[PPT] - Scalable Machine Learning 8. Recommender Systems Alex Smola Yahoo! PowerPoint Presentation

SLIDE 1

Scalable Machine Learning

8. Recommender Systems

Alex Smola Yahoo! Research and ANU

http://alex.smola.org/teaching/berkeley2012 Stat 260 SP 12

Significant content courtesy of Yehuda Koren

SLIDE 2

8. Recommender Systems

Much content courtesy of (Mr Netflix) Yehuda Koren

SLIDE 3

Outline

Neighborhood methods
User / movie similarity
Iteration on graph
Matrix Factorization
Singular value decomposition
Convex reformulation
Ranking and Session Modeling
Ordinal regression
Session models
Features
Latent dense (Bayesian Probabilistic Matrix Factorization)
Latent sparse (Dirichlet process factorization)
Coldstart problem (inferring features)
Hashing

SLIDE 4

Why

SLIDE 5

SLIDE 6

Netflix

SLIDE 7

SLIDE 8

SLIDE 9

SLIDE 10

Personalized Content

adapt to general popularity pick based on user preferences

SLIDE 11

Spam Filtering

Something went wrong!

SLIDE 12

A more formal view

User (requests content)
Objects (that can be displayed)
Context (device, location, time)
Interface (mobile browser, tablet, viewport)

u

c

interface

recommend relevant objects

SLIDE 13

Examples

Movie recommendation (Netflix)
Related product recommendation (Amazon)
Web page ranking (Google)
Social recommendation (Facebook)
News content recommendation (Yahoo)
Priority inbox & spam filtering (Google)
Online dating (OK Cupid)
Computational Advertising (Yahoo)

SLIDE 14

Running Example Netflix Movie Recommendation

score

date movie user

1 5/7/02 21 1 5 8/2/04 213 1 4 3/6/01 345 2 4 5/1/05 123 2 3 7/15/02 768 2 5 1/22/01 76 3 4 8/3/00 45 4 1 9/10/05 568 5 2 3/5/03 342 5 2 12/28/00 234 5 5 8/11/02 76 6 4 6/15/03 56 6

score date movie user

? 1/6/05 62 1 ? 9/13/04 96 1 ? 8/18/05 7 2 ? 11/22/05 3 2 ? 6/13/02 47 3 ? 8/12/01 15 3 ? 9/1/00 41 4 ? 8/27/05 28 4 ? 4/4/05 93 5 ? 7/16/03 74 5 ? 2/14/04 69 6 ? 10/3/03 83 6

Training data Test data

SLIDE 15

Challenges

Scalability
Millions of objects
100s of millions of users
Cold start
Changing user base
Changing inventory (movies, stories, goods)
Attributes
Imbalanced dataset

User activity / item reviews are power law distributed

http://www.igvita.com/2006/10/29/dissecting-the-netflix-dataset/

SLIDE 16

Netflix competition yardstick

Least mean squares prediction error
Easy to define
Wrong measure for composing sessions!
Consistent (in large sample size limit this will

converge to minimizer)

rmse(S) = s |S|−1 X

(i,u)∈S

(ˆ rui − rui)2

SLIDE 17

1 Neighborhood Methods

SLIDE 18

Basic Idea

Joe

#2 #3 #1 #4

SLIDE 19

Derive ¡unknown ¡ratings ¡from ¡those ¡of ¡“similar” ¡items ¡
Basic Idea
(user,user) similarity to recommend items
good if item base is smaller than user base
good if item base changes rapidly
traverse bipartite similarity graph
(item,item) similarity to recommend new items that

were also liked by the same users

good if the user base

is small is small

Oldest known CF method

SLIDE 20

Neighborhood based CF

12 11 10 9 8 7 6 5 4 3 2 1

4 5 5 3 1

1

3 1 2 4 4 5

2

5 3 4 3 2 1 4 2

3

2 4 5 4 2

4

5 2 2 4 3 4

5

4 2 3 3 1

6 users items

unknown rating
rating between 1 to 5

0.2 · 2 + 0.3 · 3 0.2 + 0.3 = 2.6

SLIDE 21

Neighborhood based CF

12 11 10 9 8 7 6 5 4 3 2 1

4 5 5 3 1

1

3 1 2 4 4 5

2

5 3 4 3 2 1 4 2

3

2 4 5 4 2

4

5 2 2 4 3 4

5

4 2 3 3 1

6 users items

unknown rating
rating between 1 to 5

?

