Material presented at Tokyo Web Mining Meetup, March 26, 2016. The source code is here: https://github.com/hamukazu/tokyo.webmining.2016-03-26 東京ウェブマイニング（2016年3月27）の発表資料です。すべて英語です。

1. Introduction to Algorithms for Behavior Based
Recommendation
Tokyo Web Mining Meetup
March 26, 2016
Kimikazu Kato
Silver Egg Technology Co., Ltd.
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2. About myself
加藤公一 Kimikazu Kato
Twitter: @hamukazu
LinkedIn: http://linkedin.com/in/kimikazukato
Chief Scientist at Silver Egg Technology
Ph.D in computer science, Master's degree in mathematics
Experience in numerical computation and mathematical algorithms
especially ...
Geometric computation, computer graphics
Partial differential equation, parallel computation, GPGPU
Mathematical programming
Now specialize in
Machine learning, especially, recommendation system
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3. About our company
Silver Egg Technology
Established: 1998
CEO: Tom Foley
Main Service: Recommendation System, Online Advertisement
Major Clients: QVC, Senshukai (Bellemaison), Tsutaya
We provide a recommendation system to Japan's leading web sites.
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4. Table of Contents
Introduction
Types of recommendation
Evaluation metrics
Algorithms
Conclusion
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5. Caution
This presentation includes:
State-of-the-art algorithms for recommendation systems,
But does NOT include:
Any information about the core algorithm in Silver Egg Technology
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6. Recommendation System
Recommender systems or recommendation systems (sometimes
replacing "system" with a synonym such as platform or engine) are a
subclass of information filtering system that seek to predict the
'rating' or 'preference' that user would give to an item. — Wikipedia
In this talk, we focus on collaborative filtering method, which only utilize
users' behavior, activity, and preference.
Other methods include:
Content-based methods
Method using demographic data
Hybrid
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7. Rating Prediction Problem
usermovie W X Y Z
A 5 4 1 4
B 4
C 2 3
D 1 4 ?
Given rating information for some user/movie pairs,
Want to predict a rating for an unknown user/movie pair.
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8. Item Prediction Problem
useritem W X Y Z
A 1 1 1 1
B 1
C 1
D 1 ? 1 ?
Given "who bought what" information (user/item pairs),
Want to predict which item is likely to be bought by a user.
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9. Input/Output of the systems
Rating Prediction
Input: set of ratings for user/item pairs
Output: map from user/item pair to predicted rating
Item Prediction
Input: set of user/item pairs as shopping data, integer
Output: top items for each user which are most likely to be bought by
him/her
k
k
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10. Evaluation Metrics for Recommendation
Systems
Rating prediction
The Root of the Mean Squared Error (RMSE)
The square root of the sum of squared errors
Item prediction
Precision
(# of Recommended and Purchased)/(# of Recommended)
Recall
(# of Recommended and Purchased)/(# of Purchased)
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11. RMSE of Rating Prediction
Some user/item pairs are randomly chosen to be hidden.
usermovie W X Y Z
A 5 4 1 4
B 4
C 2 3
D 1 4 ?
Predicted as 3.1 but the actual is 4, then the squared error is
.
Take the sum over the error over all the hidden items and then, take the
square root of it.
|3.1 − 4 =|
2
0.9
2
( −∑
(u,i)∈hidden
predictedui
actualui )
2
− −−−−−−−−−−−−−−−−−−−−−−−−−
√
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12. Precision/Recall of Item Prediction
If three items are recommended:
2 out of 3 recommended items are actually bought: the precision is 2/3.
2 out of 4 bought items are recommended: the recall is 2/4.
These are denoted by recall@3 and prec@3.
Ex. recall@5 = 3/5, prec@5 = 3/4
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13. ROC and AUC
# of
recom.
1 2 3 4 5 6 7 8 9 10
# of
whites
1 1 1 2 2 3 4 5 5 6
# of
blacks
0 1 2 2 3 3 3 3 4 4
Divide the first and second row by total number of white and blacks
respectively, and plot the values in xy plane.
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14. This curve is called "ROC curve." The area under this curve is called "AUC."
Higher AUC is better (max =1).
The AUC is often used in academia, but for a practical purpose...
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15. Netflix Prize
The Netflix Prize was an open competition for the best collaborative
filtering algorithm to predict user ratings for films, based on previous
ratings without any other information about the users or films, i.e.
without the users or the films being identified except by numbers
assigned for the contest. — Wikipedia
Shortly, an open competition for preference prediction.
Closed in 2009.
