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1
Joint work with
Ravi Kumar (Google) &
Andrew Tomkins (Google)
Sequences of sets
Austin R. Benson · Cornell
KDD · August 23, 2018 · London
Slides. bit.ly/SoS-KDD
Code. bit.ly/SoS-code
Data. bit.ly/SoS-data
Lots of data looks like sequences of sets.
2
EMAIL
Sequence of recipient sets in my email ⟶ one sequence of sets
Collection of email senders ⟶ sequences of sets.
Lots of data looks like sequences of sets.
3
Q&A
FORUM
TAGS
Lots of data looks like sequences of sets.
4
ACADEMIC COAUTHORSHIP
{Ravi Kumar, Andrew Tomkins} has appeared 4 times in my sequence
of coauthor sets !
Our work provides a generative model that captures
the important characteristics of sequences of sets.
5
1. email data
sequence for each account
sets are recipients on emails sent by account
2. Stack Exchange tags
sequence for each user
sets are tags on questions asked by the user
3. Coauthorship
sequence for each academic
sets are coauthors on paper
4. Proximity contact
sequence for each person
sets are people interacting with the person
tags-mathoverflow
tags-math-sx
email-Enron-core
email-Eu-core
contact-prim-school
contact-high-school
coauth-Business
coauth-Geology
Our work provides a generative model that captures
the important characteristics of sequences of sets.
6
Applications.
1. Predicting new sets.
2. Generative model ⟶ event likelihood ⟶ anomaly detection.
3. Understanding basic user behaviors.
4. Simulation.
7
What are the important
characteristics of the data?
Most sets are not entirely novel &
many are exact repeats.
8
tags-mathoverflow
tags-math-sx
email-Enron-core
email-Eu-core
contact-prim-school
contact-high-school
coauth-Business
coauth-Geology
Subsets and supersets of prior sets are common.
9
tags-mathoverflow
tags-math-sx
email-Enron-core
email-Eu-core
contact-prim-school
contact-high-school
coauth-Business
coauth-Geology
There is recencybias in the repeat behavior.
10
Consistent with previous results on sequences of single items.
[Benson-Kumar-Tomkins 16; Anderson+ 14]
size-2 subset counts size-3 subset counts
Dataset data null model data null model
email-Enron-core 5.82 4.34 ± 0.043 4.23 2.67 ± 0.038
email-Eu-core 4.46 3.11 ± 0.008 3.23 2.08 ± 0.007
contact-prim-school 2.36 1.87 ± 0.003 1.35 1.09 ± 0.002
contact-high-school 4.49 3.26 ± 0.007 2.09 1.35 ± 0.004
tags-mathoverflow 1.49 1.41 ± 0.002 1.18 1.15 ± 0.002
tags-math-sx 1.49 1.31 ± 0.001 1.21 1.12 ± 0.001
coauth-Business 1.50 1.30 ± 0.001 1.40 1.24 ± 0.001
coauth-Geology 1.29 1.15 ± 0.000 1.15 1.07 ± 0.000
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There is correlation in what gets repeated.
11
• For each sequence in each dataset, we count the number of
times each size-2 and size-3 subset appears.
• We then count the same statistics under a null model where
elements are randomly places into sets.
12
How do we model the next set in a
sequence given the history?
Our Correlated Repeat Unions (CRU) model captures
repeat behavior,recencybias,and correlations.
13
Setup.
Observe sequence of sets S1, …, Sk.
Given number r of repeated elements in Sk+1.
Model selects r elements from .
CRU model.
Start with , given r.
1. Sample set Sk-j from j steps back with recency weight wj.
2. Sample T by keeping each item x in Sk-j with correlation probability p.
3. .
4. Repeat steps 1—3 until .
(if T makes N too large, randomly drop elements from T)
[k
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N = ;<latexit 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N = N [ T<latexit 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|N| = r<latexit 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Our Correlated Repeat Unions (CRU) model captures
repeat behavior,recency bias,and correlations.
14
Setup (k = 3).
Observe S1, …, S3: {a, b}, {c}, {a, c, d}.
Given that S4 has 3 repeated elements.
Model selects three elements from {a, b, c, d}.
CRU model (p = 0.8; w1 = 0.6,w2 = 0.3 w3 = 0.1).
{a, b} w3 = 0.1{c} w2 = 0.3{a, c, d} w1 = 0.6
0. N = ;.<latexit 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1. N = {a, c}.<latexit 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2. N = {a, c}.<latexit 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3. N = {a, c, b}.<latexit 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sha1_base64="mSDh8p6Sw4A+jd6Q91R/Xl/FxcY=">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</latexit>
0. N = ;.<latexit 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1. N = {a, c, d}.<latexit 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0.8·0.8·0.8
0.8·0.8·0.2
0.8 + 0.2
0.2·0.8 + 0.8·0.8
We can learn model parameters with maximum
likelihood estimation.
