in learningCurve.m so that it returns a vector of errors for the training set and cross validation set.

 

To plot the learning curve, we need a training and cross validation set error for di erent training set sizes. To obtain di erent training set sizes, you should use di erent subsets of the original training set X. Speci cally, for a training set size of i, you should use the rst i examples (i.e., X(1:i,:) and y(1:i)).

 

You can use the trainLinearReg function to nd the parameters. Note that the lambda is passed as a parameter to the learningCurve function. After learning the parameters, you should compute the error on the train-ing and cross validation sets. Recall that the training error for a dataset is de ned as

Jtrain( ) = 2m  =1 (h (x(i))  y(i))2 # :
1   m    
  Xi    
         
         

In particular, note that the training error does not include the regular-ization term. One way to compute the training error is to use your existing cost function and set to 0 only when using it to compute the training error and cross validation error. When you are computing the training set error, make sure you compute it on the training subset (i.e., X(1:n,:) and y(1:n)) (instead of the entire training set). However, for the cross validation error, you should compute it over the entire cross validation set. You should store the computed errors in the vectors error train and error val.

 

 

When you are nished, ex5.m wil print the learning curves and produce a plot similar to Figure 3.

 

You should now submit your learning curve function.

 

In Figure 3, you can observe that both the train error and cross validation error are high when the number of training examples is increased. This re ects a high bias problem in the model { the linear regression model is too simple and is unable to t our dataset well. In the next section, you will implement polynomial regression to t a better model for this dataset.

 

 

  • Polynomial regression

 

The problem with our linear model was that it was too simple for the data and resulted in under tting (high bias). In this part of the exercise, you will

 

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Learning curve for linear regression

150            
            Train
            Cross Validation
100            
Error            
50            
0 2 4 6 8 10 12
0
      Number of training examples    

 

 

Figure 3: Linear regression learning curve

 

address this problem by adding more features.

 

For use polynomial regression, our hypothesis has the form:

 

 

h (x) =  0 + 1 (waterLevel) + 2 (waterLevel)2 + + p (waterLevel)p

  • 0+ 1x1 + 2x2 + ::: + pxp:

 

Notice that by de ning x1 = (waterLevel); x2 = (waterLevel)2; : : : ; xp = (waterLevel)p, we obtain a linear regression model where the features are the various powers of the original value (waterLevel).

 

Now, you will add more features using the higher powers of the existing feature x in the dataset. Your task in this part is to complete the code in polyFeatures.m so that the function maps the original training set X of size m 1 into its higher powers. Speci cally, when a training set X of size m 1 is passed into the function, the function should return a m p matrix X poly, where column 1 holds the original values of X, column 2 holds the values of X.^2, column 3 holds the values of X.^3, and so on. Note that you don’t have to account for the zero-eth power in this function.

 

 

Now you have a function that will map features to a higher dimension, and Part 6 of ex5.m will apply it to the training set, the test set, and the cross validation set (which you haven’t used yet).

 

You should now submit your polynomial feature mapping function.

 

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3.1 Learning Polynomial Regression

 

After you have completed polyFeatures.m, the ex5.m script will proceed to train polynomial regression using your linear regression cost function.

 

Keep in mind that even though we have polynomial terms in our feature vector, we are still solving a linear regression optimization problem. The polynomial terms have simply turned into features that we can use for linear regression. We are using the same cost function and gradient that you wrote for the earlier part of this exercise.

 

For this part of the exercise, you will be using a polynomial of degree 8. It turns out that if we run the training directly on the projected data, will not work well as the features would be badly scaled (e.g., an example with x = 40 will now have a feature x8 = 408 = 6:5 1012). Therefore, you will need to use feature normalization.

 

Before learning the parameters for the polynomial regression, ex5.m will rst call featureNormalize and normalize the features of the training set, storing the mu, sigma parameters separately. We have already implemented this function for you and it is the same function from the rst exercise.

 

After learning the parameters , you should see two plots (Figure 4,5) generated for polynomial regression with = 0.

 

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(y) 10                
dam                
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of the                
                 
out −10                
                 
flowing −20                
                 
Water −30                
                 
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Change in water level (x)

 

Figure 4: Polynomial t, = 0

 

From Figure 4, you should see that the polynomial t is able to follow the datapoints very well – thus, obtaining a low training error. However, the

 

 

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Polynomial Regression Learning Curve (lambda = 0.000000)

  100            
              Train
  90           Cross Validation
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  60            
Error 50            
             
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        Number of training examples    

 

 

Figure 5: Polynomial learning curve, = 0

 

 

polynomial t is very complex and even drops o at the extremes. This is an indicator that the polynomial regression model is over tting the training data and will not generalize well.

 

To better understand the problems with the unregularized ( = 0) model, you can see that the learning curve (Figure 5) shows the same e ect where the low training error is low, but the cross validation error is high. There is a gap between the training and cross validation errors, indicating a high variance problem.

 

One way to combat the over tting (high-variance) problem is to add regularization to the model. In the next section, you will get to try di erent

 

parameters to see how regularization can lead to a better model.

 

 

3.2 Optional (ungraded) exercise: Adjusting the reg-ularization parameter

 

In this section, you will get to observe how the regularization parameter a ects the bias-variance of regularized polynomial regression. You should now modify the the lambda parameter in the ex5.m and try = 1; 100. For each of these values, the script should generate a polynomial t to the data and also a learning curve.

