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Collaborative Filtering: Models and Algorithms
Andrea Montanari
Jose Bento, Yash Deshpande, Adel Javanmard, Raghunandan Keshavan, Sewoong Oh,
Stratis Ioannidis, Nadia Fawaz, Amy Zhang
Stanford University, Technicolor
September 15, 2012
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Problem statement
Given data on the activity of a set of users, provide personalizedrecommendations to users X, Y, Z,. . .
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Example
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An obviously useful technology
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An obviously useful technology
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Outline
1 A model
2 Algorithms and accuracy
3 Challenge #1: Privacy
4 Challenge #2: Interactivity
5 Conclusion
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A model
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Setting
Users : i 2 f1; 2; : : : ;mg
Movies : j 2 f1; 2; : : : ;ng
When user i watches movie j , she enters her rating Rij .
Want to predict ratings for missing pairs.
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Setting
Users : i 2 f1; 2; : : : ;mg
Movies : j 2 f1; 2; : : : ;ng
When user i watches movie j , she enters her rating Rij .
Want to predict ratings for missing pairs.
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Linear regression model
Movie j
vj =�genre; main actor; supporting actor; year; . . .
� 2 Rr
Rij � ci + hui ; vj i+ "ij
Want: User parameters vector ui
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Linear regression model
Movie j
vj =�genre; main actor; supporting actor; year; . . .
� 2 Rr
Rij � ci + hui ; vj i+ "ij
Want: User parameters vector ui
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Linear regression model
Movie j
vj =�genre; main actor; supporting actor; year; . . .
� 2 Rr
Rij � ci + hui ; vj i+ "ij
Want: User parameters vector ui
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Linear regression model
Movie j
vj =�genre; main actor; supporting actor; year; . . .
� 2 Rr
Rij ���ci + hui ; vj i+ "ij
Want: User parameters vector ui
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Least squares
ui = arg minxi2Rr
8<: Xj2WatchedBy(i)
�Rij � hxi ; vj i
�29=;
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Ridge regression
ui = arg minxi2Rr
8<: Xj2WatchedBy(i)
�Rij � hxi ; vj i
�2+ � kxik2
9=;
I How do we construct the vj 's?
I Ad hoc de�nitions are not suited to recommendation!
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Ridge regression
ui = arg minxi2Rr
8<: Xj2WatchedBy(i)
�Rij � hxi ; vj i
�2+ � kxik2
9=;
I How do we construct the vj 's?
I Ad hoc de�nitions are not suited to recommendation!
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If I knew the users' feature vectors
Rij � hui ; vj i+ "ij
vj = arg minyj2Rr
8<: Xi2Watched(j )
�Rij � hui ; yj i
�2+ � kyj k2
9=;
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If I knew the users' feature vectors
Rij � hui ; vj i+ "ij
vj = arg minyj2Rr
8<: Xi2Watched(j )
�Rij � hui ; yj i
�2+ � kyj k2
9=;
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Everything together
ui = arg minxi2Rr
8<: Xj2WatchedBy(i)
�Rij � hxi ; vj i
�2+ � kxik2
9=;vj = arg min
yj2Rr
8<: Xi2Watched(j )
�Rij � hui ; yj i
�2+ � kyj k2
9=;
Minimize (E =Watched)
F (X ;Y ) =X
(i ;j )2E
�Rij � hxi ; yj i
�2+ �
mXi=1
kxik22 + �nX
j=1
kyj k22
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Everything together
ui = arg minxi2Rr
8<: Xj2WatchedBy(i)
�Rij � hxi ; vj i
�2+ � kxik2
9=;vj = arg min
yj2Rr
8<: Xi2Watched(j )
�Rij � hui ; yj i
�2+ � kyj k2
9=;
Minimize (E =Watched)
F (X ;Y ) =X
(i ;j )2E
�Rij � hxi ; yj i
�2+ �
mXi=1
kxik22 + �nX
j=1
kyj k22
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Objective function
Minimize (E =Watched)
F (X ;Y ) =X
(i ;j )2E
�Rij � hxi ; yj i
�2+ �
mXi=1
kxik22 + �nX
j=1
kyj k22
� kPE (R�XY T)k2F + �kX k2F + �kY k2F
PE (A) =(Aij if (i ; j ) 2 E ;
0 otherwise
XT =hx1
���x2��� � � � ���xmiY T =
hy1
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Objective function
Minimize (E =Watched)
F (X ;Y ) =X
(i ;j )2E
�Rij � hxi ; yj i
�2+ �
mXi=1
kxik22 + �nX
j=1
kyj k22
� kPE (R�XY T)k2F + �kX k2F + �kY k2F
PE (A) =(Aij if (i ; j ) 2 E ;
0 otherwise
XT =hx1
���x2��� � � � ���xmiY T =
hy1
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Low rank structure
Low-rank M U
I V T
=m
n r
r
1. Low-rank matrix M
2. R = M+ Z
3. Observed subset E
PE (R)ij =(Mij + Zij if (i ; j ) 2 E ;0 otherwise.
