Transcript A Support Vector Method for Optimizing Rank

Learning to Rank
(part 1)
NESCAI 2008 Tutorial
Yisong Yue
Cornell University
Booming Search Industry
Goals for this Tutorial

Basics of information retrieval

What machine learning contributes

New challenges to address

New insights on developing ML algorithms
(Soft) Prerequisites

Basic knowledge of ML algorithms






Support Vector Machines
Neural Nets
Decision Trees
Boosting
Etc…
Will introduce IR concepts as needed
Outline (Part 1)

Conventional IR Methods (no learning)


Ordinal Regression


1970s to 1990s
1994 onwards
Optimizing Rank-Based Measures

2005 to present
Outline (Part 2)

Effectively collecting training data
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
Beyond independent relevance

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E.g., interpreting clickthrough data
E.g., diversity
Summary & Discussion
Disclaimer

This talk is very ML-centric



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Use IR methods to generate features
Learn good ranking functions on feature space
Focus on optimizing cleanly formulated objectives
Outperform traditional IR methods
Disclaimer

This talk is very ML-centric





Use IR methods to generate features
Learn good ranking functions on feature space
Focus on optimizing cleanly formulated objectives
Outperform traditional IR methods
Information Retrieval



Broader than the scope of this talk
Deals with more sophisticated modeling questions
Will see more interplay between IR and ML in Part 2
Brief Overview of IR
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Predated the internet
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Active research topic by the 1960’s
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As We May Think by Vannevar Bush (1945)
Vector Space Model (1970s)
Probabilistic Models (1980s)
Introduction to Information Retrieval (2008)

C. Manning, P. Raghavan & H. Schütze
Basic Approach to IR

Given query q and set of docs d1, … dn


Find documents relevant to q
Typically expressed as a ranking on d1,… dn
Basic Approach to IR

Given query q and set of docs d1, … dn



Find documents relevant to q
Typically expressed as a ranking on d1,… dn
Similarity measure sim(a,b)!R


Sort by sim(q,di)
Optimal if relevance of documents are
independent. [Robertson, 1977]
Vector Space Model
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Represent documents as vectors

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
One dimension for each word
Queries as short documents
Similarity Measures
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Cosine similarity = normalized dot product
A B
cos(A, B) 
A B
Cosine Similarity Example
Other Methods
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TF-IDF
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Okapi BM25
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[Salton & Buckley, 1988]
[Robertson et al., 1995]
Language Models
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[Ponte & Croft, 1998]
[Zhai & Lafferty, 2001]
Machine Learning
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IR uses fixed models to define similarity scores
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Many opportunities to learn models
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Appropriate training data
Appropriate learning formulation
Will mostly use SVM formulations as examples
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General insights are applicable to other techniques.
Training Data
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Supervised learning problem
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Document/query pairs
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Embedded in high dimensional feature space
Labeled by relevance of doc to query
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Traditionally 0/1
Recently ordinal classes of relevance (0,1,2,3,…)
Feature Space
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Use to learn a similarity/compatibility function
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Based off existing IR methods
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Can use raw values
Or transformations of raw values
Based off raw words
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Capture co-occurrence of words
Training Instances
xq , d
 TF (qi , d )  0.1 
 i

 TF (qi , d )  0.05 
 i

rankIR (q, d ) in top5 


  (q, d )  rankIR (q, d ) in top10
 simIR (q, d )



 w j  qi  w j  d 
 i

 wk  qi  wk  d  
 i



Learning Problem
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Given training instances:
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Learn a ranking function
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(xq,d, yq,d) for q = {1..N}, d = {1 .. Nq}
f(xq,1, … xq,Nq ) ! Ranking
Typically decomposed into per doc scores
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f(x) ! R (doc/query compatibility)
Sort by scores for all instances of a given q
How to Train?
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Classification & Regression
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Learn f(x) ! R in conventional ways
Sort by f(x) for all docs for a query
Typically does not work well
2 Major Problems
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Labels have ordering
 Additional structure compared to multiclass problems
Severe class imbalance
 Most documents are not relevant
Somewhat Relevant
Very Relevant
Not Relevant
Conventional multiclass learning does not incorporate
ordinal structure of class labels
Somewhat Relevant
Very Relevant
Not Relevant
Conventional multiclass learning does not incorporate
ordinal structure of class labels
Ordinal Regression

