Matrix Factorization Techniques for Top-N … Factorization Techniques for Top-N Recommender Systems Ernesto Diaz-Aviles Web Science 2013 L3S Research Center. Hannover,

Matrix Factorization Techniques forTop-N Recommender Systems

Ernesto Diaz-Aviles<[email protected]>

Web Science 2013L3S Research Center. Hannover, Germany

Ernesto Diaz-Aviles <[email protected]> www.L3S.de | 1/39

Summary from Last Lecture

Learning to Rank

: Supervised Machine Learning Approach forRanking

Approaches : Pointwise, Pairwise, and Listwise Approaches.

Huge impact: Web Search, E-Business, Government, etc.



Learning to Rank: Supervised Machine Learning Approach forRanking






Approaches

: Pointwise, Pairwise, and Listwise Approaches.





Approaches : Pointwise

, Pairwise, and Listwise Approaches.





Approaches : Pointwise, Pairwise

, and Listwise Approaches.













SGD Algorithm


Matrix Factorization Techniques forTop-N Recommender Systems

(Personalized Ranking)


Recommender Systems


Recommender Systems Tasks

1 Rating Prediction

2 Top-N Recommendation



1 Rating Prediction




1 Rating Prediction


(Item Prediction — Personalized Ranking)


User-based Neighborhood Method

[Y. Koren, R. Bell, and C. Volinsky. 2009. Matrix Factorization Techniques for Recommender Systems. Computer.]


User-Item Matrix

User/Item I1 I2 I3 I4 I5 I6

U1 5 3 4

U2 1 1 1

U3 1 3 1

U4 5 2 2 5 4

U5 3 2


Matrix Factorization

Wi

Hj

Feedback Matrix: X = | U | x | i |

User Latent FactorsW = | U | x | k |

Item Latent FactorsH = | k | x | i |

Xij


MF and Latent FactorsCOVER FE ATURE

COMPUTER 44

vector qi ! f, and each user u is associ-ated with a vector pu ! f. For a given item i, the elements of qi measure the extent to which the item possesses those factors, positive or negative. For a given user u, the elements of pu measure the extent of interest the user has in items that are high on the corresponding factors, again, posi-tive or negative. The resulting dot product, qi

T pu, captures the interaction between user u and item i—the user’s overall interest in the item’s characteristics. This approximates user u’s rating of item i, which is denoted by rui, leading to the estimate

rui

= qiT pu. (1)

The major challenge is computing the map-ping of each item and user to factor vectors qi, pu ! f. After the recommender system completes this mapping, it can easily esti-mate the rating a user will give to any item by using Equation 1.

Such a model is closely related to singular value decom-position (SVD), a well-established technique for identifying latent semantic factors in information retrieval. Applying SVD in the collaborative filtering domain requires factoring the user-item rating matrix. This often raises difficulties due to the high portion of missing values caused by sparse-ness in the user-item ratings matrix. Conventional SVD is undefined when knowledge about the matrix is incom-plete. Moreover, carelessly addressing only the relatively few known entries is highly prone to overfitting.

Earlier systems relied on imputation to fill in missing ratings and make the rating matrix dense.2 However, im-putation can be very expensive as it significantly increases the amount of data. In addition, inaccurate imputation might distort the data considerably. Hence, more recent works3-6 suggested modeling directly the observed rat-ings only, while avoiding overfitting through a regularized model. To learn the factor vectors (pu and qi), the system minimizes the regularized squared error on the set of known ratings:

min* *,q p ( , )u i

(rui qiTpu)

2 + (|| qi ||2 + || pu ||

2) (2)

Here, is the set of the (u,i) pairs for which rui is known (the training set).

The system learns the model by fitting the previously observed ratings. However, the goal is to generalize those previous ratings in a way that predicts future, unknown ratings. Thus, the system should avoid overfitting the observed data by regularizing the learned parameters, whose magnitudes are penalized. The constant controls

recommendation. These methods have become popular in recent years by combining good scalability with predictive accuracy. In addition, they offer much flexibility for model-ing various real-life situations.

