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Mining at scale with latent factor models for matrix completion

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PhD Thesis

F. Petroni:
"Mining at scale with latent factor models for matrix completion."
Sapienza University of Rome, 2016.

Abstract: "Predicting which relationships are likely to occur between real-world objects is a key task for several applications. For instance, recommender systems aim at predicting the existence of unknown relationships between users and items, and exploit this information to provide personalized suggestions for items to be of use to a specific user. Matrix completion techniques aim at solving this task, identifying and leveraging the latent factors that triggered the the creation of known relationships to infer missing ones.
This problem, however, is made challenging by the size of today’s datasets. One way to handle such large-scale data, in a reasonable amount of time, is to distribute the matrix completion procedure over a cluster of commodity machines. However, current approaches lack of efficiency and scalability, since, for instance, they do not minimize the communication or ensure a balance workload in the cluster.
A further aspect of matrix completion techniques we investigate is how to improve their prediction performance. This can be done, for instance, considering the context in which relationships have been captured. However, incorporating generic contextual information within a matrix completion algorithm is a challenging task.
In the first part of this thesis, we study distributed matrix completion solutions, and address the above issues by examining input slicing techniques based on graph partitioning algorithms. In the second part of the thesis, we focus on context-aware matrix completion techniques, providing solutions that can work both (i) when the revealed entries in the matrix have multiple values and (ii) all the same value."

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Mining at scale with latent factor models for matrix completion

  1. 1. Mining at scale with latent factor models for matrix completion Fabio Petroni
  2. 2. Recommender systems Marco wants to watch a movie. A B C D E F G H I J K ... Z A (1998) A-ge-man (1990) A.K.G. (2007) A Nous Amours (1983) ... many pages later ... Azumi (2003) DVD rental store 2 of 58
  3. 3. Recommender systems Marco wants to watch a movie. A B C D E F G H I J K ... Z A (1998) A-ge-man (1990) A.K.G. (2007) A Nous Amours (1983) ... many pages later ... Azumi (2003) DVD rental store But there are so many movies! Which ones will he like? 2 of 58
  4. 4. Collaborative filtering I problem B set of users B set of items (movies, books, songs, ...) B feedback I explicit (ratings, ...) I implicit (purchase, click-through, ...) I predict the preference of each user for each item B assumption: similar feedback $ similar taste I example (explicit feedback): Avatar The Matrix Up Marco 4 2 Luca 3 2 Anna 5 3 3 of 58
  5. 5. Collaborative filtering I problem B set of users B set of items (movies, books, songs, ...) B feedback I explicit (ratings, ...) I implicit (purchase, click-through, ...) I predict the preference of each user for each item B assumption: similar feedback $ similar taste I example (explicit feedback): Avatar The Matrix Up Marco ? 4 2 Luca 3 2 ? Anna 5 ? 3 3 of 58
  6. 6. Collaborative filtering taxonomy SVD PMFuser based PLS(A/I) memory based collaborative filtering item based model based probabilistic methods neighborhood models dimensionality reduction matrix completion latent Dirichlet allocation other machine learning methods Bayesian networks Markov decision processes neural networks I matrix completion is currently considered the best approach I advantages with respect to both scalability and accuracy 4 of 58