Assume class labels are ordered
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True since class labels indicate level of relevance
Learn hypothesis function f(x) ! R
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Such that the ordering of f(x) agrees with label ordering
Ex: given instances (x, 1), (y, 1), (z, 2)
 f(x) < f(z)
 f(y) < f(z)
 Don’t care about f(x) vs f(y)
Ordinal Regression

Compare with classification
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Similar to multiclass prediction
But classes have ordinal structure
Compare with regression
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Doesn’t necessarily care about value of f(x)
Only care that ordering is preserved
Ordinal Regression
Approaches
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Learn multiple thresholds
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Learn multiple classifiers
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Optimize pairwise preferences
Option 1: Multiple Thresholds
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Maintain T thresholds (b1, … bT)
b1 < b 2 < … < b T
Learn model parameters + (b1, …, bT)
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Goal

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
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Model predicts a score on input example
Minimize threshold violation of predictions
Ordinal SVM Example
[Chu & Keerthi, 2005]
Ordinal SVM Formulation
1 2 C T 

 
arg min w    i , j  i , j 
N j 1  i
w,b ,  ,  2
i

Such that for j = 0..T :
wT xi  b j  1   i, j ,
i : yi  j
wT xi  b j  1   i, j 1 , i : yi  j  1
 i, j ,  i, j 1  0,
And also:
i
b1  b2  ...  bT
[Chu & Keerthi, 2005]
Learning Multiple Thresholds
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Gaussian Processes
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Decision Trees
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[Kramer et al., 2001]
Neural Nets
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[Chu & Ghahramani, 2005]
RankProp [Caruana et al., 1996]
SVMs & Perceptrons
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PRank [Crammer & Singer, 2001]
[Chu & Keerthi, 2005]
Option 2: Voting Classifiers
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Use T different training sets
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Classifier 1 predicts 0 vs 1,2,…T
Classifier 2 predicts 0,1 vs 2,3,…T
…
Classifier T predicts 0,1,…,T-1 vs T
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Final prediction is combination
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E.g., sum of predictions
Recent work
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McRank [Li et al., 2007]
[Qin et al., 2007]
•Severe class imbalance
•Near perfect performance by always predicting 0
Option 3: Pairwise Preferences
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Most popular approach for IR applications
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Learn model to minimize pairwise
disagreements
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%(Pairwise Agreements) = ROC-Area
• 2 pairwise disagreements
Optimizing Pairwise Preferences
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Consider instances (x1,y1) and (x2,y2)
Label order has y1 > y2
Optimizing Pairwise Preferences

Consider instances (x1,y1) and (x2,y2)
Label order has y1 > y2
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Create new training instance
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(x’, +1) where x’ = (x1 – x2)
Repeat for all instance pairs with label order
preference
Optimizing Pairwise Preferences
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Result: new training set!
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Often represented implicitly
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Has only positive examples

Mispredicting means that a lower ordered
instance received higher score than higher
order instance.
Pairwise SVM Formulation
1 2 C
arg min w   i , j
2
N i, j
w,
Such that:
wT xi  wT x j  1  i , j ,
i , j  0,
i, j : yi  y j
i, j
[Herbrich et al., 1999]
Can be reduced to O(n log(n)) time [Joachims, 2005].
Optimizing Pairwise Preferences
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Neural Nets
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Boosting & Hedge-Style Methods
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RankNet [Burges et al., 2005]
[Cohen et al., 1998]
RankBoost [Freund et al., 2003]
[Long & Servidio, 2007]
SVMs



