Talwalkar mlconf (1)

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Ameet Talwalker, Postdoctoral Fellow at UC Berkeley: Divide-and-Conquer Matrix Factorization

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Talwalkar mlconf (1)

  1. 1. Divide-­‐and-­‐Conquer   Matrix  Factoriza5on Ameet  Talwalkar UC  Berkeley November  15th,  2013 Collaborators:  Lester  Mackey2,  Michael  I.  Jordan1,   Yadong  Mu3,  Shih-­‐Fu  Chang3 1UC  Berkeley            2Stanford  University            3Columbia  University  
  2. 2. Three  Converging  Trends
  3. 3. Three  Converging  Trends Big  Data
  4. 4. Three  Converging  Trends Big  Data Distributed   CompuOng
  5. 5. Three  Converging  Trends Machine   Learning Big  Data Distributed   CompuOng
  6. 6. Goal:  Extend  ML  to  the  Big  Data  SeAng   Challenge:  ML  not  developed  with  scalability  in  mind ✦ Does  not  naturally  scale  /  leverage  distributed  compuOng Machine   Learning Big  Data Distributed   CompuOng
  7. 7. Goal:  Extend  ML  to  the  Big  Data  SeAng   Challenge:  ML  not  developed  with  scalability  in  mind ✦ Does  not  naturally  scale  /  leverage  distributed  compuOng Our  approach:  Divide-­‐and-­‐conquer ✦ Apply  exisOng  base  algorithms  to  subsets  of  data  and  combine Machine   Learning Big  Data Distributed   CompuOng
  8. 8. Goal:  Extend  ML  to  the  Big  Data  SeAng   Challenge:  ML  not  developed  with  scalability  in  mind ✦ Does  not  naturally  scale  /  leverage  distributed  compuOng Our  approach:  Divide-­‐and-­‐conquer ✦ Apply  exisOng  base  algorithms  to  subsets  of  data  and  combine ✓ ✓ ✓ Build  upon  exisOng  suites  of  ML  algorithms Preserve  favorable  algorithm  properOes Naturally  leverage  distributed  compuOng Machine   Learning Big  Data Distributed   CompuOng
  9. 9. Goal:  Extend  ML  to  the  Big  Data  SeAng   Challenge:  ML  not  developed  with  scalability  in  mind ✦ Does  not  naturally  scale  /  leverage  distributed  compuOng Our  approach:  Divide-­‐and-­‐conquer ✦ Apply  exisOng  base  algorithms  to  subsets  of  data  and  combine ✓ ✓ ✓ ✦ Build  upon  exisOng  suites  of  ML  algorithms Preserve  favorable  algorithm  properOes Naturally  leverage  distributed  compuOng E.g.,   ✦ ✦ ✦ Machine   Learning Big  Data Matrix  factorizaOon  (DFC) [MTJ, NIPS11; TMMFJ, ICCV13] [KTSJ, ICML12; KTSJ, Assessing  esOmator  quality  (BLB) JRSS13; KTASJ, KDD13] Genomic  Variant  Calling [BTTJPYS13, submitted, CTZFJP13, submitted] Distributed   CompuOng
  10. 10. Goal:  Extend  ML  to  the  Big  Data  SeAng   Challenge:  ML  not  developed  with  scalability  in  mind ✦ Does  not  naturally  scale  /  leverage  distributed  compuOng Our  approach:  Divide-­‐and-­‐conquer ✦ Apply  exisOng  base  algorithms  to  subsets  of  data  and  combine ✓ ✓ ✓ ✦ Build  upon  exisOng  suites  of  ML  algorithms Preserve  favorable  algorithm  properOes Naturally  leverage  distributed  compuOng E.g.,   ✦ ✦ ✦ Machine   Learning Big  Data Matrix  factorizaOon  (DFC) [MTJ, NIPS11; TMMFJ, ICCV13] [KTSJ, ICML12; KTSJ, Assessing  esOmator  quality  (BLB) JRSS13; KTASJ, KDD13] Genomic  Variant  Calling [BTTJPYS13, submitted, CTZFJP13, submitted] Distributed   CompuOng
  11. 11. Matrix  CompleOon
  12. 12. Matrix  CompleOon
  13. 13. Matrix  CompleOon Goal: Recover a matrix from a subset of its entries
  14. 14. Matrix  CompleOon Goal: Recover a matrix from a subset of its entries
  15. 15. Matrix  CompleOon Goal: Recover a matrix from a subset of its entries
  16. 16. Matrix  CompleOon Goal: Recover a matrix from a subset of its entries Can we do this at scale? ✦ ✦ ✦ ✦ ✦ Netflix: 30M users, 100K+ videos Facebook: 1B users Pandora: 70M active users, 1M songs Amazon: Millions of users and products ...
