This is the last problem set in 6.034! To work on this problem set, you will need to get the code.

You will need to download and install an additional software package called Orange for the second part of the lab. Please download Orange first so that you get the problems worked out early. If you have downloaded and installed it, you should be able to run orange_for_6034.py and get the Orange version number. Once youve filled in the boosting part, you should be able to run lab5.py, and see the output of several classifiers on the vampire dataset.

• Orange is available for Linux (Ubuntu), Windows, and OS X.
• For Ubuntu users: Please follow the install instruction here (see section on Linux). Note: just running apt-get orange on Ubuntu will *NOT* install Orange but a completely different software package!
• To check that your Orange is properly installed, run:

python orange_for_6034.py

and you should get a version string and no errors.

Your answers for the problem set belong in the main file lab5.py, as well as neural_net.py and boost.py.

Neural Nets

In this part of Lab 5, you are to complete the API for a Neural Net. Then afterwards, you are to construct various neural nets using the API to solve abstract learning problems.

Completing the implementation

We have provided you a skeleton of the Neural Net in neural_net.py. You are to complete the unimplemented methods.

The three classes Input, PerformanceElem, and Neuron all have incomplete implementation of the following two functions:

def output(self) def dOutdX(self, elem)

Your first task is to fill in all 6 functions to complete the API.

Output

The function output(self) produces the output of each of these elements.

Be sure to use the sigmoid and ds/dz functions as discussed in class:

o = s(z) = 1.0 / (1.0 + e**(-z)) ds(o)/dz = s(z) * (1 s(z)) = o * (1 o)

and the performance function and its derivative as discussed in class:

P(o) = -0.5 (d o)**2 dP(o)/dx = (d o)

Derivatives

The function dOutdX(self, elem) generates the value of the partial derivative with respect to a given weight element.

Recall, neural nets update a given weight by computing the partial derivative of the performance function with respect to that weight. The formula we have used in class is as follows:

wi = wi + rate * dP / dwi

In our code this is represented as (see def train() you dont have to implement this):

w.set_next_value( w.get_value() + rate * network.performance.dOutdX(w) )

The element passed to the dOutdX function is always a weight. Namely it is the weight that we are doing the partial over. Your job is to figure out how to define dOutdX() in terms of recursively calling dOutdX() or output() over the inputs and weights of a network element.

For example, consider the Performance element P, P.dOutdX(w) could be defined in the following recursive fashion:

dP/d(w) = dP/do * do/dw # applying the chain rule = (d-o) * o.dOutdX(w)

Here o is the output of the Neuron that is directly connected to P.

For Neuron units, there would be more cases to consider. Namely,

1. The termination case where the weight being differentiated over is one of the (direct) weights of the current neuron.
2. The recursive case where the weight is not one that is directly connected (but is a descendant weight).

This implementation models the process of computing the chain of derivatives recursively. This topdown approach works from the output layer towards the input layers. This is in contrast to the more conventional approach (taught in class) where you computed a per-node little-delta, and then recursively computed the weight updates bottom-up, from input towards output layers.

If you are confused about how the top-down recursive chaining of derivatives work, first read the course notes to review. If you are still confused, ask the TAs for hints and clarifications.

About the API Classes

Most of the classes in the Neural Network inherit from the following two abstract classes:

ValuedElement

This interface class allows an element to have a settable value. Input (e.g. i1, i2) and Weight (e.g. w1A, wAB) are subclasses of ValueElement

Elements that are subclassed all have these methods:

• set_value(self,val) set the value of the element
• get_value(self) get the value of the element
• get_name(self) get the name of the element

DifferentiableElement

This abstract class defines the interface for elements that have outputs and are involved in partial derivatives.

• output(self): returns the output of this element
• dOutdX(self,elem): returns the partial derivative with respect to another element

Inputs, Neurons, and PerformanceElem are the three subclasses that implement DifferentiableElement. You will have to complete the interface for these classes.