0.2 · 2 + 0.3 · 3 0.2 + 0.3 = 2.6

SLIDE 22

Neighborhood based CF

12 11 10 9 8 7 6 5 4 3 2 1

4 5 5 3 1

1

3 1 2 4 4 5

2

5 3 4 3 2 1 4 2

3

2 4 5 4 2

4

5 2 2 4 3 4

5

4 2 3 3 1

6 users items

unknown rating
rating between 1 to 5

?

0.2 · 2 + 0.3 · 3 0.2 + 0.3 = 2.6

SLIDE 23

Neighborhood based CF

12 11 10 9 8 7 6 5 4 3 2 1

4 5 5 3 1

1

3 1 2 4 4 5

2

5 3 4 3 2 1 4 2

3

2 4 5 4 2

4

5 2 2 4 3 4

5

4 2 3 3 1

6 users items

unknown rating
rating between 1 to 5

similarity s13 = 0.2 s16 = 0.3 weighted average ?

0.2 · 2 + 0.3 · 3 0.2 + 0.3 = 2.6

SLIDE 24

Neighborhood based CF

12 11 10 9 8 7 6 5 4 3 2 1

4 5 5 3 1

1

3 1 2 4 4 5

2

5 3 4 3 2 1 4 2

3

2 4 5 4 2

4

5 2 2 4 3 4

5

4 2 3 3 1

6 users items

unknown rating
rating between 1 to 5

similarity s13 = 0.2 s16 = 0.3 weighted average

0.2 · 2 + 0.3 · 3 0.2 + 0.3 = 2.6

2.6

SLIDE 25

Derive ¡unknown ¡ratings ¡from ¡those ¡of ¡“similar” ¡items ¡
Properties
Intuitive
No (substantial) training
Handles new users / items
Easy to explain to user
Accuracy & scalability questionable

SLIDE 26

Normalization / Bias

Problem
Some items are significantly higher rated
Some users rate substantially lower
Ratings change over time
Bias correction is crucial for nearest neighborhood

recommender algorithm

Offset per user
Offset per movie
Time effects
Global bias

bui = µ + bu + bi

user item global

Bell & Koren ICDM 2007 http://public.research.att.com/~volinsky/netflix/BellKorICDM07.pdf

SLIDE 27

Baseline estimation

Mean rating is 3.7
Troll Hunter is 0.7 above mean
User rates 0.2 below mean
Baseline is 4.2 stars
Least mean squares problem
Jointly convex. Alternatively remove mean & iterate

minimize

b

X

(u,i)

(rui − µ − bu − bi)2 + λ "X

u

b2

u +

X

i

b2

i

#

bi = P

u∈R(i)(rui − µ − bu)

λ + |R(i)| and bu = P

i∈R(u)(rui − µ − bi)

λ + |R(u)|

SLIDE 28

Parzen Windows style CF

Similarity measure sij between items
Find set sk(i,u) of k-nearest neighbors to i that

were rated by user u

Weighted average over the set
How to compute sij?

ˆ rui = bui + P

j∈sk(i,u) sij(ruj − buj)

P

j∈sk(i,u) sij

where bui = µ + bu + bi

SLIDE 29

–
–

each item rated by a distinct set of users

1 ? ? 5 5 3 ? ? ? 4 2 ? ? ? ? 4 ? 5 4 1 ? ? ? 4 2 5 ? ? 1 2 5 ? ? 2 ? ? 3 ? ? ? 5 4 User ratings for item i: User ratings for item j:

(item,item) similarity measures
Pearson correlation coefficient
nonuniform support
compute only over shared support
shrinkage towards 0 to address problem of

small support (typically few items in common)

sij = Cov[rui, ruj] Std[rui]Std[ruj]

SLIDE 30

(item,item) similarity

Empirical Pearson correlation coefficient
Smoothing towards 0 for small support
Make neighborhood more peaked
Shrink towards baseline for small neighborhood

ˆ ρij = P

u∈U(i,j)(rui − bui)(ruj − buj)

qP

u∈U(i,j)(rui − bui)2 P u∈U(i,j)(ruj − buj)2

sij = |U(i, j)| − 1 |U(i, j)| − 1 + λ ˆ ρij sij → s2

ij

ˆ rui = bui + P

j∈sk(i,u) sij(ruj − buj)

λ + P

j∈sk(i,u) sij

SLIDE 31

Similarity for binary data

Pearson correlation meaningless
Views
Purchase behavior
Clicks
Jaccard similarity

(intersection vs. joint)

