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Set prediction three ways

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Slides from my talk at the Cornell SCAN seminar.
September 10, 2018.

Published in: Data & Analytics
  • 80% Win Rate? It's Not a BUG? [Proof Inside] ▲▲▲ http://scamcb.com/zcodesys/pdf
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Set prediction three ways

  1. 1. 1 Joint work with Ravi Kumar & Andrew Tomkins (Google) Rediet Abebe & Jon Kleinberg (Cornell) Michael Schaub & Ali Jadbabaie (MIT) Set prediction three ways Austin R. Benson · Cornell SCAN Seminar · September 10, 2018 Slides. bit.ly/arb-SCAN18
  2. 2. We usually think about predicting single-item events. 2
  3. 3. This talk looks at predicting sets from three perspectives. 3 Set-based data is common, but we don’t have a great understanding of its complexities and the associated human behavior. • Team formation (writing papers, organizational behavior). • Multiple classification codes in hospital visits. • Co-purchasing sets on Amazon. • Sets of annotations on questions on web forums.
  4. 4. 4 Set Prediction #1. Individuals repeating interactions. Given a history of an individual’s set-based interactions, which ones repeat? Who will repeat as my coauthors on my next paper? Sequences of Sets. Benson, Kumar, & Tomkins. KDD, 2018.
  5. 5. Lots of data looks like sequences of sets. 5 EMAIL Sequence of recipient sets in my email ⟶ one sequence of sets Collection of email senders ⟶ sequences of sets.
  6. 6. Lots of data looks like sequences of sets. 6 Q&A FORUM TAGS
  7. 7. Our work provides a generative model that captures the important characteristics of sequences of sets. 7 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
  8. 8. Our work provides a generative model that captures the important characteristics of sequences of sets. 8 Applications. 1. Predicting new sets. 2. Understanding basic user behaviors. 3. Generative model ⟶ event likelihood ⟶ anomaly detection. 4. Generative model ⟶ simulation. 5. Amenable to analysis.
  9. 9. 9 What are the important characteristics of the data?
  10. 10. Most sets are not entirely novel & many are exact repeats. 10 tags-mathoverflow tags-math-sx email-Enron-core email-Eu-core contact-prim-school contact-high-school coauth-Business coauth-Geology
  11. 11. Subsets and supersets of prior sets are common. 11 tags-mathoverflow tags-math-sx email-Enron-core email-Eu-core contact-prim-school contact-high-school coauth-Business coauth-Geology
  12. 12. There is recency bias in the repeat behavior. 12 Consistent with previous results on sequences of single items. [Benson-Kumar-Tomkins 16; Anderson+ 14]
  13. 13. 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. 13 • 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.
  14. 14. 14 How do we model the next set in a sequence given the history?
  15. 15. Our Correlated Repeat Unions (CRU) model captures repeat behavior,recency bias,and correlations. 15 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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  16. 16. Our Correlated Repeat Unions (CRU) model captures repeat behavior,recency bias,and correlations. 16 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
  17. 17. We can learn model parameters with maximum likelihood estimation. 17 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.
  18. 18. The optimal correlation probability is consistent within domain but differs between domains. 18 Meanper-setlikelihood x Baseline model (flat, no structure). Similar to [Anderson+ 14] CRU model.
  19. 19. Learned weights tend to decrease monotonically, which agrees with recency bias in the data. 19 100 101 102 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.
  20. 20. Asymptotic behavior depends on the recency weight model parameters. 20 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].)
  21. 21. Recap on Set Prediction #1. Individuals repeating interactions. 21 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}… Code. bit.ly/SoS-code Data. bit.ly/SoS-data
  22. 22. 22 Set Prediction #2. Subset choice models. Given a slate of alternatives, how do people choose a subset of the alternatives? What to buy after browsing Amazon? How to construct a playlist on Spotify? A discrete choice model for subset selection. Benson, Kumar, & Tomkins. WSDM, 2018.
  23. 23. Given some slate of alternatives, how do people make choices? 23 • If choosing just one thing (buying a car, picking a restaurant, etc.), there are many good ML techniques (logistic regression, deep nets, etc.) • If choosing a subset of the alternatives (what to buy after browsing Amazon, constructing a playlist on Spotify, etc.), there are not many tools. • We develop a simple and interpretable model for subset selection.
  24. 24. Our discrete choice model for subset selection as illustrated through choosing party snacks. 24 Large set of snack options and want to choose a few. {tortilla chips, potato chips, cookies, pretzels, guacamole, celery, nut mix, hummus, meatballs, cupcakes, pigs in blankets, cupcakes, potato skins, chicken wings, taquitos, …} Model 1. Independent choices. Easy computation, but not realistic. Model 2. All subsets as options. Harder computation, but more modeling power Our model. Some“special subsets”as options + independent choices. Interpolate between computation and modeling power. healthyfoodtribe.comtoday.com
  25. 25. Our model is based on a generalization of classical discrete choice and random utility maximization theory. 25 Discrete Choice Methods with Simulation,Train,2009 (https://eml.berkeley.edu/books/choice2.html) • Observe choice set C with items 1, …, c. Choose one element. • Random utility of ith item: Ui = Vi + ei Vi is base utility, ei is i.i.d. Gumbel distributed error • If choosing the item with largest random utility… • (Logistic regression assumes for feature vector x of sample) Pr[select item i | C] = eVi P j2C e Vj <latexit 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Vi = T i x<latexit 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  26. 26. A basic subset choice choice model could assume set utility is additive in the elements. 26 • Observe choice set C with items 1, …, c. Choose two elements. (repeats allowed) • Random utility of (i, j) pair: Uij = Vi + Vj + eij Vi is base utility, eij i.i.d. Gumbel distributed error • If choosing the set with largest random utility… Pr[select set {i, j} | C] = e Vi+Vj P {k,l}⇢C eVk+Vl <latexit 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Pr[select set {i, j} | C, j] = eVi P k2C eVk <latexit 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sha1_base64="ZaJhcD7ScvxQtNfQTogu245TRXE=">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</latexit><latexit sha1_base64="ZaJhcD7ScvxQtNfQTogu245TRXE=">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</latexit><latexit sha1_base64="ZaJhcD7ScvxQtNfQTogu245TRXE=">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</latexit> Observation. Probability of selecting i, given j, is the same for all j.

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