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Dual Decomposition Inference for Graphical Models over Strings

A general method for inferring strings from other strings using graphical models. Try it if you haven't observed all the words of a noisy or complex language.

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Dual Decomposition Inference for Graphical Models over Strings

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  1. Dual Decomposition Inference for Graphical Models over Strings Nanyun (Violet) Peng Ryan Cotterell Jason Eisner JohnsHopkinsUniversity

  2. Attention! • Don’t care about phonology? • Listen anyway. This is a general method for inferring strings from other strings(if you have a probability model). • So if you haven’t yet observed all the words of your noisy or complex language, try it!

  3. A Phonological Exercise Tenses 1P Pres. Sg. 3P Pres. Sg. Past Tense Past Part. TALK [tɔk] [tɔks] [tɔkt] [tɔkt] THANK [θeɪŋk] [θeɪŋks] [θeɪŋkt] [θeɪŋkt] [hæk] [hæks] [hækt] HACK [hækt] [kɹæks] [kɹækt] CRACK Verbs [slæp] [slæpt] SLAP

  4. Matrix Completion: Collaborative Filtering Movies -37 29 29 19 -36 67 22 77 -24 61 12 74 -79 -41 Users -52 -39

  5. Matrix Completion: Collaborative Filtering Movies [9,-2,1] [9,-7,2] [4,3,-2] • [-6,-3,2] -37 29 29 19 [4,1,-5] -36 67 22 77 [7,-2,0] -24 61 12 74 [6,-2,3] -79 -41 [-9,1,4] Users -52 -39 [3,8,-5]

  6. Matrix Completion: Collaborative Filtering Movies [9,-2,1] [9,-7,2] [4,3,-2] • [-6,-3,2] -37 29 29 19 [4,1,-5] -36 67 22 77 [7,-2,0] -24 61 12 74 [6,-2,3] 59 -79 -41 -80 [-9,1,4] Users -52 6 46 -39 [3,8,-5] Prediction!

  7. Matrix Completion: Collaborative Filtering [1,-4,3] [-5,2,1] Dot Product -10 Gaussian Noise -11

  8. A Phonological Exercise Tenses 1P Pres. Sg. 3P Pres. Sg. Past Tense Past Part. TALK [tɔk] [tɔks] [tɔkt] [tɔkt] THANK [θeɪŋk] [θeɪŋks] [θeɪŋkt] [θeɪŋkt] [hæk] [hæks] [hækt] HACK [hækt] [kɹæks] [kɹækt] CRACK Verbs [slæp] [slæpt] SLAP

  9. A Phonological Exercise Suffixes • /Ø/ /s/ /t/ /t/ 1P Pres. Sg. 3P Pres. Sg. Past Tense Past Part. TALK [tɔk] [tɔks] [tɔkt] [tɔkt] /tɔk/ THANK [θeɪŋk] [θeɪŋks] [θeɪŋkt] [θeɪŋkt] /θeɪŋk/ [hæk] [hæks] [hækt] HACK [hækt] /hæk/ [kɹæks] [kɹækt] CRACK Stems /kɹæk/ [slæp] [slæpt] SLAP /slæp/

  10. A Phonological Exercise Suffixes • /Ø/ /s/ /t/ /t/ 1P Pres. Sg. 3P Pres. Sg. Past Tense Past Part. TALK [tɔk] [tɔks] [tɔkt] [tɔkt] /tɔk/ THANK [θeɪŋk] [θeɪŋks] [θeɪŋkt] [θeɪŋkt] /θeɪŋk/ [hæk] [hæks] [hækt] HACK [hækt] /hæk/ [kɹæks] [kɹækt] CRACK Stems /kɹæk/ [slæp] [slæpt] SLAP /slæp/

  11. A Phonological Exercise Suffixes • /Ø/ /s/ /t/ /t/ 1P Pres. Sg. 3P Pres. Sg. Past Tense Past Part. TALK [tɔk] [tɔks] [tɔkt] [tɔkt] /tɔk/ THANK [θeɪŋk] [θeɪŋks] [θeɪŋkt] [θeɪŋkt] /θeɪŋk/ [hæk] [hæks] [hækt] HACK [hækt] /hæk/ [kɹæk] [kɹæks] [kɹækt] [kɹækt] CRACK Stems /kɹæk/ [slæp] [slæps] [slæpt] [slæpt] SLAP /slæp/ Prediction!