  17. 17. Reducing  Degrees  of  Freedom
  18. 18. Reducing  Degrees  of  Freedom ✦ Problem: Impossible without additional information ✦ mn degrees of freedom n m
  19. 19. Reducing  Degrees  of  Freedom ✦ Problem: Impossible without additional information ✦ ✦ mn degrees of freedom Solution: Assume small # of factors determine preference n m r =m n r ‘Low-rank’
  20. 20. Reducing  Degrees  of  Freedom ✦ Problem: Impossible without additional information ✦ ✦ mn degrees of freedom Solution: Assume small # of factors determine preference ✦ O(m + n) degrees of freedom ✦ Linear storage costs n m r =m n r ‘Low-rank’
  21. 21. Bad  Sampling ✦ Problem:    We  have  no  raOng   informaOon  about  
  22. 22. Bad  Sampling ✦ Problem:    We  have  no  raOng   informaOon  about ✦ SoluOon:    Assume    ˜                 + m))   ⌦(r(n observed  entries  drawn   uniformly  at  random
  23. 23. Bad  InformaOon  Spread
  24. 24. Bad  InformaOon  Spread ✦ Problem:  Other  raOngs  don’t   inform  us  about  missing  raOng bad  spread  of  informaOon
  25. 25. Bad  InformaOon  Spread ✦ Problem:  Other  raOngs  don’t   inform  us  about  missing  raOng ✦ SoluOon:    Assume   incoherence  with  standard   basis [Candes and Recht, 2009] bad  spread  of  informaOon
  26. 26. Matrix  CompleOon = In + ‘noise’ Low-rank Goal:  Recover  a  matrix  from  a  subset  of  its   entries,  assuming ✦ low-­‐rank,  incoherent ✦ uniform  sampling
  27. 27. Matrix  CompleOon = In + Low-rank ✦ Nuclear-­‐norm  heurisOc +  strong  theoreOcal  guarantees +  good  empirical  results ‘noise’
  28. 28. Matrix  CompleOon = In + Low-rank ✦ Nuclear-­‐norm  heurisOc +  strong  theoreOcal  guarantees +  good  empirical  results ⎯  very  slow  computa5on ‘noise’
  29. 29. Matrix  CompleOon = In + ‘noise’ Low-rank ✦ Nuclear-­‐norm  heurisOc +  strong  theoreOcal  guarantees +  good  empirical  results ⎯  very  slow  computa5on Goal:  Scale  MC  algorithms  and  preserve  guarantees
  30. 30. Divide-­‐Factor-­‐Combine  (DFC) [MTJ, NIPS11]
  31. 31. Divide-­‐Factor-­‐Combine  (DFC) [MTJ, NIPS11] ✦ D  step:  Divide  input  matrix  into  submatrices
  32. 32. Divide-­‐Factor-­‐Combine  (DFC) [MTJ, NIPS11] ✦ D  step:  Divide  input  matrix  into  submatrices ✦ F  step:  Factor  in  parallel  using  a  base  MC  algorithm
  33. 33. Divide-­‐Factor-­‐Combine  (DFC) [MTJ, NIPS11] ✦ D  step:  Divide  input  matrix  into  submatrices ✦ F  step:  Factor  in  parallel  using  a  base  MC  algorithm ✦ C  step:  Combine  submatrix  esOmates
  34. 34. Divide-­‐Factor-­‐Combine  (DFC) [MTJ, NIPS11] ✦ D  step:  Divide  input  matrix  into  submatrices ✦ F  step:  Factor  in  parallel  using  a  base  MC  algorithm ✦ C  step:  Combine  submatrix  esOmates Advantages: ✦ Submatrix  factorizaOon  is  much  cheaper  and  easily  parallelized ✦ Minimal  communicaOon  between  parallel  jobs ✦ Retains  comparable  recovery  guarantees  (with  proper  choice   of  division  /  combinaOon  strategies)
  35. 35. DFC-­‐Proj ✦ D  step:  Randomly  parOOon  observed  entries  into  t  submatrices:
  36. 36. DFC-­‐Proj ✦ D  step:  Randomly  parOOon  observed  entries  into  t  submatrices: ✦ F  step:  Complete  the  submatrices  in  parallel ✦ Reduced  cost:  Expect  t-­‐fold  speedup  per  iteraOon ✦ Parallel  computaOon:  Pay  cost  of  one  cheaper  MC
  37. 37. DFC-­‐Proj ✦ D  step:  Randomly  parOOon  observed  entries  into  t  submatrices: ✦ F  step:  Complete  the  submatrices  in  parallel ✦ ✦ ✦ Reduced  cost:  Expect  t-­‐fold  speedup  per  iteraOon Parallel  computaOon:  Pay  cost  of  one  cheaper  MC C  step:  Project  onto  single  low-­‐dimensional  column  space
