``What do you know?' ' Latent feature approach for the

Kaggle's GrockIt challenge

Rohan AnilAdvised by Prof. Charles Elkancollaboration with Aditya Menon

UC San DiegoMarch 19, 2012

Outline● Introduction

● Kaggle.com● GrockIt● ``What do you know?' ' Challenge

● Latent Feature Log-Linear (LFL)● Ensemble Learning● Our Results● Q/A

Kaggle.com

' 'What do you know?' ' - Competition

1st Prize : 3000$ 2nd Prize : 1500$ 3rd Prize : 500$

GrockIt.com

Dataset

Training Set

4,851,476 outcomes of students answering various questions

Outcomes

Four types:-

i) correct ii) incorrect iii) skipped iv) timed-out.

Students practicing for competitive exams

i) GMAT, ii) ACT and iii) SAT

Dataset

DatasetDifferences between training set and test set are:-

BiasBiased towards users who have answered more questions.

#ResponeOnly one response per student

TemporalOutcomes are latter in time than the training responses and validation responses of that student.

OutcomesTest set distribution is different from training set,it does not include timed-out or skipped outcomes.

Baseline

Rasch BaselineA baseline was provided by Kaggle for the dataset.

Bs - ability of the student 's'

δq - difficulty of question 'q'

For a given student 's' ( Fixed Bs )

– The probability of answering a question is only dependent on the difficult of the question q

– Consequence of this is that for every student, the ranking interms of probability of answering the question correctly is the same.

...

Dataset

Validation set Grockit created a validation set which contains responses of 80,075 students on different questions.

Test setTest set was used for ranking the teams, it contains responses of 93,100 users on different questions.

Dyadic Prediction

A dyadic prediction task is a learning task which involves predicting a class label for a pair of items ( Hoffman 1999 )

Side-Information

Sometimes there is more information in the dataset. They are

1. side-information associated with u

2. side-information associated with i

3. interaction side-information for (u,i)

Interpreting the task as a collaborative filtering problem

The dataset contains student responses for various questions.

179,107 students and 6,046 questions

....Skipped

Timed out

...

Nominal Outcomes● Correct● Incorrect● Timed-Out● Skipped

Dyadic Prediction

( , )

..... .....

.....

( , )

Training Set

Dyadic Prediction

( , ) ?

Query in Test

Side Information in the dataset

Associated with a student

Not Available

Associated with a question

Question Type, Group, Track, Subtrack, Tags

Associated with (student,question) dyad

Game, Number of Players, Started at, Answered at, Deactivated at, Question set

Side Information

Question TypeMultiple Choice, Free Response

GroupACT, GMAT, SAT

SubtrackCritical Reasoning, Data Sufficiency, English, Ientifying Sentence Errors, Improving Paragraphs, Improving Sentences, Math, Multiple Choice, Passage Based Reading, Problem Solving, Reading, Reading Comprehension, Science, Sentence Completion, Sentence Correction, Student Produced Response

Tags

describes the skill that is needed to solve the question.

Dataset

Dataset

Dataset

....

The dataset is similar to the typical dyadic dataset with a couple of key differences: ● Duplicate Dyads

There can exist duplicate dyad pairs in the training set with different outcomes, since a student can answer a question many times,

● Collaborative or Competitive AnsweringIn some games types, students can collaboratively answer questions.

Motivation for Latent feature approach

Highly successful at winning the Netflix prize 1M$ challenge (Toscher et al., 2009) where the problem was to predict ratings for movies.

Metric used to rank the teams

Binomial Capped Deviance, similar to log-likelihood

Estimated probability of correct responseCapped between [0.01,.99]

True label of the dyad

Leaderboard

Latent feature log-linear

Motivations for Latent Feature Log-Linear (LFL) (Menon & Elkan, 2010)

Well calibrated Probabilitieswe need to predict the probability of correct outcome for the dyadic pairs in the test set.

Leverage Side-InformationMost collaborative filtering algorithms do not have any principled way of including side-information

Scale WellTo be used in the industry, the method has to scale well to large datasets

Multiclass LFL model

Multiclass LFL model

Case | Y| = 3

p(y=3 | (user,item)) = exp( U3user . I

3item )

U1 I1 U1 U1 I2U2 U3 I3

Z = exp( U1user . I

1item ) +exp( U1

user . I1

item ) + exp( U3user . I

3item )

Z

Binary LFL on the dataset

Test Set contains only two types of outcomes i) correct ii) incorrect

y = 1 ( Correct Response)

y = 0 ( Incorrect Response)

The binary-LFL model has appeared in the literature before (Schein et al., 2003; Agarwal & Chen, 2009)

Training

We optimize for the negative log likelihood

We can optimize this objective function using the stochastic gradient descent method.

Regularization Terms

Stochastic Gradient Descent

LFL on GrockIt

Stochastic Gradient Descent

Grid Search

parameters

Parallel SGD Training

Was formulated independently by Gemulla et al., 2011

KDD CUP, Spring, 2011

This is us!!! =)

Parallelism

Side-Information

For a question q, let g =group(q). We can add a latent vector for each group i.e ACT, GMAT, SAT

Prediction equation after adding side information is

Categorical Features

Group – G

Track – T

Subtrack – ST

Game Type – GT

Question Type – QT

LFL Models

Training Set

Training set contains four types of outcomes

i) correct, ii) incorrect, iii) skipped and iv) timed-out.

Test set contains four types of outcomes

i) correct, ii) incorrect

We create two training sets,a) Training set with skipped and timed-out responses excluded

b) Training set with skipped and timed-out responses treated as an incorrect outcome

Results from LFL Models (a)

Results from LFL Models (b)

Observation

Throwing away data helps!Removing skipped and timed-out responses from training set improved the BCD (binomial capped deviance)

Motivates for adapting the model to the test-set distribution to win the competition.

