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Programming Exercise 3: Multi-class Classification and Neural Networks Solution
Introduction
In this exercise, you will implement one-vs-all logistic regression and neural networks to recognize hand-written digits. Before starting 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/MATLAB to change to this directory before starting this exercise.
You can also nd instructions for installing Octave/MATLAB in the \En-vironment Setup Instructions" of the course website.
Files included in this exercise
ex3.m - Octave/MATLAB script that steps you through part 1 ex3 nn.m - Octave/MATLAB script that steps you through part 2 ex3data1.mat - Training set of hand-written digits ex3weights.mat - Initial weights for the neural network exercise submit.m - Submission script that sends your solutions to our servers displayData.m - Function to help visualize the dataset fmincg.m - Function minimization routine (similar to fminunc)
sigmoid.m - Sigmoid function
[?] lrCostFunction.m - Logistic regression cost function
[?] oneVsAll.m - Train a one-vs-all multi-class classi er
[?] predictOneVsAll.m - Predict using a one-vs-all multi-class classi er
[?] predict.m - Neural network prediction function
indicates les you will need to complete
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Throughout the exercise, you will be using the scripts ex3.m and ex3 nn.m. These scripts set up the dataset for the problems and make calls to functions that you will write. You do not need to modify these scripts. You are only required to modify functions in other les, by following the instructions in this assignment.
Where to get help
The exercises in this course use Octave1 or MATLAB, a high-level program-ming language well-suited for numerical computations. If you do not have Octave or MATLAB installed, please refer to the installation instructions in the \Environment Setup Instructions" of the course website.
At the Octave/MATLAB command line, typing help followed by a func-tion name displays documentation for a built-in function. For example, help plot will bring up help information for plotting. Further documentation for Octave functions can be found at the Octave documentation pages. MAT-LAB documentation can be found at the MATLAB documentation pages.
We also strongly encourage using the online Discussions to discuss ex-ercises with other students. However, do not look at any source code written by others or share your source code with others.
Multi-class Classi cation
For this exercise, you will use logistic regression and neural networks to recognize handwritten digits (from 0 to 9). Automated handwritten digit recognition is widely used today - from recognizing zip codes (postal codes) on mail envelopes to recognizing amounts written on bank checks. This exercise will show you how the methods you’ve learned can be used for this classi cation task.
In the rst part of the exercise, you will extend your previous implemen-tion of logistic regression and apply it to one-vs-all classi cation.
1Octave is a free alternative to MATLAB. For the programming exercises, you are free to use either Octave or MATLAB.
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1.1 Dataset
You are given a data set in ex3data1.mat that contains 5000 training exam-ples of handwritten digits.2 The .mat format means that that the data has been saved in a native Octave/MATLAB matrix format, instead of a text (ASCII) format like a csv- le. These matrices can be read directly into your program by using the load command. After loading, matrices of the correct dimensions and values will appear in your program’s memory. The matrix will already be named, so you do not need to assign names to them.
Load saved matrices from file load('ex3data1.mat');
The matrices X and y will now be in your Octave environment
There are 5000 training examples in ex3data1.mat, where each training example is a 20 pixel by 20 pixel grayscale image of the digit. Each pixel is represented by a oating point number indicating the grayscale intensity at that location. The 20 by 20 grid of pixels is \unrolled" into a 400-dimensional vector. Each of these training examples becomes a single row in our data matrix X. This gives us a 5000 by 400 matrix X where every row is a training example for a handwritten digit image.
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| (x(1))T |
| (x(2))T | 7
X = 6
| (x(m...))T |
7
6
7
4
5
The second part of the training set is a 5000-dimensional vector y that contains labels for the training set. To make things more compatible with Octave/MATLAB indexing, where there is no zero index, we have mapped the digit zero to the value ten. Therefore, a \0" digit is labeled as \10", while the digits \1" to \9" are labeled as \1" to \9" in their natural order.