  7. 7. Matrix completion for collaborative filtering I the completion is driven by a factorization R P Q I associate a latent factor vector with each user and each item I missing entries are estimated through the dot product rij ⇡ piqj 5 of 58
  8. 8. Latent factor models (Koren et al., 2009) 6 of 58
  9. 9. Latent factor models - explicit feedback I discover latent factors (d = 1) Avatar (2.24) The Matrix (1.92) Up (1.18) Marco (1.98) ? 4 (3.8) 2 (2.3) Luca (1.21) 3 (2.7) 2 (2.3) ? Anna (2.30) 5 (5.2) ? 3 (2.7) I optimization criterion: minimize squared loss minimize P,Q X (i,j)2O (rij pi qj )2 7 of 58
  10. 10. Latent factor models - explicit feedback I discover latent factors (d = 1) Avatar (2.24) The Matrix (1.92) Up (1.18) Marco (1.98) 4.4 4 (3.8) 2 (2.3) Luca (1.21) 3 (2.7) 2 (2.3) 1.4 Anna (2.30) 5 (5.2) 4.4 3 (2.7) I optimization criterion: minimize squared loss minimize P,Q X (i,j)2O (rij pi qj )2 7 of 58
  11. 11. Latent factor models - explicit feedback I discover latent factors (d = 1) Avatar (2.24) The Matrix (1.92) Up (1.18) Marco (1.98) 4.4 4 (3.8) 2 (2.3) Luca (1.21) 3 (2.7) 2 (2.3) 1.4 Anna (2.30) 5 (5.2) 4.4 3 (2.7) I optimization criterion: minimize squared loss minimize P,Q X (i,j)2O (rij bµ bui bxj pi qj )2 + (||P||F + ||Q||F ) 7 of 58
  12. 12. Latent factor models - implicit feedback I discover latent factors (d = 1) Avatar The Matrix Up Marco ? 1 1 Luca 1 1 ? Anna 1 ? 1 I the squared loss minimization criterion is not effective! 8 of 58
  13. 13. Latent factor models - implicit feedback I discover latent factors (d = 1) Avatar (1) The Matrix (1) Up (1) Marco (1) 1 1 (1) 1 (1) Luca (1) 1 (1) 1 (1) 1 Anna (1) 1 (1) 1 1 (1) I the squared loss minimization criterion is not effective! I the system simply complete the matrix with all 1s 8 of 58
  14. 14. Bayesian personalized ranking I problem related with ranking more than prediction B e.g., ranked list of items that the user might like the most I the BPR criterion adopts a pairwise approach B predict whether item xj is more probable to be liked than xk B assumption: the user prefers items for which a feedback exists DT := {(ui , xj , xk)|ui 2 U ^ xj , xk 2 I, j 6= k ^ rij = 1 ^ rik =?} I user ui is assumed to prefer item xj over xk maximize P,Q X (ui ,xj ,xk )2DT  ln (pi qj pi qk) 9 of 58
  15. 15. Stochastic gradient descent I parameters ⇥ = {P, Q} I find minimum ⇥⇤ of loss function L, or maximum for BPR (ascent) I pick a starting point ⇥0 I iteratively update current estimations for ⇥ 6 7 0 5 10 15 20 25 30 loss(×107 ) iterations ⇥n+1 ⇥n ⌘ @L @⇥ I learning rate ⌘ I an update for each given training point 10 of 58
  16. 16. Overview matrix completion regularized squared loss Bayesian personalized ranking solve optimization problem model train the model P Q optimization criteria stochastic gradient descent (ascent) explicit feedback implicit feedback user latent factor vectors item latent factor vectors 11 of 58
  17. 17. Challenges of matrix completion (1) scalability: handle large scale data B 2B purchases on Amazon.com by 200M customers (2014) B parallel and distributed approaches are essential! (2) quality: improve prediction performance B Netflix awarded a 10% improvement with $1M (2006) B the performance of matrix completion can be boosted by: I feeding the system with more data I integrating contextual information in the model 12 of 58
  18. 18. Overview - state-of-the-artscalability quality centralizeddistributedparallel context-awarecontext-agnostic context-awarecontext-agnostic only positive evidence single-value revealed entries positive and negative evidence multi-value revealed entries Bayesian Personalized RankingRegularized Squared Loss Makari et al., 2014 Ahmed et al., 2013 Zhuang et al., 2013 Niu et al., 2011 Koren et al., 2009 Koren, 2008 Shi et al., 2014 Rendle et al., 2011 Riedel et al., 2013 Rendle et al., 2009 Chen et al., 2012 Menon et al., 2011 Makari et al., 2014 Recht et al., 2013 Rendle, 2012 Karatzoglou et al., 2010 Takács et al., 2009 Ricci et al., 2011 13 of 58
