Learning Similarity Measures Based on Random Walks in Graphs

Learning Similarity Measures Based on Random Walks in

Graphs William W. Cohen

Machine Learning Department and Language Technologies InstituteSchool of Computer ScienceCarnegie Mellon University

joint work with: Ni Lao, CMUGoogle

Tom Mitchell, CMUEinat Minkov, Univ Haifa

Amarnag Subramanya, Fernando Pereira, Google

Motivation: The simple and the complex

• In computer science there is a tension between– The elegant, simple and general– The messy, complex and problem-specific

• Graphs are:– Simple: so they are easy to analyze and store– General: so

• They appear in many contexts• They are often a natural representation of

important aspects of information– Well-understood


• The real world is complex…• … learning is a way to incorporate that complexity in our

models without sacrificing elegance and generality


• This talk: Learning Similarity Measures Based on Random Walks in Graphs – Many fundamental tasks in computer science map

an input to an output– … i.e., the task can be modeled as a relation between

input and output– …and further the relation can often be viewed as a

similarity relation: the desired outputs are similar to the input (query)

– we want to learn this relationship• even if (especially if) it is complex• even if it is described by a multi-step process

– Here: one line of work on learning complex relationships


• This talk:– One line of work on learning complex relationships

– Not covered here:• Minkov et al 2006, 2008, 2011: Similar framework

for personalized information management queries and NLP relationships (e.g., synonyms) using generative and reranking-based learning strategies

• Backstrom and Leskovec, 2011: Alternative, very expressive parameterization of learning complex similarity metrics in graphs with feature vectors on the edges.

Similarity Queries on Graphs

1) Given type t* and node x in G, find y:T(y)=t* and y~x.2) Given type t* and node set X, find y:T(y)=t* and y~X.

• Nearest-neighbor classification:– G contains feature nodes and instance nodes– A link (x,f) means feature f is true for instance x– x* is a query instance, y~x* means y likely of same class as x*

• Information retrieval: – G contains word nodes and document nodes– A link (w,d) means word w is in document d– X is a set of keywords, y~X means y likely to be relevant to X

• Database retrieval:– G encodes a database– … ?

BANKS: Browsing and Keyword Search

• Database is modeled as a graph– Nodes = tuples– Edges = references between tuples

• edges are directed and indicate foreign key, inclusion dependencies, ..

[Aditya et al, VLDB 2002]

MultiQuery Optimization

S. Sudarshan

Prasan Roy

writes writes

author author

paper

Query: {“sudarshan”, “roy”} Answer: subtree from graph

MultiQuery Optimization

S. Sudarshan

Prasan Roy

writes writes

author author

paper

Query: “sudarshan”, “roy” Answer: subtree from graph

y: paper(y) & ~“sudarshan” w: paper(y) & w~“roy”AND


1) Given type t* and node x in G, find y:T(y)=t* and y~x.2) Given type t* and node set X, find y:T(y)=t* and y~X.

• Nearest-neighbor classification• Information retrieval• Database retrieval• Evaluation: specific families of tasks for scientific publications:

– Citation recommendation for a paper: (given title, year, …, of paper p, what papers should be cited by p?)

– Expert-finding: (given keywords, genes, … suggest a possible author)– “Entity recommendation”: (given title, author, year, … predict entities

mentioned in a paper, e.g. gene-protein entities) – can improve NER– Literature recommendation: given researcher and year, suggest papers

to read that year

• Evaluation: Inference in a DB of automatically-extracted facts

Core tasks in CS


• Evaluation: specific families of tasks for scientific publications:– Citation recommendation for a paper: (given title, year, …, of paper p,

what papers should be cited by p?)– Expert-finding: (given keywords, genes, … suggest a possible author)– “Entity recommendation”: (given title, author, year, … predict entities

mentioned in a paper, e.g. gene-protein entities) – Literature recommendation: given researcher and year, suggest papers

to read that year

• Evaluation: Inference in a DB of automatically-extracted facts

For each task:

query 1, ans 1query 2, ans 2

….LEARNER

Sim(s,p) = mapping from query ans

variant of PPRmay use PPR

Outline

• Motivation for Learning Similarity in Graphs

• A Baseline Similarity Metric

• Some Literature-related Tasks• The Path Ranking Algorithm (Learning Method)

– Motivation– Details

• Results: BioLiterature tasks

• Results: KB Inference tasks

Defining Similarity on Graphs: PPR/RWR

Given type t* and node x, find y:T(y)=t* and y~x.

