Acknowledgements

This project is part of the Pac-man projects created by John DeNero and Dan Klein for CS188 at Berkeley EECS. We thank Pieter Abbeel, John DeNero, and Dan Klein for sharing it with us and allowing us to use as course project.

Project 5: Classification


Which Digit?
Which are Faces?


Due 12/09 at 11:59pm

Introduction

In this project, you will design three classifiers: a naive Bayes classifier, a perceptron classifier and a large-margin (MIRA) classifier. You will test your classifiers on two image data sets: a set of scanned handwritten digit images and a set of face images in which edges have already been detected. Even with simple features, your classifiers will be able to do quite well on these tasks when given enough training data.

Optical character recognition (OCR) is the task of extracting text from image sources. The first data set on which you will run your classifiers is a collection of handwritten numerical digits (0-9). This is a very commercially useful technology, similar to the technique used by the US post office to route mail by zip codes. There are systems that can perform with over 99% classification accuracy (see LeNet-5 for an example system in action).

Face detection is the task of localizing faces within video or still images. The faces can be at any location and vary in size. There are many applications for face detection, including human computer interaction and surveillance. You will attempt a simplified face detection task in which your system is presented with an image that has been pre-processed by an edge detection algorithm. The task is to determine whether the edge image is a face or not. There are several systems in use that perform quite well at the face detection task. One good system is the Face Detector by Schneiderman and Kanade. You can even try it out on your own photos in this demo.

The code for this project includes the following files and data, available as a zip file. 

Data file
data.zip Data file, including the digit and face data.
Files you will edit
naiveBayes.py The location where you will write your naive Bayes classifier.
perceptron.py The location where you will write your perceptron classifier.
mira.py The location where you will write your MIRA classifier.
dataClassifier.py The wrapper code that will call your classifiers. You will use this code to analyze the behavior of your classifier and may edit some parts of it if you want to.
answers.py Answers to Question 2 and Question 4 go here.
Files you should read but NOT edit
classificationMethod.py Abstract super class for the classifiers you will write.
(You should read this file carefully to see how the infrastructure is set up.)
samples.py I/O code to read in the classification data.
util.py Code defining some useful tools. You may be familiar with some of these by now, and they will save you a lot of time.
mostFrequent.py A simple baseline classifier that just labels every instance as the most frequent class.
runMinicontest.py The command you will use to run the minicontest, if you decide to enter.

What to submit: You will fill in portions of answers.py, naiveBayes.py, perceptron.py, mira.py and dataClassifier.py (only) during the assignment, and submit them.

Evaluation: Your code will be autograded for technical correctness. Please do not change the names of any provided functions or classes within the code, or you will wreak havoc on the autograder.

Academic Dishonesty: We will be checking your code against other submissions in the class for logical redundancy. If you copy someone else's code and submit it with minor changes, we will know. These cheat detectors are quite hard to fool, so please don't try. We trust you all to submit your own work only; please don't let us down. Instead, contact the course staff if you are having trouble.

Getting Started

To try out the classification pipeline, run dataClassifier.py from the command line. This will classify the digit data using the default classifier (mostFrequent) which blindly classifies every example with the most frequent label. 

python dataClassifier.py

As usual, you can learn more about the possible command line options by running: 

python dataClassifier.py -h

We have defined some simple features for you. Later you will design some better features. Our simple feature set includes one feature for each pixel location, which can take values 0 or 1 (off or on). The features are encoded as a Counter where keys are feature locations (represented as (column,row)) and values are 0 or 1. The face recognition data set has value 1 only for those pixels identified by a Canny edge detector.

Implementation Note: You'll find it easiest to hard-code the binary feature assumption. If you do, make sure you don't include any non-binary features. Or, you can write your code more generally, to handle arbitrary feature values, though this will probably involve a preliminary pass through the training set to find all possible feature values (and you'll need an "unknown" option in case you encounter a value in the test data you never saw during training).

Naive Bayes

A skeleton implementation of a naive Bayes classifier is provided for you in naiveBayes.py. You will fill in the trainAndTune function, the calculateLogJointProbabilities function and the findHighOddsFeatures function.