  19. 19. Overview - contributionsscalability quality centralizeddistributedparallel context-awarecontext-agnostic context-awarecontext-agnostic only positive evidence single-value revealed entries positive and negative evidence multi-value revealed entries Bayesian Personalized RankingRegularized Squared Loss Petroni et al., 2015a Petroni et al., 2014 Makari et al., 2014 Ahmed et al., 2013 Zhuang et al., 2013 Niu et al., 2011 Koren et al., 2009 Koren, 2008 Shi et al., 2014 Rendle et al., 2011 Riedel et al., 2013 Rendle et al., 2009 Petroni et al., 2015b Chen et al., 2012 Menon et al., 2011 Makari et al., 2014 Recht et al., 2013 Rendle, 2012 Karatzoglou et al., 2010 Takács et al., 2009 Ricci et al., 2011 13 of 58
  20. 20. Contents 1. Distributed Matrix Completion 1.1 Distributed Stochastic Gradient Descend 1.2 Input Partitioner 1.3 Evaluation 2. Context-Aware Matrix Completion 2.1 Open Relation Extraction 2.2 Context-Aware Open Relation Extraction 2.3 Evaluation 3. Conclusion and Outlook 14 of 58
  21. 21. Contents 1. Distributed Matrix Completion 1.1 Distributed Stochastic Gradient Descend 1.2 Input Partitioner 1.3 Evaluation 2. Context-Aware Matrix Completion 2.1 Open Relation Extraction 2.2 Context-Aware Open Relation Extraction 2.3 Evaluation 3. Conclusion and Outlook 14 of 58
  22. 22. Problems of parallel and distributed SGD I divide the training points (SGD updates) among threads I SGD updates might depend on each other! i2 u2 u4 thread1 thread2 tr22 r22 r42 r42 I both threads concurrently update the same latent vector I lock-based approaches adversely affect concurrency 15 of 58
  23. 23. SGD Taxonomy FPSGD CSGD DSGD-MRJellyfish SSGD stochastic gradient descent HogWild (Gemulla et al., 2011)(Zhuang et al., 2013) (Teflioudi et al., 2012)(Recht et al., 2013) distributed (Makari et al., 2014)(Niu et al., 2011) parallel DSGD++ ASGD (Makari et al., 2014) (Makari et al., 2014) I parallel SGD is hardly applicable to very large datasets B the time-to-convergence may be too slow B the input data may not fit into the main memory I ASGD has advantages in scalability and efficiency 16 of 58
  24. 24. Asynchronous stochastic gradient descent I distributed shared-nothing environment (cluster of machines) R computing nodes Q3 P3 Q4 P4 Q2 P2 Q1 P1 1 2 3 4 R1 R4 R3R2 I R is splitted I vectors are replicated I replicas concurrently updated I replicas deviate inconsistently I synchronization 17 of 58
  25. 25. Bulk Synchronous Processing Model computing nodes 1. local computation 2. communication 3. barrier synchronization I currently used by most of the ASGD implementations 18 of 58
  26. 26. Challenges R 1 2 3 4 R1 R4 R3R2 I 1. workload balance B ensure that computing nodes are fed with the same load B improve efficiency I 2. minimize communication B minimize vector replicas B improve scalability 19 of 58
  27. 27. Graph representation I the rating matrix describes a graph x 2 1 b c 3 a x xx 4 d x x xx x x1 2 3 4 a b c d I vertices represent users and items I edges represent training points (e.g., ratings) 20 of 58
  28. 28. Graph representation I the rating matrix describes a graph x x xx x x xx x x 2 1 b c 3 a 4 d 1 2 3 4 a b c d I find a partitioning of the graph I assign each part to a different machine 20 of 58
  29. 29. Balanced graph partitioning I partition G into smaller parts of (ideally) equal size vertex-cutedge-cut I a vertex can be cut in multiple ways and span several parts while a cut edge connects only two parts I computation steps are associated with edges I v-cut better on power-law graphs (Gonzalez et al, 2012) 21 of 58
  30. 30. Power-law graphs I characteristic of real graphs: power-law degree distribution B most vertices have few connections while a few have many 10 0 101 10 2 10 3 10 4 10 5 10 6 10 0 10 1 10 2 10 3 10 4 10 5 numberofvertices degree crawl Twitter connection 2010 log-log scale I the probability that a vertex has degree d is P(d) / d ↵ I ↵ controls the “skewness” of the degree distribution 22 of 58