• Similarity defined by “damped” version of PageRank• Similarity between nodes x and y:

– “Random surfer model”: from a node z,• with probability α, teleport back to x (“reset”)• Else pick a y uniformly from { y’ : z y’ }• repeat from node y ....

– Similarity x~y = Pr( surfer is at y | restart is always to x )

• Intuitively, x~y is sum of weight of all paths from x to y, where weight of path decreases exponentially with length (and fanout)

• Can easily extend to a “query” set X={x1,…,xk}• Disadvantages: …?

[Personalized PageRank 1999]

Some BioLiterature Retrieval Tasks

• Data used in this study– Yeast: 0.2M nodes, 5.5M links– Fly: 0.8M nodes, 3.5M links– E.g. the fly graph

Publication126,813

Author233,229

Write679,903 Gene

516,416Protein414,824

689,812

Cite 1,267,531

Bioentity5,823,376

1,785,626

Physical/Geneticinteractions1,352,820

Downstream/Uptream

Year58

Journal1,801

Transcribe293,285

before

Title Terms102,223

2,060,275

Learning Proximity Measures for BioLiterature Retrieval Tasks

• Tasks:– Gene recommendation: author, yeargene– Reference recommendation: words,yearpaper– Expert-finding: words, genesauthor– Literature-recommendation: author, [papers read in past]

• Baseline method:– Typed RWR proximity methods

• Baseline learning method:– parameterize Prob(walk edge|edge label=L) and tune the

parameters for each label L (somehow…)

Publication126,813

Author233,229

Write679,903 Gene

516,416Protein414,824

689,812

Cite 1,267,531

Bioentity5,823,376

1,785,626

Physical/Geneticinteractions1,352,820

Downstream/Uptream

Year58

Journal1,801

Transcribe293,285

before

Title Terms102,223

2,060,275

P(write)=b

P(L=cite) = a

P(NE) = c

P(bindTo) = dP(express) = d

Path-based vs Edge-label based learning

• RWR is a very robust and useful similarity metric

• Learning one-parameter-per-edge label is very limited

• In many cases, there aren’t enough parameters to express a complex relationship

Path-based vs Edge-label based learning

• Learning one-parameter-per-edge label is limited because the context in which an edge label appears is ignored– E.g. (observed from real data – task, find papers to read)

• Instead, we will learn path-specific parameters

Path Comments

Don't read about genes I’ve already read about

Do read papers from my favorite authors

• Paths will be interpreted as constrained random walks that give a similarity-like weight to every reachable node• Step 0: D0 = {a} Start at author a• Step 1: D1: Uniform over all papers p read by a• Step 2: D2: Author a’ of papers in D1 weighted by number of papers

in D1 published by a’• Step 3: D3 Papers p’ written by a’ weighted by ....• …

author –[read] paper –[contain]gene-[contain-1]paper

author –[read] paper –[write-1]author-[write]paper

A Limitation of RWR Learning Methods• Learning one-parameter-per-edge label is limited because the context

in which an edge label appears is ignored– E.g. (observed from real data – task, find papers to read)

• Instead, we will learn path-specific parameters

Path Comments

Don't read about genes I’ve already read about

Do read papers from my favorite authors

Path Comments

Do read about the genes I’m working on

Don't read papers from my own lab

author –[read] paper –[contain]gene-[contain-1]paper

author –[read] paper –[write-1]author-[write]paper

author –[write] paper –[contain]gene-[contain-1]paper

author –[write] paper –[publish-1]institute-[publlish]paper

21

Definitions• An graph G=(T,R,X,E), is

– a set of entity types T={T} and a set of relations R={R}– a set of entities (nodes) X={x}, where each node x has a type from T– a set of edges e=(x,y), where each edge has a relation label from R