Weight(ValuedElement)

Represents update-able weights in the network. In addition to ValueElement functions are the following methods, which are used for the training algorithm (you will not need them in your implementation):

• set_next_value(self,val): which sets the next weight value in self.next_value
• update(self): which sets the current weight to the value stored in self.next_value

Input(DifferentiableElement, ValuedElement)

Represents inputs to the network. These may represent variable inputs as well as fixed inputs (i.e.

threshold inputs) that are always set to -1. output() of Input units should simply return the value they are set to during training or testing.

dOutdX(self, elem) of an Input unit should return 0, since there are no weights directly connected into inputs.

Neuron(DifferentiableElement)

Represents the actual neurons in the neural net. The definitions for output and dOutdX already contains some code. Namely, weve implemented a value caching mechanism to speed up the training / testing process. So instead of implementing output and dOutdx directly you should instead implement:

• compute_output(self):
• compute_doutdx(self,elem):

PerformanceElem(DifferentiableElement)

Represents a Performance Element that allows you to set the desired output.

set_desired which sets my_desired_val

To better understand back-propagation, you should take a look at the methods train and test in neural_network.py to see how everything is put together.

Unit Testing

Once youve completed the missing functions, we have provided a python script

neural_net_tester.py to help you unit test. You will need to pass the tests in this unit tester before you can move on to the next parts. To check if your implementation works, run:

python neural_net_tester.py simple

This makes sure that your code works and can learn basic functions such as AND and OR.

Building Neural Nets

Once you have finished implementing the Neural Net API, you will be tasked to build three networks to learn various abstract data sets.

Here is an example of how to construct a basic neural network:

def make_neural_net_basic():

Returns a 1 neuron network with 2 variable inputs, and 1 fixed input.

i0 = Input(i0,-1.0) # this input is immutable i1 = Input(i1,0.0) i2 = Input(i2,0.0)

w1A = Weight(w1A,1) w2A = Weight(w2A,1) wA = Weight(wA, 1)

# the inputs must be in the same order as their associated weights

A = Neuron(A, [i1,i2,i0], [w1A,w2A,wA])

P = PerformanceElem(A, 0.0)

# Package all the components into a network

# First list the PerformanceElem P

# Then list all neurons afterwards net = Network(P,[A])

return net

Naming conventions

IMPORTANT: Be sure to use the following naming convention when creating elements for your networks:

Inputs:

• Format: i + input_number Conventions:
  • Start numbering from 1.
  • Use the same i0 for all the fixed -1 inputs Examples: i1, i2.

Weights:

• Format w + from_identifier + to_identifier Examples:
  • w1A for weight from Input i1 to Neuron A o wBC for weight from Neuron B to Neuron C.

Neurons:

• Format: alphabet_letter
• Convention: Assign neuron names in order of distance to the inputs.
• Example: A is the neuron closest to the inputs, and on the left-most (or top-most) side of the net.
• For ties, order neurons from left to right or top to bottom (depending on how you draw orient your network).

Building a 2-layer Neural Net

Now use the Neural Net API youve just completed and tested to create a two layer network that looks like the following.

Fill your answer in the function stub:

def make_neural_net_two_layer() in neural_net.py.

Your 2-layer neural net should now be able to learn slightly harder datasets, such as the classic nonlinearly separable examples such as NOT-EQUAL (XOR) and EQUAL.

When initializing the weights of the network, you should use random weights. To get deterministic random initial weights so that tests are reproducible you should first seed the random number generator, and then generate the random weights.

seed_random()

wt = random_weight()

use wt

wt2 = random_weight() use wt2

Note: the function random_weight() in neural_net.py uses the python function random.randrange(-1,2) to compute initial weights. This function generates values: -1, 0, 1 (randomly). While this may seem like a mistake, what weve found empirically is that this actually performs better than using random.uniform(-1, 1). Be our guest and play around with the

random_weight function. Youll find that Neural Nets can be quite sensitive to initialization weight settings. (Recall what happens if you set all weights to the same value.)