  38. 38. DFC-­‐Proj ✦ D  step:  Randomly  parOOon  observed  entries  into  t  submatrices: ✦ F  step:  Complete  the  submatrices  in  parallel ✦ Reduced  cost:  Expect  t-­‐fold  speedup  per  iteraOon ✦ Parallel  computaOon:  Pay  cost  of  one  cheaper  MC C  step:  Project  onto  single  low-­‐dimensional  column  space ✦ ✦ Roughly,  share  informaOon  across  sub-­‐soluOons
  39. 39. DFC-­‐Proj ✦ D  step:  Randomly  parOOon  observed  entries  into  t  submatrices: ✦ F  step:  Complete  the  submatrices  in  parallel ✦ Reduced  cost:  Expect  t-­‐fold  speedup  per  iteraOon ✦ Parallel  computaOon:  Pay  cost  of  one  cheaper  MC C  step:  Project  onto  single  low-­‐dimensional  column  space ✦ ✦ Roughly,  share  informaOon  across  sub-­‐soluOons ✦ Minimal  cost:  linear  in  n,  quadraOc  in  rank  of  sub-­‐soluOons
  40. 40. DFC-­‐Proj ✦ D  step:  Randomly  parOOon  observed  entries  into  t  submatrices: ✦ F  step:  Complete  the  submatrices  in  parallel ✦ Reduced  cost:  Expect  t-­‐fold  speedup  per  iteraOon ✦ Parallel  computaOon:  Pay  cost  of  one  cheaper  MC C  step:  Project  onto  single  low-­‐dimensional  column  space ✦ ✦ Roughly,  share  informaOon  across  sub-­‐soluOons ✦ Minimal  cost:  linear  in  n,  quadraOc  in  rank  of  sub-­‐soluOons =
  41. 41. DFC-­‐Proj ✦ D  step:  Randomly  parOOon  observed  entries  into  t  submatrices: ✦ F  step:  Complete  the  submatrices  in  parallel ✦ Reduced  cost:  Expect  t-­‐fold  speedup  per  iteraOon ✦ Parallel  computaOon:  Pay  cost  of  one  cheaper  MC C  step:  Project  onto  single  low-­‐dimensional  column  space ✦ ✦ Roughly,  share  informaOon  across  sub-­‐soluOons ✦ Minimal  cost:  linear  in  n,  quadraOc  in  rank  of  sub-­‐soluOons =
  42. 42. DFC-­‐Proj ✦ D  step:  Randomly  parOOon  observed  entries  into  t  submatrices: ✦ F  step:  Complete  the  submatrices  in  parallel ✦ Reduced  cost:  Expect  t-­‐fold  speedup  per  iteraOon ✦ Parallel  computaOon:  Pay  cost  of  one  cheaper  MC C  step:  Project  onto  single  low-­‐dimensional  column  space ✦ ✦ Roughly,  share  informaOon  across  sub-­‐soluOons ✦ Minimal  cost:  linear  in  n,  quadraOc  in  rank  of  sub-­‐soluOons =
  43. 43. DFC-­‐Proj ✦ D  step:  Randomly  parOOon  observed  entries  into  t  submatrices: ✦ F  step:  Complete  the  submatrices  in  parallel ✦ Reduced  cost:  Expect  t-­‐fold  speedup  per  iteraOon ✦ Parallel  computaOon:  Pay  cost  of  one  cheaper  MC C  step:  Project  onto  single  low-­‐dimensional  column  space ✦ ✦ Roughly,  share  informaOon  across  sub-­‐soluOons ✦ Minimal  cost:  linear  in  n,  quadraOc  in  rank  of  sub-­‐soluOons = =
  44. 44. DFC-­‐Proj ✦ D  step:  Randomly  parOOon  observed  entries  into  t  submatrices: ✦ F  step:  Complete  the  submatrices  in  parallel ✦ Reduced  cost:  Expect  t-­‐fold  speedup  per  iteraOon ✦ Parallel  computaOon:  Pay  cost  of  one  cheaper  MC C  step:  Project  onto  single  low-­‐dimensional  column  space ✦ ✦ ✦ ✦ Roughly,  share  informaOon  across  sub-­‐soluOons Minimal  cost:  linear  in  n,  quadraOc  in  rank  of  sub-­‐soluOons Ensemble: Project onto column space of each sub-solution and average
  45. 45. Does  It  Work? Yes,  with  high  probability. Theorem:    Assume:   ✦ L  0    is  low-­‐rank  and  incoherent,       ✦  ˜                                          entries  sampled  uniformly  at  random,   ⌦(r(n + m)) ✦ Nuclear  norm  heurisOc  is  base  algorithm.