Theory

A naive Bayes classifier models a joint distribution over a label $Y$ and a set of observed random variables, or features, $\{F_1, F_2, \ldots F_n\}$, using the assumption that the full joint distribution can be factored as follows (features are conditionally independent given the label):

\begin{displaymath} P(F_1 \ldots F_n, Y) = P(Y) \prod_i P(F_i \vert Y)\end{displaymath}

To classify a datum, we can find the most probable label given the feature values for each pixel, using Bayes theorem:

\begin{eqnarray*} P(y \vert f_1, \ldots, f_m) &=& \frac{P(f_1, \ldots, f_m \......&=& \textmd{arg max}_{y} P(y) \prod_{i = 1}^m P(f_i \vert y)\end{eqnarray*}

Because multiplying many probabilities together often results in underflow, we will instead compute log probabilities which have the same argmax:

\begin{eqnarray*} \textmd{arg max}_{y} log(P(y \vert f_1, \ldots, f_m) &=& \te......{arg max}_{y} (log(P(y)) + \sum_{i = 1}^m log(P(f_i \vert y)))\end{eqnarray*}

To compute logarithms, use math.log(), a built-in Python function.

Parameter Estimation

Our naive Bayes model has several parameters to estimate. One parameter is the prior distribution over labels (digits, or face/not-face), $P(Y)$.

We can estimate $P(Y)$ directly from the training data:

\begin{displaymath} \hat{P}(y) = \frac{c(y)}{n}\end{displaymath}

where $c(y)$ is the number of training instances with label y and n is the total number of training instances.

The other parameters to estimate are the conditional probabilities of our features given each label y: $P(F_i \vert Y = y)$. We do this for each possible feature value ($f_i \in {0,1}$).

\begin{eqnarray*} \hat{P}(F_i=f_i\vert Y=y) &=& \frac{c(f_i,y)}{\sum_{f_i}{c(f_i,y)}} \\\end{eqnarray*}

where $c(f_i,y)$ is the number of times pixel $F_i$ took value $f_i$ in the training examples of label y.

Smoothing

Your current parameter estimates are unsmoothed, that is, you are using the empirical estimates for the parameters $P(f_i\vert y)$. These estimates are rarely adequate in real systems. Minimally, we need to make sure that no parameter ever receives an estimate of zero, but good smoothing can boost accuracy quite a bit by reducing overfitting.

In this project, we use Laplace smoothing, which adds k counts to every possible observation value:

$P(F_i=f_i\vert Y=y) = \frac{c(F_i=f_i,Y=y)+k}{\sum_{f_i}{(c(F_i=f_i,Y=y)+k)}}$

If k=0, the probabilities are unsmoothed. As k grows larger, the probabilities are smoothed more and more. You can use your validation set to determine a good value for k. Note: don't smooth P(Y).

Question 1 (6 points) Implement trainAndTune and calculateLogJointProbabilities in naiveBayes.py. In trainAndTune, estimate conditional probabilities from the training data for each possible value of k given in the list kgrid. Evaluate accuracy on the held-out validation set for each k and choose the value with the highest validation accuracy. In case of ties, prefer the lowest value of k. Test your classifier with: 

python dataClassifier.py -c naiveBayes --autotune

Hints and observations:

Odds Ratios

One important tool in using classifiers in real domains is being able to inspect what they have learned. One way to inspect a naive Bayes model is to look at the most likely features for a given label.

Another, better, tool for understanding the parameters is to look at odds ratios. For each pixel feature $F_i$ and classes $y_1, y_2$, consider the odds ratio:

\begin{displaymath} \mbox{odds}(F_i=on, y_1, y_2) = \frac{P(F_i=on\vert y_1)}{P(F_i=on\vert y_2)}\end{displaymath}

This ratio will be greater than one for features which cause belief in $y_1$ to increase relative to $y_2$.

The features that have the greatest impact at classification time are those with both a high probability (because they appear often in the data) and a high odds ratio (because they strongly bias one label versus another).

To run the autograder for this question: 

 python autograder.py -q q1

Question 2 (2 points) Fill in the function findHighOddsFeatures(self, label1, label2). It should return a list of the 100 features with highest odds ratios for label1 over label2. The option -o activates an odds ratio analysis. Use the options -1 label1 -2 label2 to specify which labels to compare. Running the following command will show you the 100 pixels that best distinguish between a 3 and a 6. 

python dataClassifier.py -a -d digits -c naiveBayes -o -1 3 -2 6
Use what you learn from running this command to answer the following question. Which of the following images best shows those pixels which have a high odds ratio with respect to 3 over 6? (That is, which of these is most like the output from the command you just ran?)
(a)
(b)
(c)
(d)
(e)

To answer: please return 'a', 'b', 'c', 'd', or 'e' from the function q2 in answers.py.