Observed/expected ratio

Improve by counting per user (many users better than heavy users)

mi users acting on i mij users acting on both i and j m total number of users

sij = mij α + mi + mj − mij sij = observed expected ≈ mij α + mimj/m

SLIDE 32

2 Matrix Factorization

SLIDE 33

Basics

SLIDE 34

Basic Idea

~

M ≈ U · V

SLIDE 35

Latent variable view

Geared towards females Geared towards males serious escapist The Princess Diaries The Lion King Braveheart Lethal Weapon Independence Day Amadeus The Color Purple Dumb and Dumber Ocean’s ¡11 Sense and Sensibility

Gus Dave

SLIDE 36

Basic matrix factorization

4 5 5 3 1 3 1 2 4 4 5 5 3 4 3 2 1 4 2 2 4 5 4 2 5 2 2 4 3 4 4 2 3 3 1

items

.2

.4

.1 .5 .6

.5

.5 .3

.2

.3 2.1 1.1

2

2.1

.7

.3 .7

1
.9

2.4 1.4 .3

.4

.8

.5
2

.5 .3

.2

1.1 1.3

.1

1.2

.7

2.9 1.4

1

.3 1.4 .5 .7

.8

.1

.6

.7 .8 .4

.3

.9 2.4 1.7 .6

.4

2.1

~ ~

items users users A rank-3 SVD approximation

SLIDE 37

Estimate unknown ratings as inner products of latent factors

4 5 5 3 1 3 1 2 4 4 5 5 3 4 3 2 1 4 2 2 4 5 4 2 5 2 2 4 3 4 4 2 3 3 1

items

.2

.4

.1 .5 .6

.5

.5 .3

.2

.3 2.1 1.1

2

2.1

.7

.3 .7

1
.9

2.4 1.4 .3

.4

.8

.5
2

.5 .3

.2

1.1 1.3

.1

1.2

.7

2.9 1.4

1

.3 1.4 .5 .7

.8

.1

.6

.7 .8 .4

.3

.9 2.4 1.7 .6

.4

2.1

~ ~

items users A rank-3 SVD approximation users

?

SLIDE 38

Estimate unknown ratings as inner products of latent factors

4 5 5 3 1 3 1 2 4 4 5 5 3 4 3 2 1 4 2 2 4 5 4 2 5 2 2 4 3 4 4 2 3 3 1

items

.2

.4

.1 .5 .6

.5

.5 .3

.2

.3 2.1 1.1

2

2.1

.7

.3 .7

1
.9

2.4 1.4 .3

.4

.8

.5
2

.5 .3

.2

1.1 1.3

.1

1.2

.7

2.9 1.4

1

.3 1.4 .5 .7

.8

.1

.6

.7 .8 .4

.3

.9 2.4 1.7 .6

.4

2.1

~ ~

items users A rank-3 SVD approximation users

?

SLIDE 39

Estimate unknown ratings as inner products of latent factors

4 5 5 3 1 3 1 2 4 4 5 5 3 4 3 2 1 4 2 2 4 5 4 2 5 2 2 4 3 4 4 2 3 3 1

items

.2

.4

.1 .5 .6

.5

.5 .3

.2

.3 2.1 1.1

2

2.1

.7

.3 .7

1
.9

2.4 1.4 .3

.4

.8

.5
2

.5 .3

.2

1.1 1.3

.1

1.2

.7

2.9 1.4

1

.3 1.4 .5 .7

.8

.1

.6

.7 .8 .4

.3

.9 2.4 1.7 .6

.4

2.1

~ ~

items users

2.4

A rank-3 SVD approximation users

SLIDE 40

Properties

SVD is undefined for missing entries
stochastic gradient descent (faster)
alternating optimization
Overfitting without regularization

particularly if fewer reviews than dimensions

Very popular on Netflix

4 5 5 3 1 3 1 2 4 4 5 5 3 4 3 2 1 4 2 2 4 5 4 2 5 2 2 4 3 4 4 2 3 3 1 .2

.4

.1 .5 .6

.5

.5 .3

.2

.3 2.1 1.1

2

2.1

.7

.3 .7

1
.9

2.4 1.4 .3

.4

.8

.5
2

.5 .3

.2

1.1 1.3

.1

1.2

.7

2.9 1.4

1

.3 1.4 .5 .7

.8

.1

.6

.7 .8 .4

.3

.9 2.4 1.7 .6

.4

2.1

~

SVD ¡isn’t ¡defined ¡when ¡entries ¡are ¡unknown ¡

–

–

SLIDE 41

40 60 90 128 180 50 100 200 50 100 200 50 100 200 500 100 200 500 50 100 200 500 1000 1500

