Final Project Solved

    • Introduction

One of CSC311’s main objectives is to prepare you to apply machine learning algorithms to real-world tasks. The nal project aims to help you get started in this direction. You will be performing the following tasks:

Try out existing algorithms to real-world tasks.

Modify an existing algorithm to improve performance. Write a short report analyzing the result.

The nal project is not intended to be a stressful experience. It is a good chance for you to experiment, think, play, and hopefully have fun. These tasks are what you will be doing daily as a data analyst/scientist or machine learning engineer.

    • Background & Task

Online education services, such as Khan Academy and Coursera, provide a broader audience with access to high-quality education. On these platforms, students can learn new materials by watching a lecture, reading course material, and talking to instructors in a forum. However, one disadvantage of the online platform is that it is challenging to measure students’ understanding of the course material. To deal with this issue, many online education platforms include an assessment component to ensure that students understand the core topics. The assessment component is often composed of diagnostic questions, each a multiple choice question with one correct answer. The diagnostic question is designed so that each of the incorrect answers highlights a common misconception. An example of the diagnostic problem is shown in gure 1. When students incorrectly answer the diagnostic question, it reveals the nature of their misconception and, by understanding these misconceptions, the platform can o er additional guidance to help resolve them.


    • https://markus.teach.cs.toronto.edu/csc311-2021-09




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Figure 1: An example diagnostic question [1].


In this project, you will build machine learning algorithms to predict whether a student can correctly answer a speci c diagnostic question based on the student’s previous answers to other questions and other students’ responses. Predicting the correctness of students’ answers to as yet unseen diagnostic questions helps estimate the student’s ability level in a personalized education platform. Moreover, these predictions form the groundwork for many advanced customized tasks. For in-stance, using the predicted correctness, the online platform can automatically recommend a set of diagnostic questions of appropriate di culty that t the student’s background and learning status.

You will begin by applying existing machine learning algorithms you learned in this course. You will then compare the performances of di erent algorithms and analyze their advantages and dis-advantages. Next, you will modify existing algorithms to predict students’ answers with higher accuracy. Lastly, you will experiment with your modi cation and write up a short report with the results.

You will measure the performance of the learning system in terms of prediction accuracy, although you are welcome to include other metrics in your report if you believe they provide additional insight:

Prediction Accuracy = The number of correct predictions

The number of total predictions

    • Data

We subsampled answers of 542 students to 1774 diagnostic questions from the dataset provided by Eedi2, an online education platform that is currently being used in many schools [1]. The platform o ers crowd-sourced mathematical diagnostic questions to students from primary to high school (between 7 and 18 years old). The truncated dataset is provided in the folder /data.


    • https://eedi.com/



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Figure 2: An example sparse matrix [1].


3.1    Primary Data

The primary data, train_data.csv, is the main dataset you will be using to train the learning algorithms throughout the project. There is also a validation set valid_data.csv that you should use for model selection and a test set test_data.csv that you should use for reporting the nal performance. All primary data csv les are composed of 3 columns:

question id: ID of the question answered (starts from 0).

user id: ID of the student who answered the question (starts from 0).

is correct: Binary indicator whether the student’s answer was correct (0 is incorrect, 1 is correct).


We also provide a sparse matrix, sparse_matrix.npz, where each row corresponds to the user id and each column corresponds to the question id. An illustration of the sparse matrix is shown in gure 2. The correct answer given a pair of (user id, question id) will have an entry 1 and an incorrect answer will have an entry 0. Answers with no observation and held-out data (that will be used for validation and test) will have an entry NaN (np.NaN).


3.2    Question Metadata

We also provide the question metadata, question_meta.csv, which contains the following columns:

question id: ID of the question answered (starts from 0).

subject id: The subject of the question covered in an area of mathematics. The text de-scription of each subject is provided in subject_meta.csv.


3.3    Student Metadata

Lastly, we provide the student metadata, student_meta.csv, that is composed of the following columns:


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user id: ID of the student who answered the question (starts from 0).

gender: Gender of the student, when available. 1 indicates a female, 2 indicates a male, and 0 indicates unspeci ed.

data of birth: Birth date of the student, when available.

premium pupil: Student’s eligibility for free school meals or pupil premium due to being nancially disadvantaged, when available.


    • Part A

In the rst part of the project, you will implement and apply various machine learning algorithms you studied in the course to predict students’ correctness of a given diagnostic question. Review the course notes if you don’t recall the details of each algorithm. For this part, you will only be using the primary data: train_data.csv, sparse_matrix.npz, valid_data.csv, and test_data.csv. Moreover, you may use the helper functions provided in utils.py to load the dataset and evaluate your model. You may also use any functions from packages NumPy, Scipy, Pandas, and PyTorch. Make sure you understand the code instead of using it as a black box.

    1. [5pts] k-Nearest Neighbor. In this problem, using the provided code at part_a/knn.py, you will experiment with k-Nearest Neighbor (kNN) algorithm.