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16. Outline of Winner's Algorithm
Refer to the blog by E.Chen.
http://blog.echen.me/2011/10/24/winning-the-netflix-prize-a-summary/
Digest of the methods:
Neighborhood Method
Matrix Factorization
Restricted Boltzmann Machines
Regression
Regularization
Ensemble Methods
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17. Notations
Number of users:
Set of users:
Number of items (movies):
Set of items (movies):
Input matrix: ( matrix)
n
U = {1, 2, … , n}
m
I = {1, 2, … , m}
A n × m
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18. Matrix Factorization
Based on the assumption that each item is described by a small number of
latent factors
Each rating is expressed as a linear combination of the latent factors
Achieve good performance in Netflix Prize
Find such matrices , where
A ≈ YX
T
X ∈ Mat(f, n) Y ∈ Mat(f, m) f ≪ n, m
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19. Find and maximize
p (A|X, Y , σ) = N ( | , σ)∏
≠0aui
Aui X
T
u Yi
p(X| ) = N ( |0, I)σX ∏
u
Xu σX
p(Y | ) = N ( |0, I)σY ∏
i
Yi σY
X Y p (X, Y |A, σ)
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20. According to Bayes' Theorem,
Thus,
where means Frobenius norm.
How can this be computed? Use MCMC. See [Salakhutdinov et al., 2008].
Once and are determined, and the prediction for is
estimated by
p (X, Y |A, σ)
= p(A|X, Y , σ)p(X| )p(X| ) × const.σX σX
log p (U , V |A, σ, , )σU σV
= + ∥X + ∥Y + const.∑
Aui
( − )Aui X
T
u Yi
2
λX ∥
2
Fro
λY ∥
2
Fro
∥ ⋅ ∥Fro
X Y := YA
~
X
T
Aui
A
~
ui
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21. Rating
usermovie W X Y Z
A 5 4 1 4
B 4
C 2 3
D 1 4 ?
Includes negative feedback
"1" means "boring"
Zero means "unknown"
Shopping (Browsing)
useritem W X Y Z
A 1 1 1 1
B 1
C 1
D 1 ? 1 ?
Includes no negative feedback
Zero means "unknown" or
"negative"
More degree of the freedom
Difference between Rating and Shopping
Consequently, the algorithm effective for the rating matrix is not necessarily
effective for the shopping matrix.
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22. Solutions
Adding a constraint to the optimization problem
Changing the objective function itself
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23. Adding a Constraint
The problem has the too much degree of freedom
Desirable characteristic is that many elements of the product should be
zero.
Assume that a certain ratio of zero elements of the input matrix remains
zero after the optimization [Sindhwani et al., 2010]
Experimentally outperform the "zero-as-negative" method
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24. One-class Matrix Completion
[Sindhwani et al., 2010]
Introduced variables to relax the problem.
Minimize
subject to
pui
( − ) + ∥X + ∥Y∑
≠0Aui
Aui X
T
u Yi λX ∥
2
Fro
λY ∥
2
Fro
+ [ (0 − + (1 − )(1 − ]∑
=0Aui
pui X
T
u Yi )
2
pui X
T
u Yi )
2
+ T [− log − (1 − ) log(1 − )]∑
=0Aui
pui pui pui pui
= r
1
|{ | = 0}|Aui Aui
∑
=0Aui
pui
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25. Intuitive explanation:
means how likely the -element is zero.
The second term is the error of estimation considering 's.
The third term is the entropy of the distribution.
( − ) + ∥X + ∥Y∑
≠0Aui
Aui X
T
u Yi λX ∥
2
Fro
λY ∥
2
Fro
+ [ (0 − + (1 − )(1 − ]∑
=0Aui
pui X
T
u Yi )
2
pui X
T
u Yi )
2
+ T [− log − (1 − ) log(1 − )]∑
=0Aui
pui pui pui pui
pui (u, i)
pui
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26. Implicit Sparseness constraint: SLIM (Elastic Net)
In the regression model, adding L1 term makes the solution sparse:
The similar idea is used for the matrix factorization [Ning et al., 2011]:
Minimize
subject to
[ ∥Xw − y + ∥w + λρ|w ]min
w
1
2n
∥
2
2
λ(1 − ρ)
2
∥
2
2
|1
∥A − AW ∥ + ∥W + λρ|W
λ(1 − ρ)
2
∥
2
Fro
|1
diag W = 0
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27. Ranking prediction
Another strategy of shopping prediction
"Learn from the order" approach
Predict whether X is more likely to be bought than Y, rather than the
probability for X or Y.