• A path P=(R1, …,Rn) is a sequence of relations

• Path Constrained Random Walk– Given a query set S of “source” nodes– Distribution D0 at time 0 is uniform over s in S– Distribution Dt at time t>0 is formed by

• Pick x from Dt-1• Pick y uniformly from all things related to x

– by an edge labeled Rt

– Notation: fP(s,t) = Prob(st; P)

– In our examples type of t will be determined by Rn

Paper

Paper

Author

Paper

Paper

Paper

Author

Paper

WrittenBy

Write

Cite

Cite

CiteBy

CiteBy

WrittenBy

x –[AthletePlaysForTeam]y –[TeamPlaysInLeague]z

Path Ranking Algorithm (PRA)

• A PRA model scores a source-target node pair by a linear function of their path features

where P is the set of all relation paths with length ≤ L (with support on data, in some cases – see [Lao and Cohen EMNLP 2011])

• For a relation R and a set of node pairs {(si, ti)}, we construct a training dataset D ={(xi, yi)}, where xi is a vector of all the path features for (si, ti), and yi indicates whether R(si, ti) is true or not

• θ is estimated using L1,L2-regularized logistic regression

• We’ve gone from a small parameter space to a huge one

( , ) ( , )P PP

score s t f s t

P

[Lao & Cohen, ECML 2010]

( , ) Prob( ; )Pf s t s t P

25

Parameter Estimation (Details)

• Given a set of training data – D={(q(m), A(m), y(m))} m=1…M, y(m)(e)=1/0

• We can define a regularized objective function

• Use average log-likelihood as the objective om(θ)

– P(m) the index set or relevant entities, – N(m) the index set of irrelevant entities

(how to choose them will be discussed later)

1 1 2 21..

( ) ( ) | | | | / 2mm M

O o

1 ( ) 1 ( )( ) | | ln | | ln(1 )m m

m mm m i m i

i P i N

o P p N p

( )

( ) ( ) ( )( )

exp( )( 1| ; )

1 exp( )

T mm m m ii i T m

i

Ap p y q

A

Parameter Estimation (Details)

• Selecting the negative entity set Nm

– Few positive entities vs. thousands (or millions) of negative entities?– First sort all the negative entities with an uniform-weight RWR model– Then take negative entities at the k(k+1)/2-th position, for k=1,2,….

• The gradient is simple

• Use orthant-wise L-BFGS (Andrew & Gao, 2007) to estimate θ– Efficient, Can deal with L1 regularization

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.0000001 0.00001 0.001 0.1λ2 (λ1=0)

Neg

ativ

e L

og-li

kelih

ood

l=2l=3l=4

L2 Regularization

• Improves retrieval quality– On the citation recommendation task

0.20

0.25

0.30

0.35

0.40

0.45

1E-07 0.00001 0.001 0.1λ2 (λ1=0)

MA

P

l=2l=3l=4

L1 Regularization

• Does not improve retrieval quality…

1.10

1.20

1.30

1.40

1E-05 0.0001 0.001 0.01 0.1λ1 (λ2=0.00001)

Ne

ga

tive

Lo

g-l

ike

liho

od

l=2l=3l=4

0.0

0.1

0.2

0.3

0.4

0.5

1E-05 0.0001 0.001 0.01 0.1λ1 (λ2=0.00001)

MA

Pl=2l=3l=4

L1 Regularization

• … but can help reduce number of features

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1E-05 0.0001 0.001 0.01 0.1λ1 (λ2=0.00001)

MR

R

l=2l=3l=4

1

10

100

1000

1E-05 0.0001 0.001 0.01 0.1λ1 (λ2=0.00001)

No

. Act

ive

Fe

atu

res

l=2l=3l=4

Another potential “regularization: approximate RWR

31

Experiment Setup for BioLiterature

• Data sources for bio-informatics– PubMed on-line archive of over 18 million biological abstracts– PubMed Central (PMC) full-text copies of over 1 million of these papers– Saccharomyces Genome Database (SGD) a database for yeast– Flymine a database for fruit flies

• Tasks– Gene recommendation: author, yeargene– Venue recommendation: genes, title wordsjournal– Citation recommendation: title words,yearpaper– Expert-finding: title words, genesauthor