Recommender systems rely on different types of input data, which are often placed in a matrix with one dimension representing users and the other dimension representing items of interest. The most convenient data is high-quality explicit feedback, which includes explicit input by users regarding their interest in products. For example, Netflix collects star ratings for movies, and TiVo users indicate their preferences for TV shows by pressing thumbs-up and thumbs-down buttons. We refer to explicit user feedback as ratings. Usually, explicit feedback com-prises a sparse matrix, since any single user is likely to have rated only a small percentage of possible items.

One strength of matrix factorization is that it allows incorporation of additional information. When explicit feedback is not available, recommender systems can infer user preferences using implicit feedback, which indirectly reflects opinion by observing user behavior including pur-chase history, browsing history, search patterns, or even mouse movements. Implicit feedback usually denotes the presence or absence of an event, so it is typically repre-sented by a densely filled matrix.

A BASIC MATRIX FACTORIZATION MODEL Matrix factorization models map both users and items

to a joint latent factor space of dimensionality f, such that user-item interactions are modeled as inner products in that space. Accordingly, each item i is associated with a

Gearedtowardmales

Serious

Escapist

The PrincessDiaries

Braveheart

Lethal Weapon

IndependenceDay

Ocean’s 11Sense andSensibility

Gus

Dave

Gearedtoward

females

Amadeus

The Lion King Dumb andDumber

The Color Purple

Figure 2. A simpli!ed illustration of the latent factor approach, which characterizes both users and movies using two axes—male versus female and serious versus escapist. [Y. Koren, R. Bell, and C. Volinsky. 2009. Matrix Factorization Techniques for Recommender Systems. Computer.]


Matrix Factorization

Wi

Hj

Feedback Matrix: X = | U | x | i |

User Latent FactorsW = | U | x | k |

Item Latent FactorsH = | k | x | i |

Xij


CF Based on Matrix Factorization

MF estimates X : U × I by the product of two low-rankmatrices W : |U | × k and H : |I| × k as follows:

X := WHᵀ, where k is a parameter corresponding to therank of the approximation.


General Stochastic Gradient Descent Procedure for MF

SGD for MF

Input:Training data S; Regularization parameters λW and λH ;Learning rate η0; Learning rate schedule α; Number ofiterations T

Output: θ = (W,H)1: initialize W0 and H0

2: for t = 1 to T do3: (u, i, xui)← randomExample(S)4: wu ← wu − η ∂

∂wu`(xui, 〈wu,hi〉)− η λW wu

5: hi ← hi − η ∂∂hi

`(xui, 〈wu,hi〉)− η λH hi6: η = α · η7: end for8: return θT = (WT ,HT )


Stochastic Gradient Descent


Learning Algorithms: Standard Framework†

Assumption: examples are drawn independently from anunknown probability distribution P (x, y) that represents therules of Nature

Expected Risk: E(f) =∫`(f(x), y) dP (x, y)

Empirical Risk: En(f) = 1n

∑n `(f(xi), yi)

We would like f∗ that minimizes E(f) among all functions

In general f∗ /∈ FThe best we can have is f∗F that minimizes E(f) inside FBut P (x, y) is unknown by definition

Instead we compute fn ∈ F that minimized En(f)

Vapnik-Chervonenkis theory tells us when this can work

†Leon Bottou: The Tradeoffs of Large Scale Learning, NIPS tutorials, Vancouver, 2007


Learning with Approximate Optimization†

Computing fn = argmin f ∈ FEn(f) is often costly

Since we already make lots of approximations, why should wecompute fn exactly?

Let’s assume our optimizer returns fn such thatEn(fn) < En(fn) + ε

For instance, one could stop an iterative optimizationalgorithm long before its convergence



Small-scale vs. Large-scale Learning†

fn ≈ argmin Θ = w 1n

∑n `(f(xi), yi)

Simple parametric setup

F is fixed: functions fw(x) linearly parametrized by w ∈ Rd

Subject to budget constraints :

Maximal number of examples n (Small-Scale Learning)

Maximal computing time T (Large-Scale Learning)



Gradient Descent

Iterate

wt+1 ← wt − η∂En(fwt)

∂w= wt − η

∑

n

`′(fw(xi), yi)

Gradient Descent (GD)