[Herbrich et al., 1999]
SVM-perf [Joachims, 2005]
[Cao et al., 2006]
Rank-Based Measures
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Pairwise Preferences not quite right

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Assigns equal penalty for errors no matter where
in the ranking
People (mostly) care about top of ranking

IR community use rank-based measures which
capture this property.
Rank-Based Measures
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Binary relevance
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Precision@K (P@K)
Mean Average Precision (MAP)
Mean Reciprocal Rank (MRR)
Multiple levels of relevance
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Normalized Discounted Cumulative Gain (NDCG)
Precision@K
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Set a rank threshold K
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Compute % relevant in top K
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Ignores documents ranked lower than K
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Ex:
 Prec@3 of 2/3
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Prec@4 of 2/4
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Prec@5 of 3/5
Mean Average Precision
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Consider rank position of each relevance doc
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K1, K2, … KR
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Compute Precision@K for each K1, K2, … KR
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Average precision = average of P@K
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Ex:

MAP is Average Precision across multiple
queries/rankings
has AvgPrec of
1 1 2 3 
      0.76
3 1 3 5 
Mean Reciprocal Rank

Consider rank position, K, of first relevance doc

1
Reciprocal Rank score =
K
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MRR is the mean RR across multiple queries
NDCG
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Normalized Discounted Cumulative Gain
Multiple Levels of Relevance
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DCG:
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contribution of ith rank position:

Ex:
2 yi  1
log(i  1)
has DCG score of
1
3
1
0
1




 5.45
log(2) log(3) log(4) log(5) log(6)
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NDCG is normalized DCG
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best possible ranking as score NDCG = 1
Optimizing Rank-Based Measures
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Let’s directly optimize these measures

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As opposed to some proxy (pairwise prefs)
But…
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Objective function no longer decomposes
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Pairwise prefs decomposed into each pair
Objective function flat or discontinuous
Discontinuity Example
D1 D2 D3

Retrieval Score
0.9 0.6 0.3
Rank
1
2
3
Relevance
0
1
0
NDCG = 0.63
Discontinuity Example


NDCG computed using rank positions
Ranking via retrieval scores
D1 D2 D3
Retrieval Score
0.9 0.6 0.3
Rank
1
2
3
Discontinuity Example


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NDCG computed using rank positions
Ranking via retrieval scores
Slight changes to model parameters



Slight changes to retrieval scores
No change to ranking
No change to NDCG
D1 D2 D3
Retrieval Score
0.9 0.6 0.3
Rank
1
2
3
Discontinuity Example



NDCG computed using rank positions
Ranking via retrieval scores
Slight changes to model parameters



NDCG
discontinuous
Slight changes
to retrieval scores
No change to ranking
w.r.t
model
parameters!
No change to NDCG
D1 D2 D3
Retrieval Score
0.9 0.6 0.3
Rank
1
2
3
[Yue & Burges, 2007]
Optimizing Rank-Based Measures

Relaxed Upper Bound

Structural SVMs for hinge loss relaxation

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Boosting for exponential loss relaxation
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SVM-map [Yue et al., 2007]
[Chapelle et al., 2007]
[Zheng et al., 2007]
AdaRank [Xu et al., 2007]
Smooth Approximations for Gradient Descent


LambdaRank [Burges et al., 2006]
SoftRank GP [Snelson & Guiver, 2007]
Structural SVMs


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
Let x denote the set of documents/query examples for a query
Let y denote a (weak) ranking
Same objective function:
1 2 C
w 
2
N

i
i
Constraints are defined for each incorrect labeling y’ over the set of
documents x.
y'  y : w ( y, x)  w ( y' , x)  ( y' )  
T
T
After learning w, a prediction is made by sorting on wTxi
[Tsochantaridis et al., 2007]
Structural SVMs for MAP

Maximize
subject to
where
1 2 C
w 
2
N

i
i
y'  y : w ( y, x)  w ( y' , x)  ( y' )  
T
T
 ( y, x)    yij  ( xi  x j )
( yij = {-1, +1} )
i:rel j:!rel
and

( y)  1  Avgprec( y)
Sum of slacks

upper bound MAP loss.
[Yue et al., 2007]
Too Many Constraints!