 

For = 1, you should see a polynomial t that follows the data trend well (Figure 6) and a learning curve (Figure 7) showing that both the cross

 

 

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validation and training error converge to a relatively low value. This shows the = 1 regularized polynomial regression model does not have the high-bias or high-variance problems. In e ect, it achieves a good trade-o between bias and variance.

 

For = 100, you should see a polynomial t (Figure 8) that does not follow the data well. In this case, there is too much regularization and the model is unable to t the training data.

 

You do not need to submit any solutions for this optional (ungraded) exercise.

 

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dam (y) 120                
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of the                
                 
flowing out 80                
60                
Water 40                
                 
  20                
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Change in water level (x)

 

Figure 6: Polynomial t, = 1

 

 

 

3.3 Selecting using a cross validation set

 

From the previous parts of the exercise, you observed that the value of can signi cantly a ect the results of regularized polynomial regression on the training and cross validation set. In particular, a model without regular-ization ( = 0) ts the training set well, but does not generalize. Conversely, a model with too much regularization ( = 100) does not t the training set and testing set well. A good choice of (e.g., = 1) can provide a good t to the data.

 

In this section, you will implement an automated method to select the parameter. Concretely, you will use a cross validation set to evaluate

 

 

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Polynomial Regression Learning Curve (lambda = 1.000000)

  100            
              Train
  90           Cross Validation
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  70            
  60            
Error 50            
             
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  0
        Number of training examples    

 

 

Figure 7: Polynomial learning curve, = 1

 

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(y) 25                
dam                
20                
of the                
                 
out 15                
                 
flowing 10                
                 
Water 5                
                 
  0                
  −5                
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  −80

 

Change in water level (x)

 

Figure 8: Polynomial t, = 100

 

 

how good each value is. After selecting the best value using the cross validation set, we can then evaluate the model on the test set to estimate how well the model will perform on actual unseen data.

 

Your task is to complete the code in validationCurve.m. Speci cally, you should should use the trainLinearReg function to train the model using

 

 

 

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di erent values of and compute the training error and cross validation error.

 

You should try in the following range: f0; 0:001; 0:003; 0:01; 0:03; 0:1; 0:3; 1; 3; 10g.

 

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                  Train    
  18               Cross Validation  
                     
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Error 10                    
                     
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            lambda          

 

 

Figure 9: Selecting using a cross validation set

 

After you have completed the code, the next part of ex5.m will run your function can plot a cross validation curve of error v.s. that allows you select which parameter to use. You should see a plot similar to Figure 9. In this gure, we can see that the best value of is around 3. Due to randomness in the training and validation splits of the dataset, the cross validation error can sometimes be lower than the training error.

 

You should now submit your validation curve function.

 

 

3.4 Optional (ungraded) exercise: Computing test set error

 

 

In the previous part of the exercise, you implemented code to compute the cross validation error for various values of the regularization parameter . However, to get a better indication of the model’s performance in the real world, it is important to evaluate the \ nal” model on a test set that was not used in any part of training (that is, it was neither used to select the parameters, nor to learn the model parameters ).

 

For this optional (ungraded) exercise, you should compute the test error using the best value of you found. In our cross validation, we obtained a

 

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test error of 3.8599 for = 3.

 

You do not need to submit any solutions for this optional (ungraded) exercise.

 

 

3.5 Optional (ungraded) exercise: Plotting learning curves with randomly selected examples

 

In practice, especially for small training sets, when you plot learning curves to debug your algorithms, it is often helpful to average across multiple sets of randomly selected examples to determine the training error and cross validation error.

 

Concretely, to determine the training error and cross validation error for i examples, you should rst randomly select i examples from the training set and i examples from the cross validation set. You will then learn the param-eters using the randomly chosen training set and evaluate the parameters

 

on the randomly chosen training set and cross validation set. The above steps should then be repeated multiple times (say 50) and the averaged error should be used to determine the training error and cross validation error for i examples.

 

For this optional (ungraded) exercise, you should implement the above strategy for computing the learning curves. For reference, gure 10 shows the learning curve we obtained for polynomial regression with = 0:01. Your gure may di er slightly due to the random selection of examples.

 

You do not need to submit any solutions for this optional (ungraded) exercise.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Polynomial Regression Learning Curve (lambda = 0.010000)

  100            
              Train
  90           Cross Validation
  80            
  70            
  60            
Error 50            
             
  40            
  30            
  20            
  10            
  0 2 4 6 8 10 12
  0
        Number of training examples    

 

 

Figure 10: Optional (ungraded) exercise: Learning curve with randomly selected examples

 

Submission and Grading

 

After completing various parts of the assignment, be sure to use the submit function system to submit your solutions to our servers. The following is a breakdown of how each part of this exercise is scored.

 

Part Submitted File Points
     
Regularized Linear Regression Cost linearRegCostFunction.m 25 points
Function    
Regularized Linear Regression Gra- linearRegCostFunction.m 25 points
dient    
     
Learning Curve learningCurve.m 20 points
     
Polynomial Feature Mapping polyFeatures.m 10 points
Cross Validation Curve validationCurve.m 20 points
     
Total Points   100 points
     

 

You are allowed to submit your solutions multiple times, and we will take only the highest score into consideration. To prevent rapid- re guessing, the system enforces a minimum of 5 minutes between submissions.

 

 

 

 

 

 

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