• Data split– 2000 training, 2000 tuning, 2000 test

• Time variant graph – each edge is tagged with a time stamp (year)– only consider edges that are earlier than the query, during random walk

BioLiterature: Some Results

• Compare the mean average precision (MAP) of PRA to– RWR model– RWR trained with one-parameter per link

Except these† , all improvements are statistically significant at p<0.05 using paired t-test

Example Path Features and their Weights

• A PRA+qip+pop model trained for the citation recommendation task on the yeast data

6) approx. standard IR retrieval

1) papers co-cited with on-topic papers

7,8) papers cited during the past two years

9) well cited papers

12,13) papers published during the past two years

10,11) key early papers about specific genes

14) old papers

34

Extension 1: Query Independent Paths

• PageRank (and other query-independent rankings):– assign an importance score (query independent) to each web page– later combined with relevance score (query dependent)

• We generalize pagerank to heterogeneous graphs:– We include to each query a special entity e0 of special type T0 – T0 is related to all other entity types, and each type is related to all

instances of that type– This defines a set of PageRank-like query independent relation paths– Compute f(*t;P) offline for efficiency

• Example

Paper

Paper

AuthorT0

AuthorPaper

Paper

Wrote

WrittenBy

CiteBy

Citewell cited papers

productive authors

all papers

all authors

Extension 2: Entity-specific rankings

• There are entity-specific characteristics which cannot be captured by a general model– Some items are interesting to the users because of features not

captured in the data– To model this, assume the identity of the entity matters

– Introduce new features f(st; Ps,t) to account for jumping from s to t and new features f(*t; P*,t)

– At each gradient step, add a few new features of this sort with highest gradient, count on regularization to avoid overfitting

BioLiterature: Some Results

• Compare the MAP of PRA to– RWR model– query independent paths (qip) – popular entity biases (pop)

Except these† , all improvements are statistically significant at p<0.05 using paired t-test

Example Path Features and their Weights

• A PRA+qip+pop model trained for the citation recommendation task on the yeast data

6) approx. standard IR retrieval

1) papers co-cited with on-topic papers

7,8) papers cited during the past two years

9) well cited papers

12,13) papers published during the past two years

10,11) key early papers about specific genes

14) old papers

Outline

• Random Walk With Reset/Personalized PageRank– What is it?

• Similarity Queries• Learning How to “Tune” Similarity Functions for

An Application/Subdomains• Applications and Results

– BioLiterature– Knowledge Base Inference

Outline

• Motivation for Learning Similarity in Graphs

• A Baseline Similarity Metric

• Some Literature-related Tasks• The Path Ranking Algorithm (Learning Method)


• Results: BioLiterature tasks

• Results: KB Inference tasks [Lao, Mitchell, Cohen, EMNLP 2011]

Large Scale Knowledge-Bases

• Large-Scale Collections of Automatically Extracted Knowledge

– KnowItAll (Univ. Washington)

• 0.5B facts extracted from 0.1B web pages

– DBpedia (Univ. Leipzig)

• 3.5M entities 0.7B facts extracted from wikipedia

– YAGO (Max-Planck-Institute)

• 2M entities 20M facts extracted from Wikipedia and wordNet

– FreeBase

• 20M entities 0.3B links, integrated from different data sources and human judgments

– NELL (Never-Ending Language Learning, CMU)

• 0.85M facts extracted from 0.5B webpages

Inference in Noisy Knowledge Bases

• Challenges– Robustness: extracted knowledge is incomplete and noisy– Scalability: the size of knowledge base is large

American

IsA

PlaysIn

AthletePlaysInLeagueHinesWard

SteelersAthletePlaysForTeam

NFL

TeamPlaysInLeague

?

isa-1

The NELL Case Study

• Never-Ending Language Learning: “a never-ending learning system that operates 24 hours per day, for years, to continuously improve its ability to read (extract structured facts from) the web” (Carlson et al., 2010)

• Closed domain, semi-supervised extraction• Combines multiple strategies: morphological patterns,

textual context, html patterns, logical inference

• Example beliefs

A Link Prediction Task

• We consider 48 relations for which NELL database has more than 100 instances

• We create two link prediction tasks for each relation– AthletePlaysInLeague(HinesWard,?)– AthletePlaysInLeague(?, NFL)