The provided kNN code performs collaborative ltering that uses other students’ answers to predict whether the speci c student can correctly answer some diagnostic questions. In particular, the starter code implements user-based collaborative ltering: given a user, kNN nds the closest user that similarly answered other questions and predicts the correctness based on the closest student’s correctness. The core underlying assumption is that if student A has the same correct and incorrect answers on other diagnostic questions as student B, A’s correctness on speci c diagnostic questions matches that of student B.

        (a) Complete a function main located at knn.py that runs kNN for di erent values of k 2 f1; 6; 11; 16; 21; 26g. Plot and report the accuracy on the validation data as a function of k.

        (b) Choose k that has the highest performance on validation data. Report the chosen k and the nal test accuracy.

        (c) Implement a function knn_impute_by_item on the same le that performs item-based collaborative ltering instead of user-based collaborative ltering. Given a question, kNN nds the closest question that was answered similarly, and predicts the correctness basted on the closest question’s correctness. State the underlying assumption on item-based collaborative ltering. Repeat part (a) and (b) with item-based collaborative ltering.

        (d) Compare the test performance between user- and item- based collaborative ltering. State which method performs better.

        (e) List at least two potential limitations of kNN for the task you are given.

    2. [15pts] Item Response Theory. In this problem, you will implement an Item-Response Theory (IRT) model to predict students’ correctness to diagnostic questions.



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The IRT assigns each student an ability value and each question a di culty value to formulate a probability distribution. In the one-parameter IRT model, j represents the di culty of the question j, and i that represents the i-th students ability. Then, the probability that the question j is correctly answered by student i is formulated as:

p(cij = 1j i; j) =

exp ( i
j)

1 + exp ( i
j)

We provide the starter code in part_a/item_response.py.

        (a) Derive the log-likelihood log p(Cj ; ) for all students and questions. Here C is the sparse matrix. Also, show the derivative of the log-likelihood with respect to i and j (Hint: recall the derivative of the logistic model with respect to the parameters).

        (b) Implement missing functions in item_response.py that performs alternating gradient descent on and to maximize the log-likelihood. Report the hyperparameters you selected. With your chosen hyperparameters, report the training curve that shows the training and validation log-likelihoods as a function of iteration.

        (c) With the implemented code, report the  nal validation and test accuracies.

        (d) Select three questions j1; j2, and j3. Using the trained and , plot three curves on the same plot that shows the probability of the correct response p(cij = 1) as a function of given a question j. Comment on the shape of the curves and brie y describe what these curves represent.

    3. [15pts] Matrix Factorization OR Neural Networks. In this question, please read both option (i) and option (ii), but you only need to do one of the two.

        (i) Option 1: Matrix Factorization. In this problem, you will be implementing matrix factorization methods. The starter code is located at part_a/matrix_factorization.

            (a) Using a function svd_reconstruct that factorizes the sparse matrix using singular-value decomposition, try out at least 5 di erent k and select the best k using the validation set. Report the nal validation and test performance with your chosen k.

            (b) State one limitation of SVD in the task you are given. (Hint: how are you treating the missing entries?)

            (c) Implement functions als and update_u_z located at the same le that performs alternating updates. You need to use SGD to update un 2 Rk and zm 2 Rk as described in the docstring. To be noted, this is di erent from the alternating least squares (ALS) we introduced in the course slides, where we used the direct solution. As a reminder, the objective is as follows:
min
1


C
u
z   ;



O

n>
m
2
U;Z



nm




2 (n;m)







X2





where C is the sparse matrix and O = f(n; m) : entry (n; m) of matrix C is observedg.

    (d) Learn the representations U and Z using ALS with SGD. Tune learning rate and number of iterations. Report your chosen hyperparameters. Try at least 5 di erent values of k and select the best k that achieves the lowest validation accuracy.


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        (e) With your chosen k , plot and report how the training and validation squared-error-losses change as a function of iteration. Also report nal validation accuracy and test accuracy.

    (ii) Option 2: Neural Networks. In this problem, you will implement neural networks to predict students’ correctness on a diagnostic question. Speci cally, you will design an autoencoder model. Given a user v 2 RNquestions from a set of users S, our objective is:
X
min    kv    f(v;  )k22;

v2S

where f is the reconstruction of the input v. The network computes the following function:
f(v;  ) = h(W(2)g(W(1)v + b(1)) + b(2)) 2 RNquestions

for some activation functions h and g. In this question, you will be using sigmoid activation functions for both. Here, W(1) 2 R k Nquestions and W(2) 2 RNquestions k, where k 2 N is the latent dimension. We provide the starter code written in PyTorch at part_a/neural_network.

    (a) Describe at least three di erences between ALS and neural networks.

    (b) Implement a class AutoEncoder that performs a forward pass of the autoencoder following the instructions in the docstring.

    (c) Train the autoencoder using latent dimensions of k 2 f10; 50; 100; 200; 500g. Also, tune optimization hyperparameters such as learning rate and number of iterations. Select k that has the highest validation accuracy.