  12. A Model of Phonology tɔk s Concatenate tɔks “talks”

  13. A Phonological Exercise Suffixes • /Ø/ /s/ /t/ /t/ 1P Pres. Sg. 3P Pres. Sg. Past Tense Past Part. TALK [tɔk] [tɔks] [tɔkt] [tɔkt] /tɔk/ THANK [θeɪŋk] [θeɪŋks] [θeɪŋkt] [θeɪŋkt] /θeɪŋk/ [hæk] [hæks] [hækt] HACK [hækt] /hæk/ [kɹæks] [kɹækt] CRACK Stems /kɹæk/ [slæp] [slæpt] SLAP /slæp/ [koʊdz] [koʊdɪt] CODE /koʊd/ [bæt] [bætɪt] BAT /bæt/

  14. A Phonological Exercise Suffixes • /Ø/ /s/ /t/ /t/ 1P Pres. Sg. 3P Pres. Sg. Past Tense Past Part. TALK [tɔk] [tɔks] [tɔkt] [tɔkt] /tɔk/ THANK [θeɪŋk] [θeɪŋks] [θeɪŋkt] [θeɪŋkt] /θeɪŋk/ [hæk] [hæks] [hækt] HACK [hækt] /hæk/ [kɹæks] [kɹækt] CRACK Stems /kɹæk/ [slæp] [slæpt] SLAP /slæp/ [koʊdz] [koʊdɪt] CODE /koʊd/ [bæt] [bætɪt] BAT /bæt/ z instead of s ɪtinstead of t

  15. A Phonological Exercise Suffixes • /Ø/ /s/ /t/ /t/ 1P Pres. Sg. 3P Pres. Sg. Past Tense Past Part. TALK [tɔk] [tɔks] [tɔkt] [tɔkt] /tɔk/ THANK [θeɪŋk] [θeɪŋks] [θeɪŋkt] [θeɪŋkt] /θeɪŋk/ [hæk] [hæks] [hækt] HACK [hækt] /hæk/ [kɹæks] [kɹækt] CRACK Stems /kɹæk/ [slæp] [slæpt] SLAP /slæp/ [koʊdz] [koʊdɪt] CODE /koʊd/ [bæt] [bætɪt] BAT /bæt/ [it] [itən] [eɪt] EAT /it/ eɪtinstead of itɪt

  16. A Model of Phonology koʊd s Concatenate koʊd#s Apply Phonology koʊdz “codes” Modeling word forms using latent underlying morphs and phonology. Cotterell et. al. TACL 2015

  17. A Model of Phonology rizaign ation Concatenate rizaign#ation Apply Phonology • rεzɪgneɪʃn “resignation”

  18. Fragment of Our Graph for English the plural suffix • rizaign z 1) Morphemes eɪʃən dæmn Concatenation 2) Underlying words dæmn#eɪʃən dæmn#z rεzɪgn#eɪʃən rizajn#z Phonology d’æmz r,εzɪgn’eɪʃn riz’ajnz d,æmn’eɪʃn 3) Surface words “resignation” • “resigns” • “damnation” • “damns”

  19. Outline • A motivating example:phonology • General framework: • graphical models over strings • Inference on graphical models over strings • Dual decomposition inference • The general idea • Substring features and active set • Experiments and results

  20. Graphical Models over Strings? • Joint distribution over many strings • Variables • Range over Σ*  infinite set of all strings • Relations among variables • Usually specified by (multi-tape) FSTs A probabilistic approach to language change (Bouchard-Côté et. al. NIPS 2008) Graphical models over multiple strings. (Dreyer and Eisner. EMNLP 2009) Large-scale cognate recovery (Hall and Klein. EMNLP 2011)

  21. Graphical Models over Strings? • Strings are the basic units in natural languages. • Use • Orthographic (spelling) • Phonological (pronunciation) • Latent (intermediate steps not observed directly) • Size • Morphemes (meaningful subword units) • Words • Multi-word phrases, including “named entities” • URLs

  22. What relationships could you model? • spelling  pronunciation • word  noisy word (e.g., with a typo) • word  related word in another language (loanwords, language evolution, cognates) • singular  plural (for example) • root  word • underlying form  surface form