15
1. Fix correlation probability p and learn recency weights w.
⟶ single p, vector w learned for entire dataset.
2. Grid search over p, gradient descent on w.
⟶ structure of CRU model makes it easy to compute gradients.
The optimal correlation probabilityis consistent within
domain but differs between domains.
16
Meanper-setlikelihood
x Baseline model (flat, no structure). Similar to [Anderson+ 14]
CRU model.
Learned weights tend to decrease monotonically,
which agrees with recency bias in the data.
17
100
101
102
index
10 3
10 2
10 1
Recencyweightw
contact-prim-school
100
101
102
index
10 3
10 2
Recencyweightw
email-Eu-core
100
101
102
index
10 3
10 2
Recencyweightw
coauth-Geology
100
101
102
index
10 2
Recencyweightw
tags-mathoverflow
Correlation
probability p.
Asymptotic behavior depends on the
recency weight model parameters.
18
Theorem.
Let Wj =
Pj
i=1 wi.
If W1 < 1, the model tips with probability 1.
If W1 = 1, then every pair occurs infinitely often.<latexit 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We say that the model tips if
after some point, only one set appears forever more.
(Similar flavor of result to single-item sequence models [Anderson+ 14].)
Sequences of sets are a rich type of data.
19
1. The data exhibits complex repetition patterns.
2. Correlated Repeated Unions (CRU) is a model for repeat structure.
3. Optimal correlation probabilities are consistent within domain
but different across domains.
4. Optimal weights look the same across domains—fat tails.
5. Can analyze the asymptotic behavior of the model.
{a, b, c}, {a, b}, {c, d, e, f}, {a, c}, {c}, {a, b, c}, {e, g, h}, {h}, …
{a, b}, {a, x}, {a, y}, {a}, {a}, {a}, {z}, {a, b, x, y, z}, …
{j}, {j, k, l}, {a, j}, {a}, {a, k}, {a, j, k, l}, {j, k, l}, {j, k, l}, {j, k}…
Sequences of sets.
20
Austin R. Benson
http://cs.cornell.edu/~arb
@austinbenson
arb@cs.cornell.edu
THANKS!
Slides. bit.ly/SoS-KDD
Code. bit.ly/SoS-code
Data. bit.ly/SoS-data
100
101
102
index
10 3
10 2
Recencyweightw
email-Eu-core

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Sequences of Sets KDD '18

  • 1. 1 Joint work with Ravi Kumar (Google) & Andrew Tomkins (Google) Sequences of sets Austin R. Benson · Cornell KDD · August 23, 2018 · London Slides. bit.ly/SoS-KDD Code. bit.ly/SoS-code Data. bit.ly/SoS-data
  • 2. Lots of data looks like sequences of sets. 2 EMAIL Sequence of recipient sets in my email ⟶ one sequence of sets Collection of email senders ⟶ sequences of sets.
  • 3. Lots of data looks like sequences of sets. 3 Q&A FORUM TAGS
  • 4. Lots of data looks like sequences of sets. 4 ACADEMIC COAUTHORSHIP {Ravi Kumar, Andrew Tomkins} has appeared 4 times in my sequence of coauthor sets !
  • 5. Our work provides a generative model that captures the important characteristics of sequences of sets. 5 1. email data sequence for each account sets are recipients on emails sent by account 2. Stack Exchange tags sequence for each user sets are tags on questions asked by the user 3. Coauthorship sequence for each academic sets are coauthors on paper 4. Proximity contact sequence for each person sets are people interacting with the person tags-mathoverflow tags-math-sx email-Enron-core email-Eu-core contact-prim-school contact-high-school coauth-Business coauth-Geology
  • 6. Our work provides a generative model that captures the important characteristics of sequences of sets. 6 Applications. 1. Predicting new sets. 2. Generative model ⟶ event likelihood ⟶ anomaly detection. 3. Understanding basic user behaviors. 4. Simulation.
  • 7. 7 What are the important characteristics of the data?
