ML-Exercise 5 Regularized Linear Regression and Bias v.s. Variance Solved

In this exercise, you will implement regularized linear regression and use it to study models with different bias-variance properties.

1           Regularized Linear Regression
In the first half of the exercise, you will implement regularized linear regression to predict the amount of water flowing out of a dam using the change of water level in a reservoir. In the next half, you will go through some diagnostics of debugging learning algorithms and examine the effects 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 flowing out of the dam, y.

This dataset is divided into three parts:

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

•   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 fit a straight line to the data and plot learning curves. Following that, you will implement polynomial regression to find a better fit to the data.

 

Figure 1: Data

1.2         Regularized linear regression cost function
Recall that regularized linear regression has the following cost function:

  ,

where λ is a regularization parameter which controls the degree of regularization (thus, help preventing overfitting). The regularization term puts a penalty on the overal cost J. As the magnitudes of the model parameters θj increase, the penalty increases as well. Note that you should not regularize the θ0 term. (In Octave/MATLAB, the θ0 term is represented as theta(1) since indexing in Octave/MATLAB starts from 1).

You should now complete the code in the file 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 finished, 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 solutions.

1.3         Regularized linear regression gradient
Correspondingly, the partial derivative of regularized linear regression’s cost for θj is defined as

 for j = 0

for j ≥ 1

In linearRegCostFunction.m, add code to calculate the gradient, returning it in the variable grad. When you are finished, 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 solutions.

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 θ. This training function uses fmincg to optimize the cost function.

In this part, we set regularization parameter λ to zero. Because our current implementation of linear regression is trying to fit a 2-dimensional θ, regularization will not be incredibly helpful for a θ 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 fit line, resulting in an image similar to Figure 2. The best fit line tells us that the model is not a good fit to the data because the data has a non-linear pattern. While visualizing the best fit 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.

 

Figure 2: Linear Fit

2           Bias-variance
An important concept in machine learning is the bias-variance tradeoff. Models with high bias are not complex enough for the data and tend to underfit, while models with high variance overfit 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 training and cross validation error as a function of training set size. Your job is to fill 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 different training set sizes. To obtain different training set sizes, you should use different subsets of the original training set X. Specifically, for a training set size of i, you should use the first i examples (i.e., X(1:i,:) and y(1:i)).

You can use the trainLinearReg function to find the θ parameters. Note that the lambda is passed as a parameter to the learningCurve function. After learning the θ parameters, you should compute the error on the training and cross validation sets. Recall that the training error for a dataset is defined as

  .

In particular, note that the training error does not include the regularization term. One way to compute the training error is to use your existing cost function and set λ 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 finished, ex5.m wil print the learning curves and produce a plot similar to Figure 3.

You should now submit your solutions.

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 reflects a high bias problem in the model – the linear regression model is

Learning curve for linear regression

 

Figure 3: Linear regression learning curve

too simple and is unable to fit our dataset well. In the next section, you will implement polynomial regression to fit a better model for this dataset.

3           Polynomial regression
The problem with our linear model was that it was too simple for the data and resulted in underfitting (high bias). In this part of the exercise, you will address this problem by adding more features.

For use polynomial regression, our hypothesis has the form:

hθ(x) = θ0 + θ1 ∗ (waterLevel) + θ2 ∗ (waterLevel)2 + ··· + θp ∗ (waterLevel)p = θ0 + θ1x1 + θ2x2 + ... + θpxp.

Notice that by defining 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. Specifically, 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 solutions.

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 θ for the polynomial regression, ex5.m will first 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 first exercise.

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

From Figure 4, you should see that the polynomial fit is able to follow the datapoints very well - thus, obtaining a low training error. However, the polynomial fit is very complex and even drops off at the extremes. This is an indicator that the polynomial regression model is overfitting the training data and will not generalize well.

To better understand the problems with the unregularized (λ = 0) model, you can see that the learning curve (Figure 5) shows the same effect 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.

 

Figure 4: Polynomial fit, λ = 0

Polynomial Regression Learning Curve (lambda = 0.000000)

 

Figure 5: Polynomial learning curve, λ = 0

One way to combat the overfitting (high-variance) problem is to add regularization to the model. In the next section, you will get to try different λ parameters to see how regularization can lead to a better model.