  23. Chains of relations can be useful • Misspelling or pun = spelling  pronunciation  spelling • Cognate = word  historical parent  historical child

  24. Factor Graph for phonology 1) Morpheme URs rizajgn z eɪʃən dæmn 1) Morpheme rizajgn z eɪʃən dæmn Concatenation (e.g.) Concatenation (e.g.) 2) Word URs rεzɪgn#eɪʃən rizajn#z dæmn#eɪʃən dæmn#z 2) Word URs dæmn#eɪʃən dæmn#z rεzɪgn#eɪʃən rizajn#z Phonology (PFST) Phonology (PFST) r,εzɪgn’eɪʃn riz’ajnz d,æmn’eɪʃn d’æmz 3) Word SRs r,εzɪgn’eɪʃn riz’ajnz d,æmn’eɪʃn d’æmz 3) Word SRs log-probabilityLet’s maximize it!

  25. Contextual Stochastic Edit Process Stochastic contextual edit distance and probabilistic FSTs. (Cotterell et. al. ACL 2014)

  26. Inference on a Factor Graph ? ? ? ? 1) Morpheme URs ? ? ? 2) Word URs r,εzɪgn’eɪʃn riz’ajnz riz’ajnd 3) Word SRs

  27. Inference on a Factor Graph bar foo s da 1) Morpheme URs ? ? ? 2) Word URs r,εzɪgn’eɪʃn riz’ajnz riz’ajnd 3) Word SRs

  28. Inference on a Factor Graph bar foo s da 1) Morpheme URs bar#s bar#da bar#foo 2) Word URs r,εzɪgn’eɪʃn riz’ajnz riz’ajnd 3) Word SRs

  29. Inference on a Factor Graph 0.01 8e-3 0.05 0.02 bar foo s da 1) Morpheme URs bar#s bar#da bar#foo 2) Word URs r,εzɪgn’eɪʃn riz’ajnz riz’ajnd 3) Word SRs

  30. Inference on a Factor Graph 0.01 8e-3 0.05 0.02 bar foo s da 1) Morpheme URs bar#s bar#da bar#foo 2) Word URs 6e-1200 2e-1300 7e-1100 r,εzɪgn’eɪʃn riz’ajnz riz’ajnd 3) Word SRs

  31. Inference on a Factor Graph 0.01 8e-3 0.05 0.02 bar foo s da 1) Morpheme URs bar#s bar#da bar#foo 2) Word URs  6e-1200 2e-1300 7e-1100 r,εzɪgn’eɪʃn riz’ajnz riz’ajnd 3) Word SRs

  32. Inference on a Factor Graph ? far foo s da 1) Morpheme URs far#s far#da far#foo 2) Word URs r,εzɪgn’eɪʃn riz’ajnz riz’ajnd 3) Word SRs

  33. Inference on a Factor Graph ? size foo s da 1) Morpheme URs size#s size#da size#foo 2) Word URs r,εzɪgn’eɪʃn riz’ajnz riz’ajnd 3) Word SRs

  34. Inference on a Factor Graph ? … foo s da 1) Morpheme URs …#s …#da …#foo 2) Word URs r,εzɪgn’eɪʃn riz’ajnz riz’ajnd 3) Word SRs

  35. Inference on a Factor Graph • rizajn foo s da 1) Morpheme URs rizajn#s rizain#da rizajn#foo 2) Word URs r,εzɪgn’eɪʃn riz’ajnz riz’ajnd 3) Word SRs

  36. Inference on a Factor Graph • rizajn foo s da 1) Morpheme URs rizajn#s rizajn#da rizajn#foo 2) Word URs 2e-5 0.01 0.008 r,εzɪgn’eɪʃn riz’ajnz riz’ajnd 3) Word SRs

  37. Inference on a Factor Graph • rizajn • eɪʃn s d 1) Morpheme URs rizajn#s rizajn#d • rizajn#eɪʃn 2) Word URs 0.001 0.01 0.015 r,εzɪgn’eɪʃn riz’ajnz riz’ajnd 3) Word SRs