Hints:

Perceptron


A skeleton implementation of a perceptron classifier is provided for you in perceptron.py. You will fill in the train function, and the findHighWeightFeatures function.

Unlike the naive Bayes classifier, a perceptron does not use probabilities to make its decisions. Instead, it keeps a weight vector $w^y$ of each class $y$ ($y$ is an identifier, not an exponent). Given a feature list $f$, the perceptron compute the class $y$ whose weight vector is most similar to the input vector $f$. Formally, given a feature vector $f$ (in our case, a map from pixel locations to indicators of whether they are on), we score each class with:

\begin{displaymath} \mbox{score}(f,y) = \sum_i f_i w^y_i\end{displaymath}
Then we choose the class with highest score as the predicted label for that data instance. In the code, we will represent $w^y$ as a Counter.

Learning weights

In the basic multi-class perceptron, we scan over the data, one instance at a time. When we come to an instance $(f, y)$, we find the label with highest score:
\begin{displaymath} y' = \textmd{arg max}_{y''} score(f,y'')\end{displaymath}

We compare $y'$ to the true label $y$. If $y' = y$, we've gotten the instance correct, and we do nothing. Otherwise, we guessed $y'$ but we should have guessed $y$. That means that $w^y$ should have scored $f$ higher, and $w^{y'}$ should have scored $f$ lower, in order to prevent this error in the future. We update these two weight vectors accordingly:

\begin{displaymath} w^y += f\end{displaymath}
\begin{displaymath} w^{y'} -= f\end{displaymath}

Using the addition, subtraction, and multiplication functionality of the Counter class in util.py, the perceptron updates should be relatively easy to code. Certain implementation issues have been taken care of for you in perceptron.py, such as handling iterations over the training data and ordering the update trials. Furthermore, the code sets up the weights data structure for you. Each legal label needs its own Counter full of weights.

Question 3 (4 points) Fill in the train method in perceptron.py. Run your code with: 

python dataClassifier.py -c perceptron

Hints and observations:

To run the autograder for this question and visualize the output:

 python autograder.py -q q3

Visualizing weights

Perceptron classifiers, and other discriminative methods, are often criticized because the parameters they learn are hard to interpret. To see a demonstration of this issue, we can write a function to find features that are characteristic of one class. (Note that, because of the way perceptrons are trained, it is not as crucial to find odds ratios.)

Question 4 (1 point) Fill in findHighWeightFeatures(self, label) in perceptron.py. It should return a list of the 100 features with highest weight for that label. You can display the 100 pixels with the largest weights using the command: 

python dataClassifier.py -c perceptron -w
Use this command to look at the weights, and answer the following true/false question. Which of the following sequence of weights is most representative of the perceptron?
(a)
(b)
Answer the question answers.py in the method q4, returning either 'a' or 'b'.

MIRA

A skeleton implementation of the MIRA classifier is provided for you in mira.py. MIRA is an online learner which is closely related to both the support vector machine and perceptron classifiers. You will fill in the trainAndTune function.

Theory

Similar to a multi-class perceptron classifier, multi-class MIRA classifier also keeps a weight vector $w^y$ of each label $y$. We also scan over the data, one instance at a time. When we come to an instance $(f, y)$, we find the label with highest score:

We compare $y'$ to the true label $y$. If $y' = y$, we've gotten the instance correct, and we do nothing. Otherwise, we guessed $y'$ but we should have guessed $y$. Unlike perceptron, we update the weight vectors of these labels with variable step size:

where is chosen such that it minimizes
subject to the condition that

which is equivalent to
subject to and

Note that, , so the condition is always true given Solving this simple problem, we then have
However, we would like to cap the maximum possible value of by a positive constant C, which leads us to

Question 5 (6 points) Implement trainAndTune in mira.py. This method should train a MIRA classifier using each value of C in Cgrid. Evaluate accuracy on the held-out validation set for each C and choose the C with the highest validation accuracy. In case of ties, prefer the lowest value of C. Test your MIRA implementation with: 
python dataClassifier.py -c mira --autotune

Hints and observations:

To run the autograder for this question and visualize the output:

 python autograder.py -q q5

Congratulations! You're finished all projects.