0.875 0.88 0.885 0.89 0.895 0.9 0.905 0.91

10 100 1000 10000 100000

RMSE

Millions of Parameters

Factor models: Error vs. #parameters

NMF BiasSVD SVD++ SVD v.2 SVD v.3 SVD v.4 Prize: 0.8563 Netflix: 0.9514

SLIDE 42

Risk Minimization View

Objective Function
Alternating least squares

minimize

p,q

X

(u,i)∈S

(rui hpu, qii)2 + λ h kpk2

Frob + kqk2 Frob

i pu ← 2 4λ1 + X

i|(u,i)2S

qiq>

i

3 5

1

X

i

qirui qi ← 2 4λ1 + X

u|(u,i)2S

pup>

u

3 5

1

X

i

purui

good for MapReduce

SLIDE 43

Risk Minimization View

Objective Function
Stochastic gradient descent
No need for locking
Multicore updates asynchronously

(Recht, Re, Wright, 2012 - Hogwild)

minimize

p,q

X

(u,i)∈S

(rui hpu, qii)2 + λ h kpk2

Frob + kqk2 Frob

i

much faster

pu (1 ληt)pu ηtqi(rui hpu, qii) qi (1 ληt)qi ηtpu(rui hpu, qii)

SLIDE 44

Theoretical Motivation

SLIDE 45

deFinetti Theorem

Independent random variables
Exchangeable random variables
There exists a conditionally independent

representation of exchangeable r.v. This motivates latent variable models

p(X) =

m

Y

i=1

p(xi) p(X) = p(x1, . . . , xm) = p(xπ(1), . . . , xπ(m))

xi xi ϴ

p(X) = Z dp(θ)

m

Y

i=1

p(xi|θ)

SLIDE 46

Aldous Hoover Factorization

Matrix-valued set of random variable

Example - Erdos Renyi graph model

Independently exchangeable on matrix
Aldous Hoover Theorem

p(E) = Y

i,j

p(Vij)

p(E) = p(E11, E12, . . . , Emn) = p(Eπ(1)ρ(1), Eπ(1)ρ(2), . . . , Eπ(m)ρ(n))

p(E) = Z dp(θ) Z

m

Y

i=1

dp(ui)

n

Y

j=1

dp(vj) Y

i,j

p(Eij|ui, vj, θ)

SLIDE 47

Aldous Hoover Factorization

Rating matrix is (row,

column) exchangeable

Draw latent variables per

row and column

Draw matrix entries

independently given pairs

Absence / presence of

rating is a signal

Can be extended to graphs

with vertex attributes

u1 u2 u3 u4 u5 u6 v1 e11 e12 e15 e16 v2 e24 v3 e32 v4 e43 e46 v5 e55

SLIDE 48

Aldous Hoover variants

Jointly exchangeable matrix
Social network graphs
Draw vertex attributes first, then edges
Cold start problem
New user appears
Attributes (age, location, browser)
Can estimate latent variables from that
User and item factors in matrix factorization

problem can be viewed as AH-factors

SLIDE 49

Improvements

SLIDE 50

40 60 90 128 180 50 100 200 50 100 200 50 100 200 500 100 200 500 50 100 200 500 1000 1500

0.875 0.88 0.885 0.89 0.895 0.9 0.905 0.91

10 100 1000 10000 100000

RMSE

Millions of Parameters

Factor models: Error vs. #parameters

NMF BiasSVD SVD++ SVD v.2 SVD v.3 SVD v.4

Add biases

SLIDE 51

Bias

Objective Function
Stochastic gradient descent

minimize

p,q

X

(u,i)∈S

(rui (µ + bu + bi + hpu, qii))2+ λ h kpk2

Frob + kqk2 Frob + kbusersk2 + kbitemsk2i

pu (1 ληt)pu ηtqiρui qi (1 ληt)qi ηtpuρui bu (1 ληt)bu ηtρui bi (1 ληt)bi ηtρui µ (1 ληt)µ ηtρui where ρui = (rui (µ + bi + bu + hpu, qii))

SLIDE 52

40 60 90 128 180 50 100 200 50 100 200 50 100 200 500 100 200 500 50 100 200 500 1000 1500

0.875 0.88 0.885 0.89 0.895 0.9 0.905 0.91

10 100 1000 10000 100000

RMSE

Millions of Parameters

Factor models: Error vs. #parameters

NMF BiasSVD SVD++ SVD v.2 SVD v.3 SVD v.4

“who ¡rated ¡ what”