  38. Inference on a Factor Graph • rizajgn • eɪʃn s d 1) Morpheme URs • rizajgn#s • rizajgn#da • rizajgn#foo 2) Word URs  0.008 0.008 0.013 r,εzɪgn’eɪʃn riz’ajnz riz’ajnd 3) Word SRs

  39. Inference on a Factor Graph rizajgn eɪʃn s d rizajgn#s rizajgn#d rizajgn#eɪʃn  0.013 0.008 0.008 riz’ajnz riz’ajnd r,εzɪgn’eɪʃn

  40. Challenges in Inference • Global discrete optimization problem. • Variables range over a infinite set: cannot be solved by ILP or even brute force. Undecidable! • Our previous papers used approximatealgorithms: Loopy Belief Propagation, or Expectation Propagation. • Q: can we do exact inference? • A: If we can live with 1-best and not marginal inference, then we can use Dual Decomposition … which is exact. • (if it terminates! the problem is undecidable in general …)

  41. Outline • A motivating example:phonology • General framework: • graphical models over strings • Inference on graphical models over strings • Dual decomposition inference • The general idea • Substring features and active set • Experiments and results

  42. Graphical Model for Phonology 1) Morpheme URs rizajgn z eɪʃən dæmn Concatenation (e.g.) 2) Word URs dæmn#eɪʃən dæmn#z rεzɪgn#eɪʃən rizajn#z Phonology (PFST) r,εzɪgn’eɪʃn riz’ajnz d,æmn’eɪʃn d’æmz 3) Word SRs Jointly decide the value of the inter-dependent latent variables, which range over a infinite set.

  43. General Idea of Dual Decomp rizajgn • rεzign z eɪʃən eɪʃən dæmn dæmn#eɪʃən dæmn#z rεzɪgn#eɪʃən rizajn#z d’æmz r,εzɪgn’eɪʃn riz’ajnz d,æmn’eɪʃn

  44. General Idea of Dual Decomp eɪʃən rεzɪgn z dæmn eɪʃən dæmn z rizajn dæmn#eɪʃən dæmn#z rεzɪgn#eɪʃən rizajn#z d,æmn’eɪʃn d’æmz r,εzɪgn’eɪʃn riz’ajnz Subproblem 1 Subproblem 2 Subproblem 3 Subproblem 4

  45. General Idea of Dual Decomp I think it’s rεzɪgn I think it’s rizajn eɪʃən rεzɪgn z dæmn eɪʃən dæmn z rizajn dæmn#eɪʃən dæmn#z rεzɪgn#eɪʃən rizajn#z d,æmn’eɪʃn d’æmz r,εzɪgn’eɪʃn riz’ajnz Subproblem 1 Subproblem 2 Subproblem 3 Subproblem 4

  46. Outline • A motivating example:phonology • General framework: • graphical models over strings • Inference on graphical models over strings • Dual decomposition inference • The general idea • Substring features and active set • Experiments and results

  47. eɪʃən rεzɪgn z dæmn eɪʃən dæmn z rizajn dæmn#eɪʃən dæmn#z rεzɪgn#eɪʃən rizajn#z d,æmn’eɪʃn d’æmz r,εzɪgn’eɪʃn riz’ajnz Subproblem 1 Subproblem 2 Subproblem 3 Subproblem 4

  48. Substring Features and Active Set • Less i, a, j; • more ε, ɪ, g • (to match others) • I think it’s rizajn rεzɪgn rizajn eɪʃən z dæmn eɪʃən dæmn z • Lessε, ɪ, g; more i, a, j • (to match others) • I think it’s rεzɪgn dæmn#eɪʃən dæmn#z rεzɪgn#eɪʃən rizajn#z d,æmn’eɪʃn d’æmz r,εzɪgn’eɪʃn riz’ajnz Subproblem 1 Subproblem 1 Subproblem 1 Subproblem 1

  49. Features: “Active set” method • How many features? • Infinitely many possible n-grams! • Trick: Gradually increase feature set as needed. • Like Paul & Eisner (2012), Cotterell & Eisner (2015) • Only add features on which strings disagree. • Only add abcdonce abcand bcdalready agree. • Exception: Add unigrams and bigrams for free.

  50. Fragment of Our Graph for Catalan ? ? ? ? ? Stem of “grey” ? ? ? ? gris grizos grize grizes Separate these 4 words into 4 subproblems as before …

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