  31. 31. Balanced Vertex-Cut Graph Partitioning I v 2 V vertex; e 2 E edge; p 2 P part I A(v) set of parts where vertex v is replicated I 1 tolerance to load imbalance I the size |p| of part p is its edge cardinality minimize replicas reduce (1) bandwidth, (2) memory usage and (3) synchronization balance the load efficient usage of available computing resources min 1 |V | X v2V |A(v)| s.t. max p2P |p| < |E| |P| I the objective function is the replication factor (RF) B average number of replicas per vertex 23 of 58
  32. 32. Streaming Setting I input data is a list of edges, consumed in streaming fashion, requiring only a single pass partitioning algorithm stream of edges graph 3 handle graphs that don’t fit in the main memory 3 impose minimum overhead in time 3 scalable, easy parallel implementations 7 assignment decision taken cannot be later changed 24 of 58
  33. 33. Streaming algorithms history-agnostichistory-aware power-law-awarepower-law-agnostic Greedy (Gonzalez et al., 2012) DBH (Xie et al., 2014) Grid PDS Hashing (Jain et al., 2013) (Jain et al., 2013) (Gonzalez et al., 2012) lessreplicas betterbalance less replicas 25 of 58
  34. 34. Streaming algorithms - contributions history-agnostichistory-aware power-law-awarepower-law-agnostic Greedy (Gonzalez et al., 2012) DBH (Xie et al., 2014) Grid PDS Hashing (Jain et al., 2013) (Jain et al., 2013) (Gonzalez et al., 2012) lessreplicas betterbalance less replicas HDRF (Petroni et al., 2015) 25 of 58
  35. 35. HDRF: High Degree are Replicated First I favor the replication of high-degree vertices I the number of high-degree vertices in power-law graphs is very small I overall reduction of the replication factor 26 of 58
  36. 36. HDRF: High Degree are Replicated First I in the context of robustness to network failure I if few high-degree vertices are removed from a power-law graph then it is turned into a set of isolated clusters I focus on the locality of low-degree vertices 26 of 58
  37. 37. The HDRF Algorithm vertex without replicas vertex with replicas case 1 vertices not assigned to any part incoming edge e 27 of 58
  38. 38. The HDRF Algorithm vertex without replicas vertex with replicas case 1 place e in the least loaded part incoming edge e 27 of 58
  39. 39. The HDRF Algorithm case 2 only one vertex has been assigned vertex without replicas vertex with replicas case 1 place e in the least loaded part incoming edge e e 27 of 58
  40. 40. The HDRF Algorithm case 2 place e in the part vertex without replicas vertex with replicas case 1 place e in the least loaded part incoming edge e e 27 of 58
  41. 41. The HDRF Algorithm case 2 place e in the part vertex without replicas vertex with replicas case 1 place e in the least loaded part incoming edge e e case 3 vertices assigned, common part e 27 of 58
  42. 42. The HDRF Algorithm case 2 place e in the part vertex without replicas vertex with replicas case 1 place e in the least loaded part incoming edge e e case 3 place e in the intersection e 27 of 58
  43. 43. Create Replicas case 4 empty intersection e 28 of 58
  44. 44. Create Replicas case 4 least loaded part in the union case 4 empty intersection e e standard Greedy solution 28 of 58
  45. 45. Create Replicas case 4 least loaded part in the union case 4 empty intersection e case 4 replicate vertex with highest degree e δ(v1) > δ(v2) e standard Greedy solution HDRF 28 of 58
  46. 46. Experiments - Settings I standalone partitioner B VGP, a software package for one-pass vertex-cut balanced graph partitioning B measure the performance: replication and balancing I GraphLab B HDRF has been integrated in GraphLab PowerGraph 2.2 B measure the impact on the execution time of graph computation in a distributed graph computing frameworks I stream of edges in random order 29 of 58
  47. 47. Experiments - Datasets I real-word graphs I synthetic graphs 1M vertices 60M to 3M edges Dataset |V | |E| MovieLens 10M 80.6K 10M Netflix 497.9K 100.4M Tencent Weibo 1.4M 140M twitter-2010 41.7M 1.47B 10 0 10 1 102 103 10 4 10 5 10 1 10 2 10 3 10 4 10 5 numberofvertices degree α = 1.8 α = 2.2 α = 2.6 α = 3.0 α = 3.4 α = 4.0 30 of 58