To test your completed network, run:

python neural_net_tester.py two_layer

Your network should learn and classify all the datasets in the

neural_net_data.harder_data_set with 100% accuracy.

Designing For More Challenging Datasets

Now its your turn to design the network. We want to be able to classify more complex data sets.

Specifically we want you to design a new network that should theoretically be able to learn and classify the following datasets:

1. The letter-L.

4 +

3 +

2 +

1 +

0

0 1 2 3 4

2. This moat-like shape:

4

3 2 +

1

0

0 1 2 3 4

3. This patchy shape:

4

3 + + 2

1 + +

0 + +

0 1 2 3 4

We claim that a network architecture containing 5 neuron nodes or less can fully learn and classify all three shapes. In fact, we require it!

Construct a new network in:

def make_neural_net_challenging()

that can (theoretically) perfectly learn and classify all three datasets. To test your network on the first 2 of these shapes, run

python neural_net_tester.py challenging

To pass our tests, your network must get 100% accuracy within 10000 iterations.

Now try your architecture on the third dataset, patchy. Run:

python neural_net_tester.py patchy

Depending on your architecture and your initial weights, your network may either easily learn patchy or get stuck in a local maximum. Does your network completely learn the dataset within 10000 iterations? If not, take a look at the weights output at the end of the 10000 iterations. Plot the weights in terms of a linear function on a 2D graph. Do the boundaries tell you why there might be a local maximum?

Manually Setting Weights

You can have your network learn the dataset patchy perfectly and very quickly if the proper weights are set.

You can use either of these strategies to determine the proper weights.

1. You can experimentally determine the right weights by running your network until it perfectly learns the dataset. You will probably need to increase the max-iterations parameter, or playing around with different initial weight settings.
2. You can try to solve for the weights analytically. This is a good chance to put into practice the methods for solving network weights you learned in tutorial. You can review the example given in the tutorial notes on k-nearest neighbors, decision trees, and neural nets.

In either case, we want you to find preset weights to the same network that you built in the last part. Your new weight-preset network should be able to learn the patchy problem with only 1000 additional iterations of training.

After youve found the optimal weights, fill in:

def make_neural_net_with_weights()

To test, run:

python neural_net_tester.py weights

If everything tests with an accuracy of 1.0, then youve completed the Neural Networks portion of lab5. Congrats!

Now on to Boosting!

Boosting

Youre still trying to use AI to predict the votes of politicians. ID-Trees were great, but youve heard about these other magnificent learning algorithms like SVMs and Boosting. Boosting sounds easier to implement and had a pretty good reputation, so you decide to start with that.

To make sure that you interpret the results without letting your political preconceptions get in the way, you dig up some old data to work with: in particular, the data from the 4th House of Representatives, which met from 1796 to 1797. (According to the records on voteview.com, this is when the two-party system first emerged, with the two parties being designated Federalists and Republicans.)

You experiment with that data before going on to the 2007-2008 data, finding that Congressmen in 1796 were much more clear about what they were voting on than in 2008.

The framework for a boosting classifier can be found in boost.py. You need to finish coding it, and then use it to learn some classifiers and answer a few questions.

The following resources will be helpful:

• The documentation for the boosting code, which you can find embedded in py in the documentation strings.
• The Shapire paper on boosting, or the notes that summarize it (on OCW, both are available in the Readings section, under Lecture 20).
• The Lab 4 writeup, if you need to refer to how py represents legislators and votes.

A (clever|cheap) trick

The boosting code uses a trick that means it only has to try half the number of base classifiers.

It turns out that AdaBoost does not really care which side of its base classifier is +1 and which side is -1. If you choose a classifier that is the opposite of the best classifier it returns -1 for most points that should be +1, and returns +1 for most points that should be -1, and therefore has a high error rate it works the same as if you had chosen the negation of that classifier, which is the best classifier.

