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1年以上前 (2016/03/26)にアップロードinテクノロジー

Material presented at Tokyo Web Mining Meetup, March 26, 2016.

The source code is here:

https://...

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。

- Introduction to Algorithms for Behavior Based

Recommendation

Tokyo Web Mining Meetup

March 26, 2016

Kimikazu Kato

Silver Egg Technology Co., Ltd.

1 / 36 - 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

2 - 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.

3 - Table of Contents

Introduction

Types of recommendation

Evaluation metrics

Algorithms

Conclusion

4 - 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

5 - 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

6 - Rating Prediction Problem

user\movie

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.

7 - Item Prediction Problem

user\item

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.

8 - 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 k

Output: top k items for each user which are most likely to be bought by

him/her

9 - 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)

10 - RMSE of Rating Prediction

Some user/item pairs are randomly chosen to be hidden.

user\movie

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 |3.1 − 4|2 = 0.92

.

Take the sum over the error over all the hidden items and then, take the

square root of it.

−−−−−−−−−−−−−−−−−−−−−−−− −

−

∑ (predictedui − actualui)2

√(u,i)∈hidden

11 - Precision/Recal 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

12 - ROC and AUC

# of

1

2

3

4

5

6

7

8

9

10

recom.

# of

1

1

1

2

2

3

4

5

5

6

whites

# of

0

1

2

2

3

3

3

3

4

4

blacks

Divide the first and second row by total number of white and blacks

respectively, and plot the values in xy plane.

13 - 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...

14 - 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.

15 - 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

16 - Notations

Number of users: n

Set of users: U = {1, 2, … , n}

Number of items (movies): m

Set of items (movies): I = {1, 2, … , m}

Input matrix: A (n × m matrix)

17 - 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

A ≈ XT Y

Find such matrices X ∈ Mat(f, n), Y ∈ Mat(f, m) where f ≪ n, m

18 - p(A|X,Y ,σ) = ∏ N (Aui|XTu Yi,σ)

aui ≠0

p(X|σX) = ∏N (Xu|0,σXI)

u

p(Y |σY ) = ∏N (Yi|0,σY I)

i

Find X and Y maximize p (X, Y |A, σ)

19 - According to Bayes' Theorem,

p(X,Y |A,σ)

= p(A|X,Y ,σ)p(X|σX)p(X|σX) × const.

Thus,

logp(U,V |A,σ,σU ,σV )

= ∑ (A

2

ui − XT

u Yi) + λX∥X∥2

+

∥Y

+ const.

Fro

λY

∥2Fro

Aui

where ∥ ⋅ ∥

means Frobenius norm.

Fro

How can this be computed? Use MCMC. See [Salakhutdinov et al., 2008].

Once X and Y are determined, A~ := XT Y and the prediction for A is

ui

estimated by A~ui

20 - Difference between Rating and Shopping

Rating

Shopping (Browsing)

user\movie

W

X

Y

Z

user\item

W

X

Y

Z

A

5

4

1

4

A

1

1

1

1

B

4

B

1

C

2

3

C

1

D

1

4

?

D

1

?

1

?

Includes negative feedback

Includes no negative feedback

"1" means "boring"

Zero means "unknown" or

Zero means "unknown"

"negative"

More degree of the freedom

Consequently, the algorithm effective for the rating matrix is not necessarily

effective for the shopping matrix.

21 - Solutions

Adding a constraint to the optimization problem

Changing the objective function itself

22 - 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

23 - One-class Matrix Completion

[Sindhwani et al., 2010]

Introduced variables p to relax the problem.

ui

Minimize

∑ (Aui − XTu Yi) + λX∥X∥2 +

∥Y

Fro

λY

∥2Fro

Aui≠0

+ ∑ [pui(0 − XTu Yi)2 + (1 − pui)(1 − XTu Yi)2]

Aui=0

+ T ∑ [−pui logpui − (1 − pui)log(1 − pui)]

Aui=0

subject to

1

∑ pui = r

|{Aui|Aui = 0}| Aui=0

24 - ∑ (Aui − XTu Yi) + λX∥X∥2 +

∥Y

Fro

λY

∥2Fro

Aui≠0

+ ∑ [pui(0 − XTu Yi)2 + (1 − pui)(1 − XTu Yi)2]

Aui=0

+ T ∑ [−pui logpui − (1 − pui)log(1 − pui)]

Aui=0

Intuitive explanation:

p means how likely the

-element is zero.

ui

(u,i)

The second term is the error of estimation considering p 's.

ui

The third term is the entropy of the distribution.

25 - Implicit Sparseness constraint: SLIM (Elastic Net)

In the regression model, adding L1 term makes the solution sparse:

λ(1 − ρ)

min[ 1 ∥Xw − y∥2 +

∥w∥2 + λρ|w ]

w

2n

2

2

2

|1

The similar idea is used for the matrix factorization [Ning et al., 2011]:

Minimize

λ(1 − ρ)

∥A − AW∥ +

∥W∥2 + λρ|W

2

Fro

|1

subject to

diag W = 0

26 - 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.

27 - 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

u

u ∈ U

i >u j(i,j ∈ I)

means "the user u is more likely to buy i than j.

The objective is to calculate p(i >u j) such that Aui = 0 and A (which

uj

means i and j are not bought by u).

28 - Let

DA = {(u,i,j) ∈ U × I × I|Aui = 1,Auj = 0},

and define

∏ p(>u |X,Y ) := ∏ p(i >u j|X,Y )

u∈U

(u,i,j)∈DA

where we assume

p(i >u j|X,Y ) = σ(XTu Yi − XuYj)

σ(x) =

1

1 + e−x

According to Bayes' theorem, the function to be optimized becomes:

∏p(X,Y | >u) = ∏p(>u |X,Y ) × p(X)p(Y ) × const.

29 - Taking log of this,

L := log[∏p(>u |X,Y ) × p(X)p(Y )]

= log ∏ p(i >u j|X,Y ) − λX∥X∥2 −

∥Y

Fro

λY

∥2Fro

(u,i,j)∈DA

= ∑ logσ(XTu Yi − XTu Yj) − λX∥X∥2 −

∥Y

Fro

λY

∥2Fro

(u,i,j)∈DA

Now consider the following problem:

max[ ∑ logσ(XTu Yi − XTu Yj) − λX∥X∥2 −

∥Y

]

X,Y

Fro

λY

∥2Fro

(u,i,j)∈DA

This means "find a pair of matrices X, Y which preserve the order of the

element of the input matrix for each u."

30 - Computation

The function we want to optimize:

∑ logσ(XTu Yi − XTu Yj) − λX∥X∥2 −

∥Y

Fro

λY

∥2Fro

(u,i,j)∈DA

U × I × I is huge, so in practice, a stochastic method is necessary.

Let the parameters be Θ = (X, Y ).

The algorithm is the following:

Repeat the following

Choose (u, i, j) ∈ D randomly

A

Update Θ with

Θ = Θ − α ∂ (logσ(XTu Yi − XTu Yj) − λX∥X∥2Fro − λY ∥Y ∥2Fro)

∂Θ

This method is called Stochastic Gradient Descent (SGD).

31 - MyMediaLite

http://www.mymedialite.net/

Open source implemetation of recommendation systems

Written in C#

Reasonable computation time

Supports rating and item prediction

32 - 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

33 - 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

34 - 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.

35 - 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.

36