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28. Bayesian Probabilistic Ranking
[Rendle et al., 2009]
Consider matrix factorization model, but the update of elements is
according to the observation of the "orders"
The parameters are the same as usual matrix factorization, but the
objective function is different
Consider a total order for each . Suppose that
means "the user is more likely to buy than .
The objective is to calculate such that and (which
means and are not bought by ).
>u u ∈ U i j(i, j ∈ I)>u
u i j
p(i j)>u = 0Aui Auj
i j u
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29. Let
and define
where we assume
According to Bayes' theorem, the function to be optimized becomes:
= {(u, i, j) ∈ U × I × I| = 1, = 0} ,DA Aui Auj
p( |X, Y ) := p(i j|X, Y )∏
u∈U
>u ∏
(u,i,j)∈DA
>u
p(i j|X, Y )>u
σ(x)
= σ( − )X
T
u Yi Xu Yj
=
1
1 + e
−x
∏ p(X, Y | ) = ∏ p( |X, Y ) × p(X)p(Y ) × const.>u >u
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30. Taking log of this,
Now consider the following problem:
This means "find a pair of matrices which preserve the order of the
element of the input matrix for each ."
L := log[∏ p( |X, Y ) × p(X)p(Y )]>u
= log p(i j|X, Y ) − ∥X − ∥Y∏
(u,i,j)∈DA
>u λX ∥
2
Fro
λY ∥
2
Fro
= log σ( − ) − ∥X − ∥Y∑
(u,i,j)∈DA
X
T
u Yi X
T
u Yj λX ∥
2
Fro
λY ∥
2
Fro
[ log σ( − ) − ∥X − ∥Y ]max
X,Y
∑
(u,i,j)∈DA
X
T
u Yi X
T
u Yj λX ∥
2
Fro
λY ∥
2
Fro
X, Y
u
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31. Computation
The function we want to optimize:
is huge, so in practice, a stochastic method is necessary.
Let the parameters be .
The algorithm is the following:
Repeat the following
Choose randomly
Update with
This method is called Stochastic Gradient Descent (SGD).
log σ( − ) − ∥X − ∥Y∑
(u,i,j)∈DA
X
T
u Yi X
T
u Yj λX ∥
2
Fro
λY ∥
2
Fro
U × I × I
Θ = (X, Y )
(u, i, j) ∈ DA
Θ
Θ = Θ − α (log σ( − ) − ∥X − ∥Y )
∂
∂Θ
X
T
u Yi X
T
u Yj λX ∥
2
Fro
λY ∥
2
Fro
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32. MyMediaLite
http://www.mymedialite.net/
Open source implemetation of recommendation systems
Written in C#
Reasonable computation time
Supports rating and item prediction
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33. Practical Aspect of Recommendation
Problem
Computational time
Memory consumption
How many services can be integrated in a server rack?
Super high accuracy with a super computer is useless for real business
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34. Concluding Remarks: What is Important for
Good Prediction?
Theory
Machine learning
Mathematical optimization
Implementation
Algorithms
Computer architecture
Mathematics
Human factors!
Hand tuning of parameters
Domain specific knowledge
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35. References (1/2)
For beginers
比戸ら, データサイエンティスト養成読本 機械学習入門編, 技術評論社, 2016
T.Segaran. Programming Collective Intelligence, O'Reilly Media, 2007.
E.Chen. Winning the Netflix Prize: A Summary.
A.Gunawardana and G.Shani. A Survey of Accuracy Evaluation Metrics of
Recommendation Tasks, The Journal of Machine Learning Research,
Volume 10, 2009.
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36. References (2/2)
Papers
Salakhutdinov, Ruslan, and Andriy Mnih. "Bayesian probabilistic matrix
factorization using Markov chain Monte Carlo." Proceedings of the 25th
international conference on Machine learning. ACM, 2008.
Sindhwani, Vikas, et al. "One-class matrix completion with low-density
factorizations." Data Mining (ICDM), 2010 IEEE 10th International
Conference on. IEEE, 2010.
Rendle, Steffen, et al. "BPR: Bayesian personalized ranking from implicit
feedback." Proceedings of the Twenty-Fifth Conference on Uncertainty in
Artificial Intelligence. AUAI Press, 2009.
Zou, Hui, and Trevor Hastie. "Regularization and variable selection via the
elastic net." Journal of the Royal Statistical Society: Series B (Statistical
Methodology) 67.2 (2005): 301-320.
Ning, Xia, and George Karypis. "SLIM: Sparse linear methods for top-n
recommender systems." Data Mining (ICDM), 2011 IEEE 11th
International Conference on. IEEE, 2011.
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