If the data reweighting step is implemented correctly, it will produce the same weights given a classifier or its opposite. Also, a classifier with a high error rate will end up with a negative alpha value, so that in the final vote of classifiers it will act like its opposite. So the important thing about a classifier isnt that its error rate is low its that the error rate is far from 1/2.

In the boosting code, we take advantage of this. We include only classifiers that output +1 for voting YES on a particular motion, or +1 for voting NO on a particular motion, and as the best classifier we choose the classifier whose error rate is farthest from 1/2. If the error rate is high, then the result will act like a classifier that outputs +1 for voting NO or abstaining, or +1 for voting YES or abstaining, respectively. This means we dont have to include these classifiers in the base classifier list, speeding up the algorithm by a factor of 2.

Completing the code

Here are the parts that you need to complete:

• In the BoostClassifier class, the classify method is also undefined. Define it so that you can use a trained BoostClassifier as a classifier, outputting +1 or -1 based on the weighted results of its base classifiers. Complete the very similar orange_classify method as well.
• In the BoostClassifier class in py, the update_weights method is undefined. You need to define this method so that it changes the data weights in the way prescribed by the AdaBoost algorithm. Note: There are two ways of implementing this update; they happen to be mathematically equivalent.)
• In py, the most_misclassified function is undefined. You will need to define it to answer the questions.

Remember to use the supplied legislator_info(datum) to output your list of the mostmisclassified data points!

Questions

Answer the two questions republican_newspaper_vote and republican_sunset_vote in lab5.py.

When you are asked how a particular political party would vote on a particular motion, disregard the possibility of abstaining. If your classifier results indicate that the party wouldnt vote NO, consider that an indication that the party would vote YES.

Orange you glad someone else implemented these?

First things first: Have you installed Orange yet?

Now that youve installed Orange, when you run lab5.py, does it complain about Orange, or does it show you the outputs of some classifiers on vampire data?

Getting familiar with the Orange GUI

This part is to get you familiar with the Orange GUI to do a little machine learning without doing any programming. Weve given you some data files (vampires.tab, H004.tab, adult.tab, titanic.tab, breastcancer.tab, etc.) that you can play with. Try making something like the following:

Courtesy of University of Ljubljana Artificial Intelligence Laboratory. Used with permission.

Then take a look at the performance, and look at the actual predictions.

Using Orange from Python

We have given you a function called describe_and_classify that trains a bunch of classifiers that Orange provides. Some of them will be more familiar than others.

First it trains each classifier on the data, shows its output on each data point from the training data, and shows the accuracy on the training data. You know from class that the accuracy on the training data should be 1 for these classifiers. It is less than one because each classifier comes with built-in regularization to help prevent overtraining. Thats what the pruning is for the decision tree, for example. We didnt specify regularization parameters to most of the learners because they have fine defaults. You can read more about them in the Orange documentation.

Youll notice that we do one extra thing with the decision tree. We print it out. For most classifiers, their internal representations are hard for humans to read, but the decision trees internal representation can be very useful in getting to know your data.

Then describe_and_classify passes the untrained learners, without having seen the data, to cross-validation. Cross-validation hides part of the training data, trains the learner on the rest, and then tests it on the hidden part. To avoid accidentally picking just one terrible subset of the data to hide (an irreproducible subset), it divides the data evenly into some number of folds, successively tests by hiding each fold in turn, and averages over the results. You will find with cross-validation that the classifiers do much better on identifying political parties than they do on vampires. Thats because the vampire dataset has so few examples, with so little redundancy, that if you take away one example, it is very likely to remove the information you actually need in order to classify that example.

To ensure that youve gotten the right idea from each part of describe_and_classify, there are six questions just below the function.

Boosting with Orange

You may be able to do better by using AdaBoost over your Orange classifiers than you could do with any of those Orange classifiers by themselves. Then again, you may do worse. That AdaBoost doesnt tend to overfit is a somewhat conditional statement that depends a lot on the particular classifiers that boosting gets its pick of. If some of the underlying classifiers tend to overfit, then boosting will greedily choose them first, and it will be stuck with those choices for testing.