1.2 Visualizing the data
You will begin by visualizing a subset of the training set. In Part 1 of ex3.m, the code randomly selects selects 100 rows from X and passes those rows to the displayData function. This function maps each row to a 20 pixel by 20 pixel grayscale image and displays the images together. We have provided
This is a subset of the MNIST handwritten digit dataset (http://yann.lecun.com/ exdb/mnist/).
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1+e z
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the displayData function, and you are encouraged to examine the code to see how it works. After you run this step, you should see an image like Figure 1.
Figure 1: Examples from the dataset
1.3 Vectorizing Logistic Regression
You will be using multiple one-vs-all logistic regression models to build a multi-class classi er. Since there are 10 classes, you will need to train 10 separate logistic regression classi ers. To make this training e cient, it is important to ensure that your code is well vectorized. In this section, you will implement a vectorized version of logistic regression that does not employ any for loops. You can use your code in the last exercise as a starting point for this exercise.
1.3.1 Vectorizing the cost function
We will begin by writing a vectorized version of the cost function. Recall that in (unregularized) logistic regression, the cost function is
J()=1
m
y(i) log(h (x(i))) (1
y(i)) log(1 h (x(i))) :
Xi
m
=1
To compute each element in the summation, we have to compute h (x(i))
for every example i, where h (x(i)) = g( T x(i)) and g(z) = is the
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sigmoid function. It turns out that we can compute this quickly for all our examples by using matrix multiplication. Let us de ne X and as
2
| (x(2))T |
3
2
1
3
6
| (x(1))T |
7
6
0
7
X =
...
and =
...
:
6
| (x(m))T |
7
6
n
7
6
7
6
7
4
5
4
5
Then, by computing the matrix product X , we have
2
| (x(2))T |
3
2
| T (x(2)) |
3
6
| (x(1))T |
7
6
| T (x(1)) |
7
X =
...
=
...
:
6
| (x(m))T |
7
6
| T (x(m)) |
7
6
7
6
7
4
5
4
5
In the last equality, we used the fact that aT b = bT a if a and b are vectors. This allows us to compute the products T x(i) for all our examples i in one line of code.
Your job is to write the unregularized cost function in the le lrCostFunction.m Your implementation should use the strategy we presented above to calcu-
late T x(i). You should also use a vectorized approach for the rest of the cost function. A fully vectorized version of lrCostFunction.m should not contain any loops.
(Hint: You might want to use the element-wise multiplication operation (.*) and the sum operation sum when writing this function)
1.3.2 Vectorizing the gradient
Recall that the gradient of the (unregularized) logistic regression cost is a vector where the jth element is de ned as
@J
=
1 m
(i)
:
@ j
m i=1
(h (x(i)) y(i))xj
X
To vectorize this operation over the dataset, we start by writing out all
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the partial derivatives explicitly for all j,
@ 0
2
i=1
(x
)
y
)x0
3
(h
@J
m
(i)
(i)
(i)
2
3
Pm
(i)
(i)
(i)
@J
6
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i=1
(h (x )
y )x1
@ 1
@J
1
6
(i)
7
6
7
=
6 Pm
(h
(x(i)) y(i))x
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@ 2
m
2
6
.
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i=1
7
6
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6
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.
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.
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.
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P
.
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.
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@J
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m
(i)
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(i)
(i)
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@ n
4
5
6
i=1 (h (x
)xn
7
6
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1
4m P
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=
Xi
(h
(x(i)) y(i))x(i)
m
=1
=
1
XT (h (x) y):
(1)
m
where
2
h (x(1))
y(1)
3
h (x)
y =
6 h (x(2)
...)
y(2)
7
:
6 h (x(1))
y(m)
7
6
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Note that x
(i)
4
(i)
)
y
(i)
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is a vector, while (h (x
) is a scalar (single number).
To understand the last step of the derivation, let i = (h (x(i)) y(i)) and observe that:
ix(i) =
2
x(1)j
x(2)j
Xi
4
j
j
where the values i = (h (x(i))
2
3
: : : x(jm)
3
1
7
..2
= XT
;
j
6
.