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Low rank structure
Low-rank M U
I V T
=m
n r
r
1. Low-rank matrix M
2. R = M+ Z
3. Observed subset E
PE (R)ij =(Mij + Zij if (i ; j ) 2 E ;0 otherwise.
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Low rank structure
Low-rank M U
I V T
=m
n r
r
1. Low-rank matrix M
2. R = M+ Z
3. Observed subset E
PE (R)ij =(Mij + Zij if (i ; j ) 2 E ;0 otherwise.
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Low rank structure
Sample PE (R)m
n
1. Low-rank matrix M
2. R = M+ Z
3. Observed subset E
PE (R)ij =(Mij + Zij if (i ; j ) 2 E ;0 otherwise.
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Algorithms and accuracy
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Questions
I How do we minimize F (X ;Y )?
I What prediction accuracy? (RMSE = kM� M̂kF=pmn)
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How do we minimize F (X ;Y )?
I Spectral methods.
I Gradient method. [Srebro, Rennie, Jaakkola, 2003]
I Convex relaxations. [Fazel, Hindi, Boyd, 2001]
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Spectral method (for simplicity � = 0)
Replace
F (X ;Y ) = kPE (R�XY T)k2F= kPE (XY T)k2F � 2hPE (R);XY Ti+ const:
with (p = jE j=mn fraction of observed entries)
eF (X ;Y ) = p kXY Tk2F � 2hPE (R);XY Ti+ const
= p
XY T � 1
pPE (R)
2F
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Spectral method (for simplicity � = 0)
Replace
F (X ;Y ) = kPE (R�XY T)k2F= kPE (XY T)k2F � 2hPE (R);XY Ti+ const:
with (p = jE j=mn fraction of observed entries)
eF (X ;Y ) = p kXY Tk2F � 2hPE (R);XY Ti+ const
= p
XY T � 1
pPE (R)
2F
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Spectral method (for simplicity � = 0)
Replace
F (X ;Y ) = kPE (R�XY T)k2F= kPE (XY T)k2F � 2hPE (R);XY Ti+ const:
with (p = jE j=mn fraction of observed entries)
eF (X ;Y ) = p kXY Tk2F � 2hPE (R);XY Ti+ const
= p
XY T � 1
pPE (R)
2F
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Spectral method (for simplicity � = 0)
Minimize
eF (X ;Y ) =
XY T � 1
pPE (R)
2F
Solved by SVD
PE (R) = XSY T ) M̂ =1
pXm�r (S)r�rY
T
n�r
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Spectral method (for simplicity � = 0)
Minimize
eF (X ;Y ) =
XY T � 1
pPE (R)
2F
Solved by SVD
PE (R) = XSY T ) M̂ =1
pXm�r (S)r�rY
T
n�r
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Random matrix r = 4, m = n = 10000, p = 0:0012
0 10 20 30 40 50 60 70 800
10
20
30
40
�1�2�3�4
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Net�ix data (trimmed)
0 20 40 60 80 100 120 1400
20
40
60
80
100
120
140
RMSE � 0:99
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Accuracy guarantees
Theorem (Keshavan, M, Oh, 2009)
Qssume jMij j � Mmax. Then, w.h.p., rank-r projection achieves
RMSE � CMmax
qnr=jE j+C 0 kZEk2 n
pr=jE j :
E.g. Gaussian noise: C 00(1+ �z )pr n=jE j
[Improves over Achlioptas-McSherry 2003]
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Gradient descent
F (X ;Y ) =X
(i ;j )2E
�Rij � hxi ; yj i
�2+ �
mXi=1
kxik22 + �nX
j=1
kyj k22
Update rule: ( = `learning rate')
xi (1� � )xi + X
j2WatchedBy(i)
�Rij � hxi ; yj i
�yj
yj (1� � )yj + X
i2Watched(j )
�Rij � hxi ; yj i
�xi
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Gradient descent
F (X ;Y ) =X
(i ;j )2E
�Rij � hxi ; yj i
�2+ �
mXi=1
kxik22 + �nX
j=1
kyj k22
Update rule: ( = `learning rate')
xi (1� � )xi + X
j2WatchedBy(i)
�Rij � hxi ; yj i
�yj
yj (1� � )yj + X
i2Watched(j )
�Rij � hxi ; yj i
�xi
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Interpretation
xi (1� � )xi + X
j2WatchedBy(i)
wij yj
yj (1� � )yj + X
i2Watched(j )
wij xi
user avg of movies she liked
movie avg of users that liked it
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Interpretation
xi (1� � )xi + X
j2WatchedBy(i)
wij yj
yj (1� � )yj + X
i2Watched(j )
wij xi
user avg of movies she liked