SLIDE 53

Ratings are not given at random

Marlin et al. “Collaborative Filtering and the

Missing at Random Assumption” UAI 2007

B. ¡Marlin ¡et ¡al., ¡“Collaborative ¡Filtering ¡and ¡the ¡Missing ¡

at ¡Random ¡Assumption” ¡ Yahoo! survey answers Yahoo! music ratings Netflix ratings

SLIDE 54

Movie rating matrix

Characterize users by which movies they rated

Edge attributes (observed, rating)

Adding features to recommender system


4 5 5 3 1 3 1 2 4 4 5 5 3 4 3 2 1 4 2 2 4 5 4 2 5 2 2 4 3 4 4 2 3 3 1

users movies

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

users movies

 



 



rui cui

rui = µ + bu + bi + hpu, qii + hcu, xii

regression

SLIDE 55

Alternative integration

Key idea - use related ratings to average
Salakhudtinov & Mnih, 2007
Koren et al., 2008

Overparametrize items by q and x

qi ← qi + X

u

cuipu qi ← qi + X

u

cuixj

SLIDE 56

40 60 90 128 180 50 100 200 50 100 200 50 100 200 500 100 200 500 50 100 200 500 1000 1500

0.875 0.88 0.885 0.89 0.895 0.9 0.905 0.91

10 100 1000 10000 100000

RMSE

Millions of Parameters

Factor models: Error vs. #parameters

NMF BiasSVD SVD++ SVD v.2 SVD v.3 SVD v.4

temporal effects

SLIDE 57

Something Happened in Early 2004…

2004 Netflix ratings by date

Netflix changed rating labels

SLIDE 58

Are movies getting better with time?

SLIDE 59

Sources of temporal change

Items
Seasonal effects

(Christmas, Valentine’s day, Holiday movies)

Public perception of movies (Oscar etc.)
Users
Changed labeling of reviews
Anchoring (relative to previous movie)
Change of rater in household
Selection bias for time of viewing

SLIDE 60

Modeling temporal change

Time-dependent bias
Time-dependent user preferences
Parameterize functions b and p
Slow changes for items
Fast sudden changes for users
Good parametrization is key

rui(t) = µ + bu(t) + bi(t) + hqi, pu(t)i

Koren et al., KDD 2009 (CF with temporal dynamics)

SLIDE 61

Biases 33% Personalization 10% Unexplained 57%

Sources of Variance in Netflix data

1.276 (total variance) 0.732 (unexplained) 0.415 (biases) 0.129 (personalization) + +

Bias matters

SLIDE 62

40 60 90 128 180 50 100 200 50 100 200 50 100 200 500 100 200 500 50 100 200 500 1000 1500

0.875 0.88 0.885 0.89 0.895 0.9 0.905 0.91

10 100 1000 10000 100000

RMSE

Millions of Parameters

Factor models: Error vs. #parameters

NMF BiasSVD SVD++ SVD v.2 SVD v.3 SVD v.4 Prize: 0.8563 Netflix: 0.9514

( ) ( ) ( ) ( )

T ui u i i u uj j j

r t b t b t q p t b x            



T ui i u

r q p 

SLIDE 63

More ideas

Explain factorizations
Cold start (new users)
Different regularization for different parameter

groups / different users

Sharing of statistical strength between users
Hierarchical matrix co-clustering / factorization

(write a paper on that)

SLIDE 64

3 Session Modeling

SLIDE 65

Motivation

SLIDE 66

User interaction

Explicit search query
Search engine
Genre selection on movie site
Implicit search query
News site
Priority inbox
Comments on article
Viewing specific movie (see also ...)
Sponsored search (advertising)

Space, users’ time and attention are limited.

SLIDE 67

SLIDE 68

session? models?

SLIDE 69

Did the user SCROLL DOWN?

SLIDE 70

Bad ideas ...

Show items based on relevance
Yes, this user likes Die Hard.
But he likes other movies, too
Show items only for majority of users

‘apple’ vs. ‘Apple’

SLIDE 71

User response

collapse collapse

implicit user interest log it!