5 6
m
7
6
7
4
5
y(i)).
The expression above allows us to compute all the partial derivatives without any loops. If you are comfortable with linear algebra, we encourage you to work through the matrix multiplications above to convince yourself that the vectorized version does the same computations. You should now implement Equation 1 to compute the correct vectorized gradient. Once you are done, complete the function lrCostFunction.m by implementing the gradient.
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Debugging Tip: Vectorizing code can sometimes be tricky. One com-mon strategy for debugging is to print out the sizes of the matrices you are working with using the size function. For example, given a data ma-trix X of size 100 20 (100 examples, 20 features) and , a vector with dimensions 20 1, you can observe that X is a valid multiplication oper-ation, while X is not. Furthermore, if you have a non-vectorized version of your code, you can compare the output of your vectorized code and non-vectorized code to make sure that they produce the same outputs.
1.3.3 Vectorizing regularized logistic regression
After you have implemented vectorization for logistic regression, you will now add regularization to the cost function. Recall that for regularized logistic regression, the cost function is de ned as
J()=1
m
y(i) log(h (x(i))) (1 y(i)) log(1 h (x(i))) +
n
j2:
Xi
X
m
2m
=1
j=1
Note that you should not be regularizing 0 which is used for the bias term.
Correspondingly, the partial derivative of regularized logistic regression 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
Now modify your code in lrCostFunction to account for regularization.
Once again, you should not put any loops into your code.
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Octave/MATLAB Tip: When implementing the vectorization for reg-ularized logistic regression, you might often want to only sum and update certain elements of . In Octave/MATLAB, you can index into the matri-ces to access and update only certain elements. For example, A(:, 3:5)
B(:, 1:3) will replaces the columns 3 to 5 of A with the columns 1 to 3 from B. One special keyword you can use in indexing is the end keyword in indexing. This allows us to select columns (or rows) until the end of the matrix. For example, A(:, 2:end) will only return elements from the 2nd to last column of A. Thus, you could use this together with the sum and
.^ operations to compute the sum of only the elements you are interested in (e.g., sum(z(2:end).^2)). In the starter code, lrCostFunction.m, we have also provided hints on yet another possible method computing the regularized gradient.
You should now submit your solutions.
1.4 One-vs-all Classi cation
In this part of the exercise, you will implement one-vs-all classi cation by training multiple regularized logistic regression classi ers, one for each of the K classes in our dataset (Figure 1). In the handwritten digits dataset, K = 10, but your code should work for any value of K.
You should now complete the code in oneVsAll.m to train one classi er for each class. In particular, your code should return all the classi er parameters in a matrix 2 RK (N+1) , where each row of corresponds to the learned logistic regression parameters for one class. You can do this with a \for"-loop from 1 to K, training each classi er independently.
Note that the y argument to this function is a vector of labels from 1 to 10, where we have mapped the digit \0" to the label 10 (to avoid confusions with indexing).
When training the classi er for class k 2 f1; :::; Kg, you will want a m-dimensional vector of labels y, where yj 2 0; 1 indicates whether the j-th training instance belongs to class k (yj = 1), or if it belongs to a di erent class (yj = 0). You may nd logical arrays helpful for this task.
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Octave/MATLAB Tip: Logical arrays in Octave/MATLAB are arrays which contain binary (0 or 1) elements. In Octave/MATLAB, evaluating the expression a == b for a vector a (of size m 1) and scalar b will return a vector of the same size as a with ones at positions where the elements of a are equal to b and zeroes where they are di erent. To see how this works for yourself, try the following code in Octave/MATLAB:
a = 1:10; % Create a and b
b = 3;
a == b % You should try different values of b here
Furthermore, you will be using fmincg for this exercise (instead of fminunc). fmincg works similarly to fminunc, but is more more e cient for dealing with a large number of parameters.
After you have correctly completed the code for oneVsAll.m, the script ex3.m will continue to use your oneVsAll function to train a multi-class clas-si er.
You should now submit your solutions.