We have set up a learner that uses the BoostClassifier from the first part, but its underlying classifiers are the Orange classifiers we were just looking at. When you combine classifiers in this way, you create whats called an ensemble classifier. You will notice, as you run this new classifier on the various data sets weve provided, that the ensemble frequently performs worse in cross-validation than some (or most) of its underlying classifiers.

Your job is to find a set of classifiers for the ensemble that get at least 74% accuracy on the breastcancer dataset. You may use any subset of the classifiers weve provided. Put the short names of your classifiers into the list classifiers_for_best_ensemble. Classifier performance appears to be architecture dependent, so you might be able to get to 74% with just one classifier on your machine, but that wont be enough on the server in this case, try to get even better performance at home. Hints

Neural Nets

If you are having problems with getting your network to convergence on certain problems, try the following:

1. Order your weight initialization (i.e. calls to random_weight()) from the bottom-most weights to the top-most weights in you network. While this ordering theoretically irrelevant, weve found that this ordering worked well in practice (in conjunction with 1 above). NN are unfortunately quite sensitive to initial weight settings.
2. Play around with the seed_random function to try different starting random seeds, although seeding the random function with 0 is what worked for us.
3. If none of these work, try setting weights that are close to what you want in terms of the final expected solution.

FAQs

Q: I cant figure out how to write compute_doutdx for Neurons.

A: Heres a more step-by-step explanation of what this function does:

So you have a neural net, and youre at a particular neuron, call it D. This neuron D has some inputs for example, maybe it has an input y_B from neuron B with weight w_BD, and an input y_C from neuron C with weight w_CD, and a constant input (-1) with weight w_D. So its total input lets call it z is (y_B*w_BD + y_C*w_CD + -1*w_D). Its going to put that input z through its sigmoid function to get its output lets call that y_D.

Now, we want to compute the derivative of y_D with respect to some weight w. There are two cases.

In the first case, w is one of the weights on something that goes directly into D that is, w is w_BD or w_CD or w_D. In that case, we just do chain rule once, and we see that d(y_D)/d(w) = d(y_D)/dz * dz/dw.

We know the derivative of the sigmoid function, so d(y_D)/dz is y_D*(1-y_D). And the derivative of z with respect to w is just whatever thing that weight w was being multiplied by so for example, if w was w_BD, then dz/dw would be y_B.

In the second case, w is not one of the weights going directly into D, its probably something in an earlier layer, so we have to do some more chain rule to find dz/dw. We know z is (y_B*w_BD + y_C*w_CD + -1*w_D) (because we defined it to be that), so to find dz/dw, we can find the derivative of each of those terms: dz/dw = w_BD*(d(y_B)/dw) + w_CD*(d(y_C)/dw) + w_D*(d(-1)/dw).

And how do we get the d(y_B)/dw etc? Well, y_B is the output of neuron B, so thats just B.dOutdX(w). However, w might not have come from any path through B, so we only want to include the terms for which w is actually relevant. For that we can use the isa_descendant_weight_of function. Q: How do I figure out how the boost classifier thinks a Republican would vote on the newspapers thing?

A: The classifiers are supposed to return +1 for Republican and -1 for anything else (theres a comment right above the definition of BaseVoteClassifier in boost.py that says this).

boost_1796.classifiers is a list of the weak classifiers and their corresponding weights in the trained classifier. You can look through those and find the newspaper one, and look at whether the weights on it are positive or negative.

Q: How can I tell which input goes with which weight, in the my_inputs and my_weights lists in a Neuron?

A: The two lists are in the same order my_weights[0] is the weight for my_inputs[0], etc.

Q: I cant get my challenging neural net to correctly recognize all of the data sets. What can I do?

A: Make sure you read the hints. Youll just need to play around with the initial weights.

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6.034 Artificial Intelligence

Fall 2010

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