movie avg of users that liked it
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A variant (stochastic gradient)
Pick (i ; j ) 2 E :
xi (1� � )xi + wij yj
yj (1� � )yj + wij xi
[Srebro, Rennie, Jaakkola, 2003]
[Srebro, Jaakkola, 2005]
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A variant (stochastic gradient)
Pick (i ; j ) 2 E :
xi (1� � )xi + wij yj
yj (1� � )yj + wij xi
[Srebro, Rennie, Jaakkola, 2003]
[Srebro, Jaakkola, 2005]
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In the words of SimonFunk
[Ne�ix challenge, 2006-2009]
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And his results
Target RMSE � 0:8564
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Three variants
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
0 5 10 15 20 25 30 35
Message PassingAlt. Min.OptSpace
iterations
RMSE
OptSpace � Gradient descent on Grassmannian [Keshavan-M.-Oh 2009]Alternating Least Squares [Koren-Bell 2008]
Message Passing [Keshavan-M. 2011]
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Accuracy guarantees: Vignette (m = n = 2000 rank-8)
low-rank matrix M sampled matrix ME
OptSpace output M̂ squared error (M� M̂)2
0:25% sampled
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Accuracy guarantees: Vignette (m = n = 2000 rank-8)
low-rank matrix M sampled matrix ME
OptSpace output M̂ squared error (M� M̂)2
0:50% sampled
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Accuracy guarantees: Vignette (m = n = 2000 rank-8)
low-rank matrix M sampled matrix ME
OptSpace output M̂ squared error (M� M̂)2
0:75% sampled
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Accuracy guarantees: Vignette (m = n = 2000 rank-8)
low-rank matrix M sampled matrix ME
OptSpace output M̂ squared error (M� M̂)2
1:00% sampled
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Accuracy guarantees: Vignette (m = n = 2000 rank-8)
low-rank matrix M sampled matrix ME
OptSpace output M̂ squared error (M� M̂)2
1:25% sampled
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Accuracy guarantees: Vignette (m = n = 2000 rank-8)
low-rank matrix M sampled matrix ME
OptSpace output M̂ squared error (M� M̂)2
1:50% sampled
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Accuracy guarantees: Vignette (m = n = 2000 rank-8)
low-rank matrix M sampled matrix ME
OptSpace output M̂ squared error (M� M̂)2
1:75% sampled
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Accuracy guarantees: Unstructured factors
Incoherence
kuik2 � � hku�k2iav ; kvj k22 � � hkv�k2iav :
[Candés, Recht 2008]
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Accuracy guarantees
Theorem (Keshavan, M, Oh, 2009)
Assume jMij j � Mmax. Then, w.h.p., rank-r projection achieves
RMSE � CMmax
qnr=jE j+C 0 kZEk2 n
pr=jE j :
Theorem (Keshavan, M, Oh, 2009)
Let M be incoherent with �1(M )=�r (M ) = O(1). IfjE j � Cn minfr(logn)2; r2 logng then, w.h.p., OptSpace achieves
RMSE � C 00npr
jE j kZEk2 ;
with complexity O(nr3(logn)2).
E.g. Gaussian noise: C 00�zpr n=jE j
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Accuracy guarantees
Theorem (Keshavan, M, Oh, 2009)
Assume jMij j � Mmax. Then, w.h.p., rank-r projection achieves
RMSE � CMmax
qnr=jE j+C 0 kZEk2 n
pr=jE j :
Theorem (Keshavan, M, Oh, 2009)
Let M be incoherent with �1(M )=�r (M ) = O(1). IfjE j � Cn minfr(logn)2; r2 logng then, w.h.p., OptSpace achieves
RMSE � C 00npr
jE j kZEk2 ;
with complexity O(nr3(logn)2).
E.g. Gaussian noise: C 00�zpr n=jE j
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Two surprises
Can do much better than SVD !
Error = Noise = Sampling factor
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Analogous guarantees for convex relaxations
Candés, Recht, 2008 �
Candés, Plan, 2009
Candés, Tao, 2009
Gross, 2010
Negahbahn, Wainwright, 2010
Koltchinskii, Lounici, Tsybakov, 2011
. . .