SLIDE 72

hover on link

SLIDE 73

Response is conditioned on available options

User search for ‘chocolate’
What the user really would have wanted
User can only pick from

available items

Preferences are often relative

user picks this

SLIDE 74

Models

SLIDE 75

Independent click model

Each object has click probability
Object is viewed independently
Used in computational advertising (with some position correction)
Horribly wrong assumption
OK if probability is very small (OK in ads)

p(x|s) =

n

Y

i=1

1 1 + e−xisi

SLIDE 76

Logistic click model

User picks at most one object
Exponential family model for click
Ignores order of objects
Assumes that the user looks at all before taking action

p(x|s) = esx es0 + P

x0 esx0 = exp (sx − g(s))

no click

SLIDE 77

Sequential click model

User traverses list
At each position some probability of clicking
When user reaches end of the list he aborts
This assumes that a patient user viewed all items

no click click

p(x = j|s) = "j−1 Y

i=1

1 1 + esi # 1 1 + e−sj

SLIDE 78

Skip click model

User traverses list
At each position some probability of clicking
At each position the user may abandon the process
This assumes that user traverses list sequentially

click no no no no click click click

SLIDE 79

Context skip click model

User traverses list
At each position some probability of clicking which depends on previous content
At each position the user may abandon the process
User may click more than once

SLIDE 80

Context skip click model

SLIDE 81

Context skip click model

Viewing probability
Click probability (only if viewed)

p(vi = 1|vi−1 = 0) = 0 p(vi = 1|vi−1 = 1, ci−1 = 0) = 1 1 + e−αi p(vi = 1|vi−1 = 1, ci−1 = 1) = 1 1 + e−βi

nctional form: p(ci = 1|vi = 1, ci−1, di) = 1 1 + e−f(|ci−1|,di,di−1)

user is gone user returns prior context

SLIDE 82

Incremental gains score

Submodular gain per additional document
Relevance score per document
Coverage over different aspects
Position dependent score
Score dependent on number of previous clicks

f(|ci−1|, di, di−1) :=ρ(S, di|a, b) ρ(S, di−1|a, b) + γ|ci−1| + δi := X

s∈S

X

j

[s]j aj X

d∈di

[d]j + bj “ ρj(di) ρj(di−1) ”! + γ|ci−1| + δi

SLIDE 83

Latent variables

We don’t know v whether user viewed result

Use variational inference to integrate out v

(more next week in graphical models)

Optimization

log p(c)  log p(c) + D(q(v)kp(v|c)) = Ev∼q(v) [ log p(c) + log q(v) log p(v|c)] = Ev∼q(v) [ log p(c, v)] H(q(v)).

SLIDE 84

Optimization

Compute latent viewing probability given clicks
Easy since we only have one transition from

views to no views (no DP needed)

Expected log-likelihood under viewing model
Convex expected log-likelihood
Stochastic gradient descent
Parametrization uses personalization, too

(user, position, viewport, browser)

SLIDE 85

SLIDE 86

SLIDE 87

SLIDE 88

SLIDE 89

4 Feature Representation

SLIDE 90

Bayesian Probabilistic Matrix Factorization

SLIDE 91

Statistical Model

Aldous-Hoover factorization
normal distribution for

user and item attributes

rating given by inner product
Ratings
Latent factors

U V j

i

Rij

j=1,...,M i=1,...,N V

σ

U

σ σ

r s

p(Rij|Ui, Vj, σ2) = N (Rij|U T

i Vj, σ2)

p(U|σ2

U) = N

i=1

N (Ui|0, σ2

UI),

p(V |σ2

V ) = M

j=1

N (Vj|0, σ2

V I)

Salakhudtinov & Mnih, ICML 2008 BPMF

SLIDE 92

Details

U Vj

i

Rij

j=1,...,M i=1,...,N

σ Θ

V U

Θ α αV

U

eature

Priors on all factors
Wishart prior is conjugate

to Gaussian, hence use it

Allows us to adapt the

variance automatically

Inference (Gibbs sampler)
Sample user factors (parallel)
Sample movie factors (parallel)
Sample hyperparameters (parallel)

SLIDE 93

Making it fancier (constrained BPMF)

i

Y V j Rij

j=1,...,M

U i

i

I

i=1,...,N V

σ

U

σ W

k=1,...,M

k

W

σ σ

t .

who rated what

SLIDE 94

Results (Mnih & Salakthudtinov)

1−5 6−10 −20 −40 −80 −160 −320 −640 >641 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Number of Observed Ratings RMSE PMF Constrained PMF Movie Average

1−5 6−10 −20 −40 −80 −160 −320 −640 >641 5 10 15 20 Number of Observed Ratings Users (%)

helps for infrequent users

SLIDE 95

Multiple Sources

SLIDE 96

Social Network Data

Data: users, connections, features Goal: suggest connections

SLIDE 97

Social Network Data

Data: users, connections, features Goal: suggest connections

SLIDE 98

Social Network Data

Data: users, connections, features Goal: suggest connections

x x’ y y’ e

SLIDE 99

Social Network Data

Data: users, connections, features Goal: model/suggest connections

x x’ y y’ e

p(x, y, e) = Y

i∈Users

p(yi)p(xi|yi) Y

i,j∈Users

p(eij|xi, yi, xj, yj)

Direct application of the Aldous-Hoover theorem. Edges are conditionally independent.