Iterate

• wt+1! wt " !"En(fwt)

"w

Gradient J

Best speed achieved with fixed learning rate ! = 1"max

.(e.g., Dennis & Schnabel, 1983)

Cost per Iterations Time to reach Time to reachiteration to reach # accuracy # E(fn)! E(f"F ) < $

GD O(nd) O!% log 1

#

"O

!nd% log 1

#

"O

!d2 %$1/& log2 1

$

"

– In the last column, n and # are chosen to reach # as fast as possible.– Solve for # to find the best error rate achievable in a given time.– Remark: abuses of the O() notation


Stochastic Gradient Descent

Iterate

Draw Random Example (xt, yy)

wt+1 ← wt − ηt `′(fwt(xt), yt)

Stochastic Gradient Descent (SGD)

Iterate

• Draw random example (xt, yt).

• wt+1! wt "!

t

"#(fwt(xt), yt)

"w

Total Gradient <J(x,y,w)>

Partial Gradient J(x,y,w)

Best decreasing gain schedule with ! = 1"min

.(see Murata, 1998; Bottou & LeCun, 2004)

Cost per Iterations Time to reach Time to reachiteration to reach # accuracy # E(fn)! E(f"F ) < $

SGD O(d) % k# + o

!1#

"O

!d % k

#

"O

!d % k

$

"

With 1 # k # &2

– Optimization speed is catastrophic.– Learning speed does not depend on the statistical estimation rate $.– Learning speed depends on condition number % but scales very well.

More info. at L. Bottou webpage: http://leon.bottou.org/


General Stochastic Gradient Descent Procedure for MF

SGD for MF

Input:Training data S; Regularization parameters λW and λH ;Learning rate η0; Learning rate schedule α; Number ofiterations T

Output: θ = (W,H)1: initialize W0 and H0

2: for t = 1 to T do3: (u, i, xui)← randomExample(S)4: wu ← wu − η ∂

∂wu`(xui, 〈wu,hi〉)− η λW wu

5: hi ← hi − η ∂∂hi

`(xui, 〈wu,hi〉)− η λH hi6: η = α · η7: end for8: return θT = (WT ,HT )


Pairwise Learning

(Bob, #EHEC)(Alice, #EHEC #Hannover) (Bob, #HUS)(Charlie, #HUS) (Charlie, #RKI) (Alice, #EHEC)... Monitor

#EHEC #Hannover #HUS #RKI

Alice 2 1 ? ?

Bob 1 ? 1 ?

Charlie ? ? 1 1

E.g, Pairwise Preferences for Alice: #EHEC > {#Hannover, #HUS, # RKI}#Hannover > {#HUS, #RKI}



Ranking Loss (pairwise):

L(P,W,H) =1

|P |∑

p∈Ph(yuij · 〈wu,hi − hj〉) , (1)

argminθ=(W,H)

L(P,W,H) +λW2||W||22 +

λH2||H||22 . (2)

where

h(z) = max(0, 1− z) is the hinge-loss;

yuij = sign(xui − xuj) is the sign(z) function, which returns+1 if z > 0, i.e., xui > xuj , and −1 if z < 0.

The prediction function 〈wu,hi − hj〉 = 〈wu,hi〉 − 〈wu,hj〉corresponds to the difference of predictor values xui − xuj .Gradient

−∇h(pt, θt) =

yuij · (hi − hj) if θr = wu,

yuij ·wu if θt = hi,

yuij · (−wu) if θt = hj ,

0 otherwise.




L(P,W,H) =1

|P |∑


argminθ=(W,H)

L(P,W,H) +λW2||W||22 +

λH2||H||22 . (2)

where




−∇h(pt, θt) =




0 otherwise.




L(P,W,H) =1

|P |∑


argminθ=(W,H)

L(P,W,H) +λW2||W||22 +

λH2||H||22 . (2)

where



The prediction function 〈wu,hi − hj〉 = 〈wu,hi〉 − 〈wu,hj〉corresponds to the difference of predictor values xui − xuj .

Gradient

−∇h(pt, θt) =




0 otherwise.