  48. 48. Results - Synthetic Graphs Replication Factor I 128 parts 4 8 1.8 2.0 2.4 2.8 3.2 3.6 4.0 replicationfactor alpha HDRF PDS |P|=133 grid |P|=121 greedy DBH skewed homogeneous 31 of 58
  49. 49. Results - Synthetic Graphs Replication Factor I 128 parts 4 8 1.8 2.0 2.4 2.8 3.2 3.6 4.0 replicationfactor alpha HDRF PDS |P|=133 grid |P|=121 greedy DBH skewed homogeneous less dense less edges easyer to partition 31 of 58
  50. 50. Results - Synthetic Graphs Replication Factor I 128 parts 4 8 1.8 2.0 2.4 2.8 3.2 3.6 4.0 replicationfactor alpha HDRF PDS |P|=133 grid |P|=121 greedy DBH skewed homogeneous power-law-agnostic more dense more edges difficult to partition 31 of 58
  51. 51. Results - Synthetic Graphs Replication Factor I 128 parts 4 8 1.8 2.0 2.4 2.8 3.2 3.6 4.0 replicationfactor alpha HDRF PDS |P|=133 grid |P|=121 greedy DBH skewed homogeneous power-law-aware 31 of 58
  52. 52. Results - Real-Word Graphs Replication Factor I 133 parts 1 2 4 8 16 Tencent Weibo Netflix MovieLens 10M twitter-2010 replicationfactor PDS 7.9 11.3 11.5 8.0 greedy 2.8 8.1 10.7 5.9 DBH 1.5 7.1 12.9 6.8 HDRF 1.3 4.9 9.1 4.8 1 2 4 8 16 Tencent Weibo Netflix MovieLens 10M twitter-2010 replicationfactor PDS greedy DBH HDRF 32 of 58
  53. 53. Results - Load Relative Standard Deviation I MovieLens 10M 0 1 2 3 4 5 6 7 8 16 32 64 128 256 loadrelativestandarddeviation(%) parts PDS grid greedy DBH HDRF hashing 0 1 2 3 4 5 6 7 8 16 32 64 128 256 loadrelativestandarddeviation(%) parts PDS grid greedy DBH HDRF hashing 33 of 58
  54. 54. Results - Load Relative Standard Deviation I MovieLens 10M 0 1 2 3 4 5 6 7 8 16 32 64 128 256 loadrelativestandarddeviation(%) partitions PDS grid greedy DBH HDRF hashing history-agnostic 0 1 2 3 4 5 6 7 8 16 32 64 128 256 loadrelativestandarddeviation(%) parts PDS grid greedy DBH HDRF hashing 33 of 58
  55. 55. Results - Load Relative Standard Deviation I MovieLens 10M 0 1 2 3 4 5 6 7 8 16 32 64 128 256 loadrelativestandarddeviation(%) partitions PDS grid greedy DBH HDRF hashing history-aware 0 1 2 3 4 5 6 7 8 16 32 64 128 256 loadrelativestandarddeviation(%) parts PDS grid greedy DBH HDRF hashing 33 of 58
  56. 56. Results - Graph Algorithm Runtime Speedup I ASGD algorithm for collaborative filtering on Tencent Weibo 1 1.5 2 2.5 3 32 64 128 speedup parts greedy PDS 1 1.5 2 2.5 3 32 64 128 speedup parts greedy PDS |P|=31,57,133 I the speedup is proportional to both: B the advantage in replication factor B the actual network usage of the algorithm 34 of 58
  57. 57. Summary I HDRF is a simple and remarkably effective one-pass vertex-cut graph partitioning algorithm B achieves on average a replication factor I about 40% smaller than DBH I more than 50% smaller than greedy I almost 3⇥ smaller than PDS I more than 4⇥ smaller than grid I almost 14⇥ smaller than hashing I close to optimal load balance I ASGD execution time is up to 2⇥ faster when using HDRF I HDRF has been included in GraphLab! 35 of 58
  58. 58. Contents 1. Distributed Matrix Completion 1.1 Distributed Stochastic Gradient Descend 1.2 Input Partitioner 1.3 Evaluation 2. Context-Aware Matrix Completion 2.1 Open Relation Extraction 2.2 Context-Aware Open Relation Extraction 2.3 Evaluation 3. Conclusion and Outlook 36 of 58
  59. 59. Knowledge bases I New generation algorithms for web information retrieval make use of a knowledge base (KB) to increase their accuracy I match the words in a query to real world entities (e.g., a person, a location, etc) I use real world connections among these entities I improve the task of providing the user with the proper content he was looking for I KB represented as a graph 37 of 58
