Slides of my paper presentation at RecSys 2015. The full paper can be found (for free) at http://dl.acm.org/citation.cfm?id=2800185

1. Gaussian
Ranking
by
Matrix
Factoriza5on
Harald
Steck
hsteck@netflix.com
RecSys
2015

2. Overview
• Matrix
Factoriza<on
Model
• asymmetric
MF
• Objec<ve:
op<mize
various
Ranking
Metrics
•
exploit
proper<es
of
MF
model
&
implicit
data
• Training:
pointwise
&
listwise
• Related
Work
• Experiments

3.
Basic
Idea:
data
.
items
i
users
u
≈
users
u
Low-­‐rank
Matrix
Factoriza<on
Model

4. Basic
Idea:
-­‐
latent
user
vector:
-­‐
by
[Paterek
07],
extended
to
SVD++
[Koren
08]
Asymmetric
Matrix
Factoriza<on

5. Overview
• Matrix
Factoriza<on
Model
• asymmetric
MF
• Objec5ve:
op5mize
various
Ranking
Metrics
•
exploit
proper5es
of
MF
model
&
implicit
data
• Training:
pointwise
&
listwise
• Related
Work
• Experiments

6. AMF
as
Neural
Network
rank
loss
=
f
(ranks)
items
i
…
click
history
…
user
vec.
…
scores
…
ranks

7. AMF
as
Neural
Network
rank
loss
=
f
(ranks)
items
i
…
click
history
…
user
vec.
…
scores
…
ranks

8. 1st
term:
Rank
Loss
example
1:
AUC
• pairwise
comparisons
!
(linear)
sum
of
ranks

9. example
2:
nDCG
(for
binary
relevance)
• emphasizes
top
of
ranked
list
• also
a
func<on
of
the
ranks
of
the
posi<ves
1st
term:
Rank
Loss

10. 2nd
term:
Ac<va<on
Func<on
T
Scores
!
Ranks:
+
+
+
-­‐
binary
data:
nega<ves
and
posi<ves
-­‐
sparse
data:
many
few
!
MF
scores:
Gaussian
distrib.
assumed
scores
i

11.
score
rank
1
N
Scores
!
Ranks:
2nd
term:
Ac<va<on
Func<on
score

12.
score
…
piecewise
quadra<c
2nd
term:
Ac<va<on
Func<on

13. 3rd
term
• score:
• deriva<ve:

14. Pueng
it
All
Together
training
objec<ve
func<on:
rank
prior
on
param’s
scores
of
loss
"
lambda
nega<ves
"gamma
-­‐
minimized
by
stochas<c
gradient
descent

15. Overview
• Matrix
Factoriza<on
Model
• asymmetric
MF
• Objec<ve:
op<mize
various
Ranking
Metrics
•
exploit
proper<es
of
MF
model
&
data
• Training:
pointwise
&
listwise
• Related
Work
• Experiments

16. Listwise
Approach
• consider
ALL
items
for
each
user:
-­‐ es<mate
standard
devia<on
of
scores
for
each
user
!
width
of
ac<va<on
func<on

17. Listwise
Approach
• consider
ALL
items
for
each
user:
-­‐
sort
by
scores
!
exact
ranks
-­‐
using
logis<c
ac<va<on
func<on:
2nd
term
in
chain
rule

18. AUC
nDCG
Listwise
Approach
deriva5ves
L’:
1st
&
2nd
terms
top
of
ranked
list

19. !
between
nDCG
and
AUC:
L’
=
constant
!
use
very
large
std.
for
ac<va<on
func<on
in
pointwise
approach
AUC
nDCG
Pointwise
Approach
deriva5ves
L’:
top
of
ranked
list

20. Overview
• Matrix
Factoriza<on
Model
• asymmetric
MF
• Objec<ve:
op<mize
various
Ranking
Metrics
•
exploit
proper<es
of
MF
model
&
data
• Training
• Related
Work
• Experiments

21. Related
Work
• various
learning-­‐to-­‐rank
approaches
exist
• ogen
tailored
to
specific
ranking
losses
• mostly
pairwise
approaches,
eg:
• AUC:
BPR
[Rendle
et
al.
’09]
• MRR:
CLiMF
[Shi
et
al.
’12]
used
as
• MAP:
TFMAP
[Shi
et
al.
‘12]
baselines
• listwise
approaches,
eg:
•
top-­‐1
[Shi
et
al.
’10]
...
like
neural
network
• …
addi<onal
references
in
the
paper

22. Overview
• Matrix
Factoriza<on
Model
• basic
MF
!
asymmetric
MF
!
Neural
Network
• Objec<ve:
op<mize
various
Ranking
Metrics
•
exploit
proper<es
of
MF
model
&
data
• Training
• Related
Work
• Experiments

23. 10
m
MovieLens
Data
• 10k
movies
&
70k
users
• 1%
dense
data
• binarized:
3+
star
ra<ng
!
1,
otherwise
0
• 5-­‐fold
cross-­‐valida<on

24. 10
m
MovieLens
Data
5-­‐fold
cross-­‐valida<on
std
:
0.001

25. 10
m
MovieLens
Data
std=0.002

26. Nellix
Play
Data
• Test
day:
4/9/2014
• rela(ve
improvement
to
RMSE
training
std=1%

27. Nellix
Play
Data
std=2%

28. Conclusions
• learning-­‐to-­‐rank
approach:
– implicit
feedback
data
– proper<es
of
MF
model
! Gaussian
distribu<on
of
scores
! non-­‐linear
ac<va<on
func<ons
derived
for
ranking
• pointwise
and
listwise
training
• various
ranking
metrics
can
be
used:
– compe<<ve
for
op<mizing
AUC
– par<cularly
effec<ve
at
head
of
ranked
list

29. Thank
You
!
Ques5ons
?