L(P,W,H) =1

|P |∑


argminθ=(W,H)

L(P,W,H) +λW2||W||22 +

λH2||H||22 . (2)

where




−∇h(pt, θt) =




0 otherwise.


Example of Model Update based on SGD for MF

Input: Stream representative sample at time t: St; Regularizationparameters λW , λH+ , and λH− ; Learning rate η0; Learning rateschedule α; Number of iterations Tθ

Output: θ = (W,H)1: procedure updateModel(St, λW , λH+ , λH− , η0, α, Tθ)2: for t = 1 to Tθ do3: ((u, i), (u, j))← randomPair(St) ∈ P4: yuij ← sign(xui − xuj)5: wu ← wu + η yuij (hi − hj)− η λW wu

6: hi ← hi + η yuij wu − η λH+ hi7: hj ← hj + η yuij (−wu)− η λH− hj8: η = α · η9: end for

10: return θ = (WTθ ,HTθ )11: end procedure


ScenarioTowards Real-time Collaborative Filtering for Big Fast Data


Scenario: #-tags Recommendation in Twitter

Top-N Recommendations

Recommendation of Interesting Topics ⇒ #-tags

Online Collaborative Filtering












Pairwise Learning

(Bob, #EHEC)(Alice, #EHEC #Hannover) (Bob, #HUS)(Charlie, #HUS) (Charlie, #RKI) (Alice, #EHEC)... Monitor

#EHEC #Hannover #HUS #RKI

Alice 2 1 ? ?

Bob 1 ? 1 ?

Charlie ? ? 1 1

E.g, Pairwise Preferences for Alice: #EHEC > {#Hannover, #HUS, # RKI}#Hannover > {#HUS, #RKI}


RMFO: Main Steps

Social Media Stream, e.g., Twitter

Sample the Stream

Real-time Personalized Recommendations

Online Matrix Factorization: Pairwise Approach for Personalized Rank Learning


RMFO: Sampling Strategies

Social Media Stream, e.g., Twitter

Sample the Stream


Online Matrix Factorization: Pairwise Approach for Personalized Rank Learning

Sampling Strategies

RMFO-SP: Single Pass

RMFO-UB: User Buffer

RMFO-RSV: ReservoirSampling


RMFO-SP: Single Pass

TwitterMonitor

Online Collaborative Filtering Pairwise MF


Model Update Using Single Pair at the Time


RMFO-UB: User Buffer

TwitterMonitor



User 1

User 2

User m

.

.

.

Individual User Buffers of

Limited Size


RMFO-RSV: Reservoir Sampling

TwitterMonitor

Reservoir Sampling




Experimental Evaluation for Top-N Recommendation

Our three RMFO variations: RMFO-SP, RMFO-UB, andRMFO-RSV

SNAP@Stanford 476 million Twitter tweets DatasetDataset statistics (5-core):

Events (tweets): 35,350,508Users: 413,987Items (#-tags): 37,297

Leave-One-Out Protocol with a time sensitive split ofdatasetMetric recall also known as hit-rate :

recall :=

∑u∈Utest 1[iu∈Top-Nu]

|Utest|

Baselines: (1) Trending Topics (TT) and (2) WeightedRegularized Matrix Factorization (WRMF)1

1WRMF: Collaborative Filtering for Implicit Feedback Datasets Authors:Hu, Y.; Koren, Y.; Volinsky, C. IEEEInternational Conference on Data Mining (ICDM 2008)MyMediaLite Implementation







recall :=


|Utest|








Leave-One-Out Protocol with a time sensitive split ofdataset

Metric recall also known as hit-rate :

recall :=


|Utest|









recall :=


|Utest|









recall :=


|Utest|




Recommendation PerformanceTT (previous month)WRMFRankMF-SPRankMF-UBRankMF-RSV

Recall@1 Recall@5 Recall@10 Recall@15 Recall@20 Recall@30 Recall@40 [email protected] 0.0522 0.0780 0.0929 0.0943 0.1061 0.1132 0.12100.0885 0.1896 0.2573 0.3045 0.3406 0.3943 0.4331 0.46370.0357 0.1003 0.1469 0.1807 0.2078 0.2510 0.2859 0.31650.0377 0.1070 0.1555 0.1897 0.2169 0.2605 0.2955 0.32580.1040 0.2458 0.3215 0.3694 0.4048 0.4562 0.4942 0.5247

