Collaborative Filtering: Models and Algorithms

Accuracy guarantees: Vignette (m = n = 2000 rank-8)

low-rank matrix M sampled matrix ME

OptSpace output M^ squared error (M   M^ )2

Andrea Montanari (Stanford) 1:25% sampled September 15, 2012 31 / 58

Collaborative Filtering

Accuracy guarantees: Vignette (m = n = 2000 rank-8)

low-rank matrix M sampled matrix ME

OptSpace output M^ squared error (M   M^ )2

Andrea Montanari (Stanford) 1:50% sampled September 15, 2012 31 / 58

Collaborative Filtering

Accuracy guarantees: Vignette (m = n = 2000 rank-8)

low-rank matrix M sampled matrix ME

OptSpace output M^ squared error (M   M^ )2

Andrea Montanari (Stanford) 1:75% sampled September 15, 2012 31 / 58

Collaborative Filtering

Accuracy guarantees: Unstructured factors

Incoherence

k k hk k ii ;u 2 k k hk k ijv
:v¡
u¡ 2 2 2
av 2 av

[Candés, Recht 2008]

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 32 / 58

Accuracy guarantees

Theorem (Keshavan, M, Oh, 2009)

jM j MAssume ij max. Then, w.h.p., rank-r projection achieves

q

j j + k k p j jM =C max nr E
RMSE ZC H E 2 n r = E :

Theorem (Keshavan, M, Oh, 2009)

( ) ( ) = ( )MLet
be incoherent with 1 M =r M O 1 . If
j j ! f ( ) gE Cn min r log n 2; r 2 log n then, w.h.p., OptSpace achieves
p
ZC HH n r E 2;
j j k kRMSEE

( ( ) )logwith complexity O nr 3
n 2.

j jp

E.g. Gaussian noise: C HHz r n= E

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 33 / 58

Accuracy guarantees

Theorem (Keshavan, M, Oh, 2009)

jM j MAssume ij max. Then, w.h.p., rank-r projection achieves

q

j j + k k p j jM =C max nr E
RMSE ZC H E 2 n r = E :

Theorem (Keshavan, M, Oh, 2009)

( ) ( ) = ( )MLet
be incoherent with 1 M =r M O 1 . If
j j ! f ( ) gE Cn min r log n 2; r 2 log n then, w.h.p., OptSpace achieves
p
ZC HH n r E 2;
j j k kRMSEE

( ( ) )logwith complexity O nr 3
n 2.

j jp

E.g. Gaussian noise: C HHz r n= E

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 33 / 58

Two surprises

Can do much better than SVD !

Error = Noise = Sampling factor

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 34 / 58

Analogous guarantees for convex relaxations

Candés, Recht, 2008 £
Candés, Plan, 2009
Candés, Tao, 2009
Gross, 2010
Negahbahn, Wainwright, 2010
Koltchinskii, Lounici, Tsybakov, 2011
...

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 35 / 58

A noisy example

n = 500; r = 4; z = 1, example from [Candés, Plan, 2009]

1.4 rank-r projection
SDP
1.2
ADMiRA
1 OptSpace
Oracle Bound
RMSE 0.8

0.6

0.4

0.2

0

j j0 100 200 300 400 500

E =n

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 36 / 58

Challenge #1: Privacy

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 37 / 58

Research question
User ratings 3 User feature vector

User ratings 3 User private attributes ???

Let us try to do it!

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 38 / 58

Research question
User ratings 3 User feature vector

User ratings 3 User private attributes ???

Let us try to do it!

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 38 / 58

Research question
User ratings 3 User feature vector

User ratings 3 User private attributes ???

Let us try to do it!

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 38 / 58

Which attribute?

Number of persons in the household

Non-obvious [ no prize :-( we won it :-) ]
Recommender has incentive
2011 CAMRa Challenge

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 39 / 58

Which attribute?