  60. 60. Knowledge bases challenges I large-scale example: LOD contains over 60B facts (sub-rel-obj) I despite the size, KBs are far from being complete B 75% people have unknown nationality in Freebase B 71% people with place of birth attribute missing in Freebase I data not only incomplete but also uncertain, noisy or false I challenges: B fill missing information B remove incorrect facts I idea: scan the web extracting new information 38 of 58
  61. 61. Open relation extraction I open relation extraction is the task of extracting new facts for a potentially unbounded set of relations from various sources natural language text knowledge bases 39 of 58
  62. 62. Input data: facts from natural language text Enrico Fermi was a professor in theoretical physics at Sapienza University of Rome. "professor at"(Fermi,Sapienza) tuple sub objrel surface fact open information extractor extract all facts in textsurface relation 40 of 58
  63. 63. Input data: facts from knowledge bases natural language text Fermi Sapienza employee(Fermi,Sapienza) KB fact employee KB relation "professor at"(Fermi,Sapienza) surface fact 41 of 58
  64. 64. Matrix completion for open relation extraction (Caesar,Rome) (Fermi,Rome) (Fermi,Sapienza) (de Blasio,NY) 1 1 1 1 1 KB relationsurface relation employeeborn in professor at mayor of tuples x relations 42 of 58
  65. 65. Matrix completion for open relation extraction (Caesar,Rome) (Fermi,Rome) (Fermi,Sapienza) (de Blasio,NY) 1 1 1 1 1 KB relationsurface relation employeeborn in professor at mayor of tuples x relations ? ? ?? ? ?? ? ? ? ? 42 of 58
  66. 66. Contributions (Caesar,Rome) (Fermi,Rome) (Fermi,Sapienza) (de Blasio,NY) 1 1 1 1 1 KB relationsurface relation employeeborn in professor at mayor of tuples x relations ? ? ?? ? ?? ? ? ? ? we propose CORE (context-aware open relation extraction) that integrates contextual information into such models to improve prediction performance 43 of 58
  67. 67. Contextual information Tom Peloso joined Modest Mouse to record their fifth studio album. person organization surface relation "join"(Peloso,Modest Mouse) unspecific relation entity types article topic words record album Contextual information named entity recognizer label entity with coarse- grained type 44 of 58
  68. 68. CORE - latent representation of variables I associates latent representations fv with each variable v 2 V Peloso (Peloso,Modest Mouse) Modest Mouse join person organization record album tuple relation entities context latent factor vectors 45 of 58
  69. 69. CORE - modeling facts 1 0 0 1 0 0 0.5 0.5 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0.5 0.5 1 0 1 0 1 0 0 0 1 0 0 0.5 0.5 0 0 1 0.6 0.4 0 0 1 0 0 1 0 0 0.5 0.5 1 0 1 0 “born in”(x,y) employee(x,y) Caesar,Rome Fermi,Rome Fermi,Sapienza Caesar Rome Fermi Sapienza person, organization person, location physics history relations tuples entities tuple types tuple topics x1 x2 x3 x4 Surface KB Context … “professor at”(x,y) I models the input data in terms of a matrix in which each row corresponds to a fact x and each column to a variable v I groups columns according to the type of the variables I in each row the values of each column group sum up to unity 46 of 58
  70. 70. CORE - modeling context 1 0 0 1 0 0 0.5 0.5 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0.5 0.5 1 0 1 0 1 0 0 0 1 0 0 0.5 0.5 0 0 1 0.6 0.4 0 0 1 0 0 1 0 0 0.5 0.5 1 0 1 0 “born in”(x,y) employee(x,y) Caesar,Rome Fermi,Rome Fermi,Sapienza Caesar Rome Fermi Sapienza person, organization person, location physics history relations tuples entities tuple types tuple topics x1 x2 x3 x4 Surface KB Context … “professor at”(x,y) I aggregates and normalizes contextual information by tuple B a fact can be observed multiple times with different context B there is no context for new facts (never observed in input) I this approach allows us to provide comprehensive contextual information for both observed and unobserved facts 47 of 58