0

0.1

0.2

0.3

0.4

0.5

0.6

Top-1 Top-5 Top-10 Top-15 Top-20 Top-30 Top-40 Top-50

recall @ N

reca

ll

TT (previous month) WRMF RMFO-SP RMFO-UB RMFO-RSV


Recommendation Performance for Different Reservoir Sizes

recall@10 Reservoir Size

RMFO-RSV RankMF-UB-512

Trending Topics

(previous month)

WRMF (Batch)

0.5 0.0621339040754 0.1555 0.0780177217 0.25731384921 0.1143406622964 0.1555 0.0780177217 0.25731384922 0.1845008184564 0.1555 0.0780177217 0.25731384924 0.2611452241341 0.1555 0.0780177217 0.25731384928 0.3215161808443 0.1555 0.0780177217 0.2573138492

Recall@10Test Set Size (= n test users; leave one out)

260246

128 factors

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.5 2 3.5 5 6.5 8

Top-10: recall vs Reservoir Sizere

call

Reservoir Size (Millions)

RMFO-RSVRMFO-UB-512WRMF (Batch)Trending Topics (previous month)


Recommendation Performance and Dimensionality

RMFO-RSV 8M

8163264128WRMF-128

RMFO-RSV-8RMFO-RSV-16RMFO-RSV-32RMFO-RSV-64RMFO-RSV-128

Recall@1 Recall@5 Recall@10 Recall@15 Recall@20 Recall@30 Recall@40 Recall@50

0.0468 0.1529 0.2289 0.2811 0.3207 0.3787 0.4216 0.45570.0624 0.1830 0.2627 0.3146 0.3529 0.4080 0.4483 0.48030.0793 0.2118 0.2912 0.3407 0.3771 0.4297 0.4683 0.49880.0913 0.2284 0.3076 0.3570 0.3924 0.4431 0.4804 0.51010.1040 0.2458 0.3215 0.3694 0.4048 0.4562 0.4942 0.52470.0885 0.1896 0.2573 0.3045 0.3406 0.3943 0.4331 0.4637

8 0.2289 0.2573 0.0780177216 0.2627 0.2573 0.0780177232 0.2912 0.2573 0.0780177264 0.3076 0.2573 0.07801772

128 0.3215 0.2573 0.07801772

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

8 20 32 44 56 68 80 92 104 116 128

Top-10: Recall vs Number of Factorsre

call

Number of Factors

RMFO-RSVWRMF-128TT


Time and Space Savings vs Recommendation Quality

Method Time recall@10 Space Gain Gain in(128 factors) (seconds) in speed recall

WRMF (Baseline) 23127.34 0.2573 100.00% – –

RMFO-RSV 0.5 M 47.97 0.0621 1.41% 482.16 -75.85%

RMFO-RSV 1 M 89.15 0.1143 2.83% 259.42 -55.56%

RMFO-RSV 2 M 171.18 0.1845 5.66% 135.11 -28.30%

RMFO-RSV 4 M 329.60 0.2611 11.32% 70.17 +1.49%

RMFO-RSV 8 M 633.85 0.3215 22.63% 36.49 +24.95%

RMFO-RSV INF 1654.52 0.3521 100.00% 13.98 +36.84%


Conclusions

Matrix Factorization: Powerful Machine Learning approach forTop-N Recommendation

SGD for MF : Large Scale Datasets

RMFO approach for recommending topics to users in presenceof streaming data

RMFO achieves state-of-the-art performance in terms ofrecommendation quality with significant improvements inspeed and space efficiency


Conclusions






Conclusions






Conclusions






Thank you!

Your Software Project and Thesis at L3S Research Center :)

Ernesto Diaz-Aviles

[email protected]


www.L3S.de

More Info

www.L3S.de

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Matrix Factorization Techniques for Top-N … Factorization Techniques for Top-N Recommender Systems Ernesto Diaz-Aviles Web Science 2013 L3S Research Center. Hannover,