Number of persons in the household

Non-obvious [ no prize :-( we won it :-) ]
Recommender has incentive
2011 CAMRa Challenge

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 39 / 58

CAMRa dataset

m % 2 ¡ 105 users, n % 2 ¡ 104 movies
jE j % 4:5 ¡ 106 ratings (p % 0:001)

272 households of size 2
14 households of size 3
4 households of size 4

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 40 / 58

CAMRa dataset

m % 2 ¡ 105 users, n % 2 ¡ 104 movies
jE j % 4:5 ¡ 106 ratings (p % 0:001)

272 households of size 2
(14(h(ou(se(ho(ld(s (of(si(ze((3
(4 (ho(us(eh(ol(ds(o(f s(iz(e(4

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 41 / 58

CAMRa dataset

m % 2 ¡ 105 users, n % 2 ¡ 104 movies
jE j % 4:5 ¡ 106 ratings (p % 0:001)

272 households of size 2

Can you identify whether two users shared an account?
Can you identify which user watched a movie?

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 42 / 58

Short answer

Yes: For a signicant fraction of the accounts.

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 43 / 58

Two examples (Netix dataset, no ground truth)

No movie info used!

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 44 / 58

How did you do it?

Rij $ hui ; vj i + "ij

( )  Pn Rr +1; j P WatchedBy(i)o Hyperplane

Rij ; vj

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 45 / 58

How did you do it?

Rij $ hui ; vj i + "ij

( )  Pn Rr +1; j P WatchedBy(i)o Hyperplane

Rij ; vj

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 45 / 58

How did you do it? Subspace clustering

One user 3 One hyperplane
Two users 3 Two hyperplanes

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 46 / 58

Example: A two-user household (CAMRa)

Number of movie rating eventsDistance Difference Distribution for household 172, with similarity score 0.898117
80
User 1
User 2
70 Normal fit (−)
Normal fit (+)

60

50

40

30

20

10

0 2
−2 −1.5 −1 −0.5 0 0.5 1 1.5

Difference of distances of xj from the two hyperplanes

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 47 / 58

Example: Another two-user household (CAMRa)

120 Distance Difference Distribution for household 124, with similarity score 0.500787
100
User 1
User 2
Normal fit in µ ± 1.5σ

Number of movie rating events 80

60

40

20

0 −2 −1.5 −1 −0.5 0 0.5 1 1.5
−2.5 Difference of distances of xj from the two hyperplanes

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 48 / 58

Classifying households Size 1, Empirical
Size 2, Empirical
0.08 Size 1, Fitted Gamma
MOVIEX Size 2, Fitted Gamma

0.06PDF

0.04

0.02

00 (M5S0E1−MSE120)0× m / lo1g5(m0) 200

MSE1 3 MSE using one hyperplane
MSE2 3 MSE using two hyperplanes

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 49 / 58

Challenge #2: Interactivity

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 50 / 58

We want to design the system

User Recommender

We took time out of the picture.

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 51 / 58

We want to design the system

User Recommender

We took time out of the picture.

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 51 / 58

Interactive system Recom.
Recom.
t = 1 User Recom.
t = 2 User
t = 3 User

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 52 / 58

Abstract from other users

At time t
Recommender suggests movie vt ;
User gives feedback Rt

Focus on user u

Rt $ hu; vt i + "t

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 53 / 58

Abstract from other users

At time t
Recommender suggests movie vt ;
User gives feedback Rt

Focus on user u

Rt $ hu; vt i + "t

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 53 / 58

Linear bandits

Decision:

vt

Observations:

Rt $ hu; vt i + "t

Reward: =yt h iˆt

Andrea Montanari (Stanford) u ; v`

`=1

[Rusmevichientong, Tsitsiklis, 2008]

Collaborative Filtering September 15, 2012 54 / 58

Important dierences

Number of observations $ Dimensions

Cannot explore completely at random.

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 55 / 58

Simulating an interactive system (Netix data)

10 8 8
8
6Cumulative Reward 6 6
4 Cumulative Reward
2 Cumulative Reward4 4
0
2 2
0
5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30
t 0 t 0 t

3 `typical' users
Constant-optimal policy

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 56 / 58

Conclusion

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 57 / 58

Conclusion

A crucial technology for modern information networks.
Only scratched the surface.

Thanks!

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 58 / 58

Conclusion

A crucial technology for modern information networks.
Only scratched the surface.

Thanks!

Andrea Montanari (Stanford) Collaborative Filtering September 15, 2012 58 / 58