Ensemble Learning

No Single Model works well on every dyad.Combining predictions from multiple models can outperform each of the individual models (Takcas et al., 2009 )

1M$ Netflix Prize was won by a blend of multiple models

Intuition for Ensemble LearningTrue labels for four samples(1,1,0,0)

Predictions from four different models.(0,1,0,0) – accuracy 75%(1,0,0,0) – accuracy 75%(1,1,1,0) – accuracy 75%(1,1,0,1) – accuracy 75%

Average of different models(.75,.75,.25,.25)

Threshold the average at 0.5(1,1,0,0) – accuracy = 100%

Using Linear Regression for combining predictions

For a set with known labels,

{ (s,q) – > y(s,q) } , where y can take 0 or 1

pi = pi ( y=1) | (s,q) ) is the estimated probability of a correct response from the ith model

Define matrix P and column matrix Y,

where each row of P contains predictions from n models, ( p1 .., pi , .. pn )

and Y contains the target value y(s,q)

Similarly using predictions for every dyad in the set, we create matrix P with predictions and Y with target values.

We solve,

Pw = Y

To predict the probability of a correct response of an example in the test set,

We combine predictions from n models using the weight vector w

pestimated = wj pj

....

Which set to use?

Step 1

for each of the n models

Train on the training setPredict on the validation setsave parameters

Step 2: Estimate w using linear regression on the validation set predictions

Step 3:

for each of the n models

Train on the training set + validation setPredict on the test set

Step 4:

Combine predictions of the test set using w

Results

After combining predictions using linear regression

2 weeks later

some weeks later..

Gradient Boosted Decision Trees

Leverage Side-Information in Ensemble learning

Gradient Boosted Decision Trees (GBDT) (Friedman, 1999) algorithm can be used to combine predictions and side information together.

Popular algorithm

GBDT is a powerful learning algorithm that is widely used (see Li & Xu, 2009, chap. 6)

The core of the algorithm is a decision tree learner

Decision Tree

Decision tres can handle both i) Numeric, and ii) categorical variables.

It can also handle missing information.

Decision Tree

Prediction ( Y6 + Y7 + Y9 ) / 3

Prediction ( Y1 + Y3 ) / 2 ................... .................

Decision function

Gradient Boosting

Select the base learner, and loss function.● Decision Tree as the base learner, and Squared

Loss as the loss function Gradient boosting is an iterative-procedure

● Iteratively fit a base learner on the gradient of the previous iteration

Gradient Boosting

We can add the a regularization parameter as follows

Side-Information for GBDT

Meta-Features

Preprocessing Tags

Each question has a set of tags that is associated with it. Some are listed below

Statistics (incl. mean median mode),259

Strengthen Hypothesis,260

Student Produced Response,261

System of Linear Equations,262

Systems of Linear Equations,263

Systems of linear equations and inequalities,264

We manually merge the tags that we feel are very similar.

We cluster the tags into 40 clusters using spectral clustering (Ng et al., 2001) with normalized co-ocurrence of tags as the similarity measure to generate the affinity matrix A.

Results from GBDT

● GBDT only improved the bcd marginally.

Including Temporal Features

...

GBDT Results after including temporal features

Feb 23, Week, competition end

Last day

Combined predictions from GBDT models using linear regression, improved slightly.

Last day of competition

Final Private set ranks

Post competition analysis

Latent feature approach is a good approach for this dataset.

LFL performs really well on the dataset

Code will be available soon @ http:/ / code.google.com/p/ latent-feature-log-linear/

Questions

References

Agarwal, Deepak and Chen, Bee-Chung. Regression based latent factor models. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, KDD ’09, pp. 19– 28, New York, NY, USA, 2009. ACM. ISBN 978- 1-60558-495-9.Friedman, Jerome H. Stochastic gradient boosting. Computational Statistics and Data Analysis, 38: 367– 378, 1999.Gemulla, Rainer, Nijkamp, Erik, Haas, Peter J., and Sismanis, Yannis. Large-scale matrix factorization with distributed stochastic gradient descent. In Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining, KDD ’11, New York, NY, USA, 2011. ACM. ISBN 978-1-4503-0813-7.Hofmann, Thomas, Puzicha, Jan, and Jordan Michael I. Learning from dyadic data. In Proceedings of the 1998 conference on Advances in neural information processing systems II, pp. 466– 472, Cambridge, MA, USA, 1999. MIT Press. ISBN 0-262-11245-0.Li, Xiaochun and Xu, Ronghui (eds.). High dimensional data analysis in cancer research. Springer, CA, U.S.A, 2009.Menon, Aditya Krishna and Elkan, Charles. A log linear model with latent features for dyadic predic-tion. In ICDM’10, pp. 364– 373, 2010.Ng, Andrew Y., Jordan, Michael I., and Weiss, Yair. On spectral clustering: Analysis and an algorithm.In Advances in Nueral Information Processing Systems, pp. 849– 856. MIT Press, 2001.

References

Rasch, Georg. Estimation of parameters and control of the model for two response categories, 1960.

Schein, Andrew I., Lawrence, Andrew I., Saul, Lawrence K., and Ungar, Lyle H. A generalized linear model for principal component analysis of binary data, 2003.

Takcas, G abor, Pilaszy, Istvan, Nemeth, Bottyan, and Tikk, Domonkos. Scalable ́collaborative filtering approaches for large recommender systems. J. Mach. Learn. Res., 10:623– 656, June 2009. ISSN 1532- 4435.

Tscher, Andreas, Jahrer, Michael, and Bell, Robert M. The bigchaos solution to the netflix grand prize, 2009.