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A noisy example
I n = 500; r = 4; �z = 1, example from [Candés, Plan, 2009]
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 100 200 300 400 500
rank-r projectionSDP
ADMiRAOptSpace
Oracle Bound
RMSE
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Challenge #1: Privacy
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Research question
User ratings ! User feature vector
User ratings ! User private attributes ???
Let us try to do it!
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Research question
User ratings ! User feature vector
User ratings ! User private attributes ???
Let us try to do it!
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Research question
User ratings ! User feature vector
User ratings ! User private attributes ???
Let us try to do it!
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Which attribute?
Number of persons in the household
I Non-obvious
I Recommender has incentive
I 2011 CAMRa Challenge [ no prize :-( we won it :-) ]
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Which attribute?
Number of persons in the household
I Non-obvious
I Recommender has incentive
I 2011 CAMRa Challenge [ no prize :-( we won it :-) ]
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CAMRa dataset
I m � 2 � 105 users, n � 2 � 104 movies
I jE j � 4:5 � 106 ratings (p � 0:001)
I 272 households of size 2
I 14 households of size 3
I 4 households of size 4
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CAMRa dataset
I m � 2 � 105 users, n � 2 � 104 movies
I jE j � 4:5 � 106 ratings (p � 0:001)
I 272 households of size 2
I((((((((((((14 households of size 3
I(((((((((((4 households of size 4
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CAMRa dataset
I m � 2 � 105 users, n � 2 � 104 movies
I jE j � 4:5 � 106 ratings (p � 0:001)
I 272 households of size 2
I Can you identify whether two users shared an account?
I Can you identify which user watched a movie?
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Short answer
Yes: For a signi�cant fraction of the accounts.
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Two examples (Net�ix dataset, no ground truth)
No movie info used!
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How did you do it?
Rij � hui ; vj i+ "ij
n(Rij ;�vj ) 2 Rr+1; j 2WatchedBy(i)
o� Hyperplane
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How did you do it?
Rij � hui ; vj i+ "ij
n(Rij ;�vj ) 2 Rr+1; j 2WatchedBy(i)
o� Hyperplane
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How did you do it? Subspace clustering
One user ! One hyperplaneTwo users ! Two hyperplanes
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Example: A two-user household (CAMRa)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
10
20
30
40
50
60
70
80
Difference of distances of xj from the two hyperplanes
Nu
mb
er
of
mo
vie
ra
tin
g e
ve
nts
Distance Difference Distribution for household 172, with similarity score 0.898117
User 1
User 2
Normal fit (−)
Normal fit (+)
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Example: Another two-user household (CAMRa)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.50
20
40
60
80
100
120
Difference of distances of xj from the two hyperplanes
Num
ber
of m
ovie
rating e
vents
Distance Difference Distribution for household 124, with similarity score 0.500787
User 1
User 2
Normal fit in µ ± 1.5σ
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Classifying households
0 50 100 150 2000
0.02
0.04
0.06
0.08
(MSE1−MSE
2) × m / log(m)
PD
F
MOVIEX
Size 1, Empirical
Size 2, Empirical
Size 1, Fitted Gamma
Size 2, Fitted Gamma
MSE1 ! MSE using one hyperplaneMSE2 ! MSE using two hyperplanes
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Challenge #2: Interactivity
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We want to design the system
User Recommender
We took time out of the picture.
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We want to design the system
User Recommender
We took time out of the picture.
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Interactive system
User Recom.
User Recom.
User Recom.
t = 1
t = 2
t = 3
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Abstract from other users
At time t
I Recommender suggests movie vt ;
I User gives feedback Rt
Focus on user u
Rt � hu ; vt i+ "t
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Abstract from other users
At time t
I Recommender suggests movie vt ;
I User gives feedback Rt
Focus on user u
Rt � hu ; vt i+ "t
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Linear bandits
Decision:
vt
Observations:
Rt � hu ; vt i+ "t
Reward:
Ot =tX
`=1
hu ; v`i
[Rusmevichientong, Tsitsiklis, 2008]
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Important di�erences
I Number of observations � Dimensions
I Cannot explore completely at random.
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Simulating an interactive system (Net�ix data)
0 5 10 15 20 25 30
0
2
4
6
8
10
t
Cum
ula
tive
Rew
ard
0 5 10 15 20 25 30
0
2
4
6
8
t
Cum
ula
tive
Rew
ard
0 5 10 15 20 25 30
0
2
4
6
8
t
Cum
ula
tive
Rew
ard
I 3 `typical' users
I Constant-optimal policy
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Conclusion
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Conclusion
I A crucial technology for modern information networks.
I Only scratched the surface.
Thanks!
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Conclusion
I A crucial technology for modern information networks.
I Only scratched the surface.
Thanks!
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