SLIDE 100

Applications

SLIDE 101

Applications

SLIDE 102

Applications

SLIDE 103

Applications

social network = friendship + interests

SLIDE 104

Applications

social network = friendship + interests

recommend users based

n friendship & interests

recommend apps based

n friendship & interests

SLIDE 105

Social Recommendation

recommend users based

n friendship & interests

recommend apps based

n friendship & interests
boost traffic
make the user

graph more dense

increase user

population

stickiness
boost traffic
increased revenue
increased user

participation

make app graph

more dense

... usually addressed by separate tools ...

SLIDE 106

Homophily

recommend users based

n friendship & interests

recommend apps based

n friendship & interests
users with similar

interests are more likely to connect

friends install similar

applications

Highly correlated. Estimate both jointly

SLIDE 107

Model

x x’ y y’ e v u s

(latent) app features

(latent) user features

app install

SLIDE 108

Model

x x’ y y’ e v u a

Social interaction
App install

SLIDE 109

Model

Social interaction
App install

i xj + y> i Wyj)

aij ∼ p(a|x>

i vj + y> i Muj)

cold start latent features bilinear features

SLIDE 110

Optimization Problem

minimize λe X

(i,j)

l(eij, x>

i xj + y> i Wyj)+

λa X

(i,j)

l(aij, x>

i vj + y> i Muj)+

λx X

i

γ(xi|yi) + λv X

i

γ(vi|ui)+ λW W2 + λM M2 + λA A2 + λB B2

minimize

SLIDE 111

Optimization Problem

minimize λe X

(i,j)

l(eij, x>

i xj + y> i Wyj)+

λa X

(i,j)

l(aij, x>

i vj + y> i Muj)+

λx X

i

γ(xi|yi) + λv X

i

γ(vi|ui)+ λW W2 + λM M2 + λA A2 + λB B2

minimize social

SLIDE 112

Optimization Problem

minimize λe X

(i,j)

l(eij, x>

i xj + y> i Wyj)+

λa X

(i,j)

l(aij, x>

i vj + y> i Muj)+

λx X

i

γ(xi|yi) + λv X

i

γ(vi|ui)+ λW W2 + λM M2 + λA A2 + λB B2

minimize social app

SLIDE 113

Optimization Problem

minimize λe X

(i,j)

l(eij, x>

i xj + y> i Wyj)+

λa X

(i,j)

l(aij, x>

i vj + y> i Muj)+

λx X

i

γ(xi|yi) + λv X

i

γ(vi|ui)+ λW W2 + λM M2 + λA A2 + λB B2

minimize social app

reconstruction

SLIDE 114

Optimization Problem

minimize λe X

(i,j)

l(eij, x>

i xj + y> i Wyj)+

λa X

(i,j)

l(aij, x>

i vj + y> i Muj)+

λx X

i

γ(xi|yi) + λv X

i

γ(vi|ui)+ λW W2 + λM M2 + λA A2 + λB B2

minimize social app regularizer

reconstruction

SLIDE 115

Loss Function

SLIDE 116

Loss

Much more evidence of application non-install

(i.e. many more negative examples)

Few links between vertices in friendship

network (even within short graph distance)

Generate ranking problems (link, non-link) with

non-links drawn from background set

SLIDE 117

Loss

application recommendation social recommendation

SLIDE 118

Optimization

Nonconvex optimization problem
Large set of variables
Stochastic gradient descent
n x, v, ε for speed
Use hashing to reduce

memory load, i.e.

xi = Ayi + i vj = Buj + ˜ j eij ∼ p(e|x>

i xj + y> i Wyj)

aij ∼ p(a|x>

i vj + y> i Muj)

xij = σ(i, j)X[h(i, j)]

binary hash hash

SLIDE 119

Y! Pulse

SLIDE 120

Y! Pulse Data

1.2M users, 386 items 6.1M friend connections 29M interest indications

SLIDE 121

App Recommendation

SIM: similarity based model; RLFM: regression based latent factor model (Chen&Agarwal); NLFM: SIM&RLFM

SLIDE 122

Social recommendation

SLIDE 123

app recommendation L2 penalty

SLIDE 124

Multiple relations

(user, user) (user, app) (app, advertisement)

Users visiting several properties

news, mail, frontpage, social network, etc.

