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Sequences of Sets KDD '18

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Slides from my talk at KDD 2018 in London.

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Sequences of Sets KDD '18

  1. 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. 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. 3. Lots of data looks like sequences of sets. 3 Q&A FORUM TAGS
  4. 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. 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. 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. 7 What are the important characteristics of the data?
  8. 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. 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. 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. 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. 12 How do we model the next set in a sequence given the history?
  13. 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. 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. 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. 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. 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. 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. 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. 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

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