• The actual nodes y known to satisfy R(x; ?) are treated as labeled positive examples, and all other nodes are treated as negative examples

Current NELL method (baseline)

• FOIL (Quinlan and Cameron-Jones, 1993) is a learning algorithm similar to decision trees, but in relational domains

• NELL implements two assumptions for efficient learning– The predicates are functional --e.g. an athlete plays in at

most one league– Only find clauses that correspond to bounded-length paths

of binary relations -- relational pathfinding (Richards & Mooney, 1992)

• FOL not great for handling uncertainty– FOIL can only combine rules with disjunctions, therefore cannot

leverage low accuracy rules– E.g. rules for teamPlaysSports

High

accuracy

but low

recall

Current NELL method (baseline)

Experiments - Cross Validation on KB data(for parameter setting, etc)

RWR: Random Walk with Restart (PPR)

†Paired t-test give p-values 7x10-3, 9x10-4, 9x10-8, 4x10-4

†

†

†

†

Example Paths

Synonyms of the query

team

Evaluation by Mechanical Turk

• There are many test queries per predicate– All entities of a predicate’s domain/range, e.g.

• WorksFor(person, organization)– On average 7,000 test queries for each functional predicate, and

13,000 for each non-functional predicate

• Sampled evaluation– We only evaluate the top ranked result for each query– We sort the queries for each predicate according to the scores

of their top ranked results, and then evaluate precisions at top 10, 100 and 1000 queries

• Each belief is voted by 5 workers– Workers are given assertions like “Hines Ward plays for the

team Steelers”, as well as Google search links for each entity

Evaluation by Mechanical Turk• On 8 functional predicates where N-FOIL can successfully learn

– PRA is comparable to N-FOIL for p@10, but has significantly better p@100

• On 8 randomly sampled non-functional (one-many) predicates – Slightly lower accuracy than functional predicates

Task #Rules

N-FOILp@10 p@10

0#Path

s

PRAp@10 p@10

0

Functional Predicates 2.1(+37) 0.76 0.380 43 0.79 0.668Non-functional

Predicates ---- ---- ---- 92 0.65 0.620

PRA: Path Ranking Algorithm

Beyond Pure KB Inference

• Following Minkov et al, 2008:– Learn paths in a graph composed of multiple

dependency trees—to find synonyms, etc.

Learning Lexico-Syntactic Patterns

• Following Minkov et al, 2008:– Learn paths in a graph composed of text and

knowledge [Lao et al, EMNLP 2011]

Beyond Pure KB Inference• Following Minkov et al, 2008:

– Learn paths in a graph composed of text and knowledge [Lao et al, EMNLP 2011]



Outline

• Motivation for Learning Similarity in Graphs• A Baseline Similarity Metric• Some Literature-related Tasks• The Path Ranking Algorithm (Learning Method)


• Results: BioLiterature tasks• Results: KB Inference tasks • Conclusions

Summary/Conclusion

• Learning is the way to make a clean, elegant formulation of a task work in the messy, complicated real world

• Learning how to navigate graphs is a significant, core task that models– Recommendation, expert-finding, …– Information retrieval– Inference in KBs– …

• It includes significant, core learning problems– Regularization/search of huge feature space– Discovery: long paths, lexicalized paths, …– Incorporating knowledge of graph structure …– ….

Looking Forward

• PRA learns very restricted “inference rules”

desiredResult(Query,Result)

p1(Query,X1), p2(X1,X2), … pk(Xk-1,Result)

• Can you generalize from these to a larger set of inference rules?• Can you generalize from binary to n-ary relationships?• Can you jointly learn several relationships at once?

• PRA learns to navigate “real” graphs– What about graphs that are built on-the-fly?

• E.g., Graphs that summarize a program’s execution, or a theorem-prover’s behavior?

• Future work?

58

• Thanks to:– My co-authors on this work– All of you for being here

– NSF grant IIS-0811562– NIH grant R01GM081293– Gifts from Google

– CIKM Organizers!

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Learning Similarity Measures Based on Random Walks in Graphs