Different statistical models
Latent Dirichlet Allocation for latent factors
Indian Buffet Process

Extensions

x x’ y y’ e

SLIDE 125

Multiple relations

(user, user) (user, app) (app, advertisement)

Users visiting several properties

news, mail, frontpage, social network, etc.

Different statistical models
Latent Dirichlet Allocation for latent factors
Indian Buffet Process

v u a

Extensions

x x’ y y’ e

SLIDE 126

v u a

Multiple relations

(user, user) (user, app) (app, advertisement)

Users visiting several properties

news, mail, frontpage, social network, etc.

Different statistical models
Latent Dirichlet Allocation for latent factors
Indian Buffet Process

v u a

Extensions

x x’ y y’ e

SLIDE 127

v u a

Multiple relations

(user, user) (user, app) (app, advertisement)

Users visiting several properties

news, mail, frontpage, social network, etc.

Different statistical models
Latent Dirichlet Allocation for latent factors
Indian Buffet Process

v u a

Extensions

x x’ y y’ e

SLIDE 128

More strategies

SLIDE 129

Multiple factor LDA

Discrete set of preferences

(Porteous, Bart, Welling, 2008)

User picks one to assess movie
Movie represented by a discrete attribute
Inference by Gibbs sampler
Works fairly well
Extension by Lester Mackey and coworkers to

combine with BPMF model

SLIDE 130

More state representations

Indian Buffet Process

(Griffiths & Ghahramani, 2005)

Attribute vector is binary string
Models preferences naturally & very compact

(Inference is costly)

Hierarchical attribute representation and

clustering over users ... TO DO

SLIDE 131

5 Hashing

SLIDE 132

Parameter Storage

We have millions of users
We have millions of products
Storage - for 100 factors this requires

106 x 106 x 8 = 8TB

We want a model that can be kept in RAM (<16GB)
Instant response for each user
Disks have 20 IOP/s at best (SSDs much better)
Privacy (what if parameter vector leaks)

SLIDE 133

Recall - Hash Kernels

Hey, please mention subtly during your talk that people should use Yahoo mail more often. Thanks, Someone instance: task/user (=barney):

⇥ xi ∈ RN×(U+1)

1 3 2

1

h()

h(‘mention’) h(‘mention_barney’)

s(m_b) s(m)

{-1, 1}

Similar to count hash (Charikar, Chen, Farrach-Colton, 2003)

X

i

¯ w[h(i)]σ(i)xi

SLIDE 134

Hashing compression
Approximation is O(1/n)
To show that estimate is unbiased take expectation
ver Rademacher hash.

Collaborative Filtering

ui =

j,k:h(j,k)=i

ξ(j, k)Ujk and vi =

j,k:h(j,k)=i

ξ(j, k)Vjk. Xij :=

k

ξ(k, i)ξ(k, j)uh(k,i)vh(k,j).

SLIDE 135

Hashing compression
Expectation

Collaborative Filtering

Xij :=

k

ξ(k, i)ξ(k, j)uh(k,i)vh(k,j).

Xij := X

k

ξ(k, i)ξ0(k, j) X

l,k:h(k,l)=h(k,i)

X

,k:h0(k,o)=h0(k,j)

ξ(k, l)ξ0(k, o)UklVko

ui = X

j,k:h(k,j)=i

ξ(k, j)Ukj and vi = X

j,k:h0(k,j)=i

ξ0(k, j)Vkj.

expectation vanishes expectation vanishes

SLIDE 136

Collaborative Hashing

Combine with stochastic gradient descent
Random access in memory is expensive

(we now have to do k lookups per pair)

Feistel networks can accelerate this
Distributed optimization without locking

SLIDE 137

Examples

Thousands of elements in M Thousands of elements in U

1225 840 720 520 400 240 120 60 32 16 10 9 8 7 6 5 1.20 1.22 1.24 1.26 1.28 1.30 1.32 rows in M rows in U

983 500 450 400 350 300 250 200 150 100 50 1682 500 450 400 350 300 250 200 150 100 50 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16

Eachmovie MovieLens

SLIDE 138

Summary

Neighborhood methods
User / movie similarity
Iteration on graph
Matrix Factorization
Singular value decomposition
Convex reformulation
Ranking and Session Modeling
Ordinal regression
Session models
Features
Latent dense (Bayesian Probabilistic Matrix Factorization)
Latent sparse (Dirichlet process factorization)
Coldstart problem (inferring features)
Hashing

SLIDE 139