  71. 71. CORE - factorization model 1 0 0 1 0 0 0.5 0.5 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0.5 0.5 1 0 1 0 1 0 0 0 1 0 0 0.5 0.5 0 0 1 0.6 0.4 0 0 1 0 0 1 0 0 0.5 0.5 1 0 1 0 “born in”(x,y) employee(x,y) Caesar,Rome Fermi,Rome Fermi,Sapienza Caesar Rome Fermi Sapienza person, organization person, location physics history relations tuples entities tuple types tuple topics x1 x2 x3 x4 Surface KB Context … “professor at”(x,y) I uses factorization machines as underlying framework I associates a score s(x) with each fact x s(x) = X v12V X v22V {v1} xv1 xv2 f T v1 fv2 I weighted pairwise interactions of latent factor vectors 48 of 58
  72. 72. CORE - parameter estimation I parameters: ⇥ = {fv | v 2 V } I Bayesian personalized ranking, all observations are positive I goal: produce a ranked list of tuples for each relation professor at location tuple entities tuple contextfact (Fermi,Sapienza) Fermi Sapienza organization physics person x (Caesar,Rome) professor at Caesar locationRome history tuple entities tuple contextfact person x-sampled negative evidenceobserved fact I pairwise approach, x is more likely to be true than x- maximize X x f (s(x) s(x-)) I stochastic gradient ascent ⇥ ⇥ + ⌘r⇥ ⇣ ⌘ 49 of 58
  73. 73. Experiments - dataset 440k facts extracted from corpus 15k facts from entity mentions linked using string matching I Contextual information article metadata bag-of-word sentences where the fact has been extracted entity type person organization location miscellaneous news desk (e.g., foreign desk) descriptors (e.g., finances) online section (e.g., sports) section (e.g., a, d) publication year m t w I letters to indicate contextual information considered 50 of 58
  74. 74. Experiments - methodology I we consider (to keep experiments feasible): 10k tuples 10 surface relations19 Freebase relations I for each relation and method: B we rank the tuples subsample B we consider the top-100 predictions and label them manually I evaluation metrics: number of true facts MAP (quality of the ranking) I methods: B PITF, tensor factorization method (designed to work in-KB) B NFE, matrix completion method (best context-agnostic) B CORE, uses relations, tuples and entities as variables B CORE+m, +t, +w, +mt, +mtw 51 of 58
  75. 75. Results - Freebase relations Relation # PITF NFE CORE CORE+m CORE+t CORE+w CORE+mt CORE+mtw person/company 208 70 (0.47) 92 (0.81) 91 (0.83) 90 (0.84) 91 (0.87) 92 (0.87) 95 (0.93) 96 (0.94) person/place_of_birth 117 1 (0.0) 92 (0.9) 90 (0.88) 92 (0.9) 92 (0.9) 89 (0.87) 93 (0.9) 92 (0.9) location/containedby 102 7 (0.0) 63 (0.47) 62 (0.47) 63 (0.46) 61 (0.47) 61 (0.44) 62 (0.49) 68 (0.55) parent/child 88 9 (0.01) 64 (0.6) 64 (0.56) 64 (0.59) 64 (0.62) 64 (0.57) 67 (0.67) 68 (0.63) person/place_of_death 71 1 (0.0) 67 (0.93) 67 (0.92) 69 (0.94) 67 (0.93) 67 (0.92) 69 (0.94) 67 (0.92) person/parents 67 20 (0.1) 51 (0.64) 52 (0.62) 51 (0.61) 49 (0.64) 47 (0.6) 53 (0.67) 53 (0.65) author/works_written 65 24 (0.08) 45 (0.59) 49 (0.62) 51 (0.69) 50 (0.68) 50 (0.68) 51 (0.7) 52 (0.67) person/nationality 61 21 (0.08) 25 (0.19) 27 (0.17) 28 (0.2) 26 (0.2) 29 (0.19) 27 (0.18) 27 (0.21) neighbor./neighborhood_of 39 3 (0.0) 24 (0.44) 23 (0.45) 26 (0.5) 27 (0.47) 27 (0.49) 30 (0.51) 30 (0.52) ... Average MAP100 # 0.09 0.46 0.47 0.49 0.47 0.49 0.49 0.51 Weighted Average MAP100 # 0.14 0.64 0.64 0.66 0.67 0.66 0.70 0.70 0.5 0.55 0.6 0.65 0.7 NFE CORE CORE+m CORE+t CORE+w CORE+mt CORE+mtw Weighted Average MAP 0.64 0.64 0.66 0.67 0.66 0.70 0.70 52 of 58