1.4.1 One-vs-all Prediction
After training your one-vs-all classi er, you can now use it to predict the digit contained in a given image. For each input, you should compute the \probability" that it belongs to each class using the trained logistic regression classi ers. Your one-vs-all prediction function will pick the class for which the corresponding logistic regression classi er outputs the highest probability and return the class label (1, 2,..., or K) as the prediction for the input example.
You should now complete the code in predictOneVsAll.m to use the one-vs-all classi er to make predictions.
Once you are done, ex3.m will call your predictOneVsAll function using the learned value of . You should see that the training set accuracy is about 94.9% (i.e., it classi es 94.9% of the examples in the training set correctly).
You should now submit your solutions.
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Neural Networks
In the previous part of this exercise, you implemented multi-class logistic re-gression to recognize handwritten digits. However, logistic regression cannot form more complex hypotheses as it is only a linear classi er.3
In this part of the exercise, you will implement a neural network to rec-ognize handwritten digits using the same training set as before. The neural network will be able to represent complex models that form non-linear hy-potheses. For this week, you will be using parameters from a neural network that we have already trained. Your goal is to implement the feedforward propagation algorithm to use our weights for prediction. In next week’s ex-ercise, you will write the backpropagation algorithm for learning the neural network parameters.
The provided script, ex3 nn.m, will help you step through this exercise.
2.1 Model representation
Our neural network is shown in Figure 2. It has 3 layers { an input layer, a hidden layer and an output layer. Recall that our inputs are pixel values of digit images. Since the images are of size 20 20, this gives us 400 input layer units (excluding the extra bias unit which always outputs +1). As before, the training data will be loaded into the variables X and y.
You have been provided with a set of network parameters ( (1); (2)) already trained by us. These are stored in ex3weights.mat and will be loaded by ex3 nn.m into Theta1 and Theta2 The parameters have dimensions that are sized for a neural network with 25 units in the second layer and 10 output units (corresponding to the 10 digit classes).
Load saved matrices from file load('ex3weights.mat');
The matrices Theta1 and Theta2 will now be in your Octave
environment
Theta1 has size 25 x 401
Theta2 has size 10 x 26
3You could add more features (such as polynomial features) to logistic regression, but that can be very expensive to train.
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Figure 2: Neural network model.
2.2 Feedforward Propagation and Prediction
Now you will implement feedforward propagation for the neural network. You will need to complete the code in predict.m to return the neural network’s prediction.
You should implement the feedforward computation that computes h (x(i)) for every example i and returns the associated predictions. Similar to the one-vs-all classi cation strategy, the prediction from the neural network will be the label that has the largest output (h (x))k.
Implementation Note: The matrix X contains the examples in rows. When you complete the code in predict.m, you will need to add the column of 1’s to the matrix. The matrices Theta1 and Theta2 contain the parameters for each unit in rows. Speci cally, the rst row of Theta1 corresponds to the rst hidden unit in the second layer. In Octave/MAT-LAB, when you compute z(2) = (1)a(1), be sure that you index (and if necessary, transpose) X correctly so that you get a(l) as a column vector.
Once you are done, ex3 nn.m will call your predict function using the loaded set of parameters for Theta1 and Theta2. You should see that the
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accuracy is about 97.5%. After that, an interactive sequence will launch dis-playing images from the training set one at a time, while the console prints out the predicted label for the displayed image. To stop the image sequence, press Ctrl-C.
You should now submit your solutions.
Submission and Grading
After completing this assignment, be sure to use the submit function to sub-mit your solutions to our servers. The following is a breakdown of how each part of this exercise is scored.
Part
Submitted File
Points
Regularized Logisic Regression
lrCostFunction.m
30 points
One-vs-all classi er training
oneVsAll.m
20
points
One-vs-all classi er prediction
predictOneVsAll.m
20
points
Neural Network Prediction Function
predict.m
30 points
Total Points
100 points
You are allowed to submit your solutions multiple times, and we will take only the highest score into consideration.
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