  • 8. Most sets are not entirely novel & many are exact repeats. 8 tags-mathoverflow tags-math-sx email-Enron-core email-Eu-core contact-prim-school contact-high-school coauth-Business coauth-Geology
  • 9. Subsets and supersets of prior sets are common. 9 tags-mathoverflow tags-math-sx email-Enron-core email-Eu-core contact-prim-school contact-high-school coauth-Business coauth-Geology
  • 10. There is recencybias in the repeat behavior. 10 Consistent with previous results on sequences of single items. [Benson-Kumar-Tomkins 16; Anderson+ 14]
  • 11. size-2 subset counts size-3 subset counts Dataset data null model data null model email-Enron-core 5.82 4.34 ± 0.043 4.23 2.67 ± 0.038 email-Eu-core 4.46 3.11 ± 0.008 3.23 2.08 ± 0.007 contact-prim-school 2.36 1.87 ± 0.003 1.35 1.09 ± 0.002 contact-high-school 4.49 3.26 ± 0.007 2.09 1.35 ± 0.004 tags-mathoverflow 1.49 1.41 ± 0.002 1.18 1.15 ± 0.002 tags-math-sx 1.49 1.31 ± 0.001 1.21 1.12 ± 0.001 coauth-Business 1.50 1.30 ± 0.001 1.40 1.24 ± 0.001 coauth-Geology 1.29 1.15 ± 0.000 1.15 1.07 ± 0.000 <latexit 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There is correlation in what gets repeated. 11 • For each sequence in each dataset, we count the number of times each size-2 and size-3 subset appears. • We then count the same statistics under a null model where elements are randomly places into sets.
  • 12. 12 How do we model the next set in a sequence given the history?
  • 13. Our Correlated Repeat Unions (CRU) model captures repeat behavior,recencybias,and correlations. 13 Setup. Observe sequence of sets S1, …, Sk. Given number r of repeated elements in Sk+1. Model selects r elements from . CRU model. Start with , given r. 1. Sample set Sk-j from j steps back with recency weight wj. 2. Sample T by keeping each item x in Sk-j with correlation probability p. 3. . 4. Repeat steps 1—3 until . (if T makes N too large, randomly drop elements from T) [k j=1Sj<latexit 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N = ;<latexit 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N = N [ T<latexit 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|N| = r<latexit 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  • 14. Our Correlated Repeat Unions (CRU) model captures repeat behavior,recency bias,and correlations. 14 Setup (k = 3). Observe S1, …, S3: {a, b}, {c}, {a, c, d}. Given that S4 has 3 repeated elements. Model selects three elements from {a, b, c, d}. CRU model (p = 0.8; w1 = 0.6,w2 = 0.3 w3 = 0.1). {a, b} w3 = 0.1{c} w2 = 0.3{a, c, d} w1 = 0.6 0. N = ;.<latexit 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1. N = {a, c}.<latexit 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2. N = {a, c}.<latexit 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3. N = {a, c, b}.<latexit 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0. N = ;.<latexit 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1. N = {a, c, d}.<latexit 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0.8·0.8·0.8 0.8·0.8·0.2 0.8 + 0.2 0.2·0.8 + 0.8·0.8
  • 15. We can learn model parameters with maximum likelihood estimation. 15 1. Fix correlation probability p and learn recency weights w. ⟶ single p, vector w learned for entire dataset. 2. Grid search over p, gradient descent on w. ⟶ structure of CRU model makes it easy to compute gradients.
  • 16. The optimal correlation probabilityis consistent within domain but differs between domains. 16 Meanper-setlikelihood x Baseline model (flat, no structure). Similar to [Anderson+ 14] CRU model.
  • 17. Learned weights tend to decrease monotonically, which agrees with recency bias in the data. 17 100 101 102 index 10 3 10 2 10 1 Recencyweightw contact-prim-school 100 101 102 index 10 3 10 2 Recencyweightw email-Eu-core 100 101 102 index 10 3 10 2 Recencyweightw coauth-Geology 100 101 102 index 10 2 Recencyweightw tags-mathoverflow Correlation probability p.
  • 18. Asymptotic behavior depends on the recency weight model parameters. 18 Theorem. Let Wj = Pj i=1 wi. If W1 < 1, the model tips with probability 1. If W1 = 1, then every pair occurs infinitely often.<latexit 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We say that the model tips if after some point, only one set appears forever more. (Similar flavor of result to single-item sequence models [Anderson+ 14].)
  • 19. Sequences of sets are a rich type of data. 19 1. The data exhibits complex repetition patterns. 2. Correlated Repeated Unions (CRU) is a model for repeat structure. 3. Optimal correlation probabilities are consistent within domain but different across domains. 4. Optimal weights look the same across domains—fat tails. 5. Can analyze the asymptotic behavior of the model. {a, b, c}, {a, b}, {c, d, e, f}, {a, c}, {c}, {a, b, c}, {e, g, h}, {h}, … {a, b}, {a, x}, {a, y}, {a}, {a}, {a}, {z}, {a, b, x, y, z}, … {j}, {j, k, l}, {a, j}, {a}, {a, k}, {a, j, k, l}, {j, k, l}, {j, k, l}, {j, k}…
  • 20. Sequences of sets. 20 Austin R. Benson http://cs.cornell.edu/~arb @austinbenson arb@cs.cornell.edu THANKS! Slides. bit.ly/SoS-KDD Code. bit.ly/SoS-code Data. bit.ly/SoS-data 100 101 102 index 10 3 10 2 Recencyweightw email-Eu-core