  76. 76. Results - surface relations Relation # PITF NFE CORE CORE+m CORE+t CORE+w CORE+mt CORE+mtw head 162 34 (0.18) 80 (0.66) 83 (0.66) 82 (0.63) 76 (0.57) 77 (0.57) 83 (0.69) 88 (0.73) scientist 144 44 (0.17) 76 (0.6) 74 (0.55) 73 (0.56) 74 (0.6) 73 (0.59) 78 (0.66) 78 (0.69) base 133 10 (0.01) 85 (0.71) 86 (0.71) 86 (0.78) 88 (0.79) 85 (0.75) 83 (0.76) 89 (0.8) visit 118 4 (0.0) 73 (0.6) 75 (0.61) 76 (0.64) 80 (0.68) 74 (0.64) 75 (0.66) 82 (0.74) attend 92 11 (0.02) 65 (0.58) 64 (0.59) 65 (0.63) 62 (0.6) 66 (0.63) 62 (0.58) 69 (0.64) adviser 56 2 (0.0) 42 (0.56) 47 (0.58) 44 (0.58) 43 (0.59) 45 (0.63) 43 (0.53) 44 (0.63) criticize 40 5 (0.0) 31 (0.66) 33 (0.62) 33 (0.7) 33 (0.67) 33 (0.61) 35 (0.69) 37 (0.69) support 33 3 (0.0) 19 (0.27) 22 (0.28) 18 (0.21) 19 (0.28) 22 (0.27) 23 (0.27) 21 (0.27) praise 5 0 (0.0) 2 (0.0) 2 (0.01) 4 (0.03) 3 (0.01) 3 (0.02) 5 (0.03) 2 (0.01) vote 3 2 (0.01) 3 (0.63) 3 (0.63) 3 (0.32) 3 (0.49) 3 (0.51) 3 (0.59) 3 (0.64) Average MAP100 # 0.04 0.53 0.53 0.51 0.53 0.53 0.55 0.59 Weighted Average MAP100 # 0.08 0.62 0.61 0.63 0.63 0.61 0.65 0.70 0.5 0.6 0.7 NFE CORE CORE+m CORE+t CORE+w CORE+mt CORE+mtw Weighted Average MAP 0.62 0.61 0.63 0.63 0.61 0.65 0.70 53 of 58
  77. 77. Anecdotal results author(x,y) ranked list of tuples 1 (Winston Groom, Forrest Gump) 2 (D. M. Thomas, White Hotel) 3 (Roger Rosenblatt, Life Itself) 4 (Edmund White, Skinned Alive) 5 (Peter Manso, Brando: The Biography) similar relations 0.98 “reviews x by y”(x,y) 0.97 “book by”(x,y) 0.95 “author of”(x,y) 0.95 ” ‘s novel”(x,y) 0.95 “ ‘s book”(x,y) similar relations 0.87 “scientist”(x,y) 0.84 “scientist with”(x,y) 0.80 “professor at”(x,y) 0.79 “scientist for”(x,y) 0.78 “neuroscientist at”(x,y) ranked list of tuples 1 (Riordan Roett, Johns Hopkins University) 2 (Dr. R. M. Roberts, University of Missouri) 3 (Linda Mayes, Yale University) 4 (Daniel T. Jones, Cardiff Business School) 5 (Russell Ross, University of Iowa) “scientist at”(x,y) I semantic similarity of relations is one aspect of our model I similar relations treated differently in different contexts 54 of 58
  78. 78. Contents 1. Distributed Matrix Completion 1.1 Distributed Stochastic Gradient Descend 1.2 Input Partitioner 1.3 Evaluation 2. Context-Aware Matrix Completion 2.1 Open Relation Extraction 2.2 Context-Aware Open Relation Extraction 2.3 Evaluation 3. Conclusion and Outlook 55 of 58
  79. 79. Conclusion I we tackle the scalability and the quality of MC I we investigate graph partitioning techniques for ASGD I we propose HDRF, one-pass v-cut graph partitioning alg B exploit power-law nature of real-word graphs B provides minimum replicas with close to optimal load balance B significantly reduces the time needed to perform computation I we propose CORE, a matrix completion model for open relation extraction that incorporates contextual information B based on factorization machines and BPR B extensible model, additional information can be integrated B exploiting context significantly improve prediction quality I all code released https://github.com/fabiopetroni 56 of 58
  80. 80. Overview - Future workscalability quality centralizeddistributedparallel context-awarecontext-agnostic context-awarecontext-agnostic only positive evidence single-value revealed entries positive and negative evidence multi-value revealed entries Bayesian Personalized RankingRegularized Squared Loss Petroni et al., 2015a Petroni et al., 2014 Makari et al., 2014 Ahmed et al., 2013 Zhuang et al., 2013 Niu et al., 2011 Koren et al., 2009 Koren, 2008 Shi et al., 2014 Rendle et al., 2011 Riedel et al., 2013 Rendle et al., 2009 Petroni et al., 2015b Chen et al., 2012 Menon et al., 2011 Makari et al., 2014 Recht et al., 2013 Rendle, 2012 Karatzoglou et al., 2010 Takács et al., 2009 Ricci et al., 2011 56 of 58
  81. 81. Future Directions I distribute training for context-aware matrix completion B asynchronous approach: local vector copies, synchronization B challenging input placement, the input describes an hypergraph I adaptation of the HDRF algorithm to hypergraphs B do high degree vertices play a crucial role? “born in”(x,y) Caesar,Rome Fermi,Rome Caesar Rome Fermi person, location physics history e1 e2 I distributed Bayesian personalized ranking B sampling of a negative counterpart for each training point B sampling from the local portion of the dataset in current node? 57 of 58
  82. 82. Thank you! Questions? Fabio Petroni Sapienza University of Rome, Italy 58 of 58

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