For Average Precision, the true labeling is a ranking where the
relevant documents are all ranked in the front, e.g.,

An incorrect labeling would be any other ranking, e.g.,

This ranking has Average Precision of about 0.8 with (y,y’) ¼ 0.2

Intractable number of rankings, thus an intractable number of
constraints!
Structural SVM Training

STEP 1: Solve the SVM objective function using only the current
working set of constraints.

STEP 2: Using the model learned in STEP 1, find the most
violated constraint from the exponential set of constraints.

STEP 3: If the constraint returned in STEP 2 is more violated
than the most violated constraint the working set by some small
constant, add that constraint to the working set.

Repeat STEP 1-3 until no additional constraints are added.
Return the most recent model that was trained in STEP 1.
STEP 1-3 is guaranteed to loop for at most a polynomial number of
iterations. [Tsochantaridis et al., 2005]
Illustrative Example
Original SVM Problem


Exponential constraints
Most are dominated by a small
set of “important” constraints
Structural SVM Approach


Repeatedly finds the next most
violated constraint…
…until set of constraints is a good
approximation.
Illustrative Example
Original SVM Problem


Exponential constraints
Most are dominated by a small
set of “important” constraints
Structural SVM Approach


Repeatedly finds the next most
violated constraint…
…until set of constraints is a good
approximation.
Illustrative Example
Original SVM Problem


Exponential constraints
Most are dominated by a small
set of “important” constraints
Structural SVM Approach


Repeatedly finds the next most
violated constraint…
…until set of constraints is a good
approximation.
Illustrative Example
Original SVM Problem


Exponential constraints
Most are dominated by a small
set of “important” constraints
Structural SVM Approach


Repeatedly finds the next most
violated constraint…
…until set of constraints is a good
approximation.
Finding Most Violated Constraint

Required for structural SVM training




Depends on structure of loss function
Depends on structure of joint discriminant
Efficient algorithms exist despite intractable
number of constraints.
More than one approach


[Yue et al., 2007]
[Chapelle et al., 2007]
Gradient Descent

Objective function is discontinuous



We only need the gradient!



Difficult to define a smooth global approximation
Upper-bound relaxations (e.g., SVMs, Boosting)
sometimes too loose.
But objective is discontinuous…
… so gradient is undefined
Solution: smooth approximation of the gradient

Local approximation
LambdaRank

Assume implicit objective function C

Goal: compute dC/dsi


si = f(xi) denotes score of document xi
Given gradient on document scores

Use chain rule to compute gradient on model
parameters (of f)
[Burges et al., 2006]
[Burges, 2007]
Intuition:
•Rank-based measures emphasize top of ranking
•Higher ranked docs should have larger derivatives
(Red Arrows)
•Optimizing pairwise preferences emphasize bottom
of ranking (Black Arrows)
LambdaRank for NDCG

The pairwise derivative of pair i,j is


1

ij  NDCG (i, j )

1

exp(
s

s
)
i
j 


Total derivative of output si is
C
 i 
si
   
jDi
ij
jDi
ji
Properties of LambdaRank
 1There
exists a cost function C if
i  j
i, j :

s j si

Amounts to the Hessian of C being symmetric

If Hessian also positive semi-definite, then C is
convex.
1Subject
to additional assumptions – see [Burges et al., 2006]
Summary (Part 1)

Machine learning is a powerful tool for designing
information retrieval models

Requires clean formulation of objective

Advances




Ordinal regression
Dealing with severe class imbalances
Optimizing rank-based measures via relaxations
Gradient descent on non-smooth objective functions