    (d) With your chosen k , plot and report how the training and validation objectives changes as a function of epoch. Also, report the nal test accuracy.

    (e) Modify a function train so that the objective adds the L2 regularization. The objective is as follows:
min X kv    f(v;  )k22 + 2 (kW(1)k2F + kW(2)k2F )


v2S

You may use a method get_weight_norm to obtain the regularization term. Using the k and other hyperparameters selected from part (d), tune the regularization penalty 2 f0:001; 0:01; 0:1; 1g. With your chosen , report the nal validation and test accuracy. Does your model perform better with the regularization penalty?

    4. [15pts] Ensemble. In this problem, you will be implementing bagging ensemble to im-prove the stability and accuracy of your base models. Select and train 3 base models with bootstrapping the training set. You may use the same or di erent base models. Your imple-mentation should be completed in part_a/ensemble.py. To predict the correctness, generate 3 predictions by using the base model and average the predicted correctness. Report the nal validation and test accuracy. Explain the ensemble process you implemented. Do you obtain better performance using the ensemble? Why or why not?





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    • Part B

In the second part of the project, you will modify one of the algorithms you implemented in part A to hopefully predict students’ answers to the diagnostic question with higher accuracy. In particular, consider the results obtained in part A, reason about what factors are limiting the performance of one of the methods (e.g. over tting? under tting? optimization di culties?) and come up with a proposed modi cation to the algorithm which could help address this problem. Rigorously test the performance of your modi ed algorithm, and write up a report summarizing your results as described below.

You will not be graded on how well the algorithm performs (i.e. its accuracy); rather, your grade will be based on the quality of your analysis. Try to be creative! You may also optionally use the provided metadata (question_meta.csv and student_meta.csv) to improve the accuracy of the model. At last, you are free to use any third-party ideas or code as long as it is publicly available. You must properly provide references to any work that is not your own in the write-up.

The length of your report for part B should be 3-4 pages. Don’t be afraid to keep the text short and to include large illustrative gures. The guidelines and marking schemes are as follows:

    1. [15pts] Formal Description: Precisely de ne the way in which you are extending the algorithm. You should provide equations and possibly an algorithm box. Describe way your proposed method should be expected to perform better. For instance, are you intending it to improve the optimization, reduce over tting, etc.?

    2. [10pts] Figure or Diagram: that shows the overall model or idea. The idea is to make your report more accessible, especially to readers who are starting by skimming your report.

    3. [15pts] Comparison or Demonstration: Include:

A comparison of the accuracies obtained by your model to those from baseline models. Include a table or a plot for an illustrative comparison.

Based on the argument you gave for why your extension should help, design and carry out an experiment to test your hypothesis. E.g., consider how you would disentangle whether the bene ts are due to optimization or regularization.

    4. [15pts] Limitations: of your approach.

Describe some settings in which we’d expect your approach to perform poorly, or where all existing models fail.

Try to guess or explain why these limitations are the way they are.

Give some examples of possible extensions, ways to address these limitations, or open problems.

    • Optional: Competition

Optionally, you will take part in a competition where you will submit the predictions of your model on a private test dataset. We will be using a platform called Kaggle3, which is a popular online community of data scientists and machine learning practitioners to solve data science challenges.


    • https://www.kaggle.com/



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You will rst form a team on Kaggle and submit your prediction as a csv le on the project website, which can be found here:

https://www.kaggle.com/t/51c62bffb0f348008333b29ab4385b23

Your submission should be a csv le with two columns id and is correct. You should use the provided functions load_private_test_csv to load the dataset and save_private_test_csv to save the predictions as a csv le both located at utils.py. Once you save the csv le, you can directly upload it to the competition page. After the submission, you can nd your model’s performance on the public leaderboard, which displays an accuracy of your predictions on 70% of private data. The full leaderboard that uses all private data will be released after the competition.


You are not required to use your model from Part B for this part.

Performance on the competition will not a ect your grade, not even as extra credit. The top 3 teams with the highest performance on all private data will receive a special gift from the instructors.

    • Friendly Advice

Read carefully! Make sure you are following the guidelines. Read this document carefully to understand the dataset, what you are asked to implement and demonstrate, etc. Ask questions on Piazza and visit o ce hours if you are unsure of any requirements.

Be honest! You are not being marked on how good the results are. It doesn’t matter if your method is better or worse than the ones you compare to. What matters is that you clearly describe the problem, your method, what you did, and what the results were. Just be scienti c.

Be careful! Don’t do things like test on your training data, set hyperparameters using test accuracy, compare unfairly against other methods, include plots with unlabeled axes, use unde ned symbols in equations, etc. Do sensible crosschecks like running your algorithms several times to understand the between-run variability, performing gradient checking, etc.


References

    [1] Wang, Zichao, et al. Diagnostic Questions: The NeurIPS 2020 Education Challenge. arXiv preprint arXiv:2007.12061 (2020)



















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