  46. 46. Does  It  Work? Yes,  with  high  probability. Theorem:    Assume:   ✦ L  0    is  low-­‐rank  and  incoherent,       ✦  ˜                                          entries  sampled  uniformly  at  random,   ⌦(r(n + m)) ✦ Nuclear  norm  heurisOc  is  base  algorithm. ˆ             Then    L    =    L0    with  (slightly  less)  high  probability.      
  47. 47. Does  It  Work? Yes,  with  high  probability. Theorem:    Assume:   ✦ L  0    is  low-­‐rank  and  incoherent,       ✦  ˜                                          entries  sampled  uniformly  at  random,   ⌦(r(n + m)) ✦ Nuclear  norm  heurisOc  is  base  algorithm. ˆ             Then    L    =    L0    with  (slightly  less)  high  probability.       ✦ Noisy  seang:  (2          ✏)  approximaOon  of  original  bound       +       ✦ Can  divide  into  an  increasing  number  of  subproblems   ˜ (  t    !    1  )  when  number  of  observed  entries  in ! (r2 (n + m))              
  48. 48. DFC  Noisy  Recovery MC 0.25 Proj−10% Proj−Ens−10% Base−MC RMSE 0.2 0.15 0.1 0.05 0 0 2 4 6 8 10 % revealed entries ✦ Noisy recovery relative to base algorithm ( n = 10K, r = 10 )
  49. 49. DFC Speedup MC 3500 Proj−10% Proj−Ens−10% Base−MC 3000 time (s) 2500 2000 1500 1000 500 0 1 2 3 m ✦ 4 5 4 x 10 Speedup over APG for random matrices with 4% of entries revealed and r = 0.001n
  50. 50. Matrix  CompleOon NeIlix  Prize:   ✦ 100  million  raOngs  in  {1,  ...  ,  5} ✦ 18K  movies,  480K  user ✦ Issues:  Full-­‐rank;  Noisy,  non-­‐uniform   observaOons
  51. 51. Matrix  CompleOon NeIlix  Prize:   ✦ 100  million  raOngs  in  {1,  ...  ,  5} ✦ 18K  movies,  480K  user ✦ Issues:  Full-­‐rank;  Noisy,  non-­‐uniform   observaOons NeIlix Method Error Time Nuclear  Norm DFC,  t=4 DFC,  t=10 DFC-­‐Ens,  t=4 DFC-­‐Ens,  t=10 0.8433 2653.1s
  52. 52. Matrix  CompleOon NeIlix  Prize:   ✦ 100  million  raOngs  in  {1,  ...  ,  5} ✦ 18K  movies,  480K  user ✦ Issues:  Full-­‐rank;  Noisy,  non-­‐uniform   observaOons NeIlix Method Error Time Nuclear  Norm DFC,  t=4 DFC,  t=10 DFC-­‐Ens,  t=4 DFC-­‐Ens,  t=10 0.8433 0.8436 0.8484 0.8411 0.8433 2653.1s 689.5s 289.7s 689.5s 289.7
  53. 53. Matrix  CompleOon NeIlix  Prize:   ✦ 100  million  raOngs  in  {1,  ...  ,  5} ✦ 18K  movies,  480K  user ✦ Issues:  Full-­‐rank;  Noisy,  non-­‐uniform   observaOons NeIlix Method Error Time Nuclear  Norm DFC,  t=4 DFC,  t=10 DFC-­‐Ens,  t=4 DFC-­‐Ens,  t=10 0.8433 0.8436 0.8484 0.8411 0.8433 2653.1s 689.5s 289.7s 689.5s 289.7
  54. 54. Robust  Matrix  FactorizaOon [Chandrasekaran, Sanghavi, Parrilo, and Willsky, 2009; Candes, Li, Ma, and Wright, 2011; Zhou, Li, Wright, Candes, and Ma, 2010] Matrix   Comple5on = In + Low-rank ‘noise’
  55. 55. Robust  Matrix  FactorizaOon [Chandrasekaran, Sanghavi, Parrilo, and Willsky, 2009; Candes, Li, Ma, and Wright, 2011; Zhou, Li, Wright, Candes, and Ma, 2010] Matrix   Comple5on = In Principal   Component   Analysis + + ‘noise’ Low-rank = In ‘noise’ Low-rank
  56. 56. Robust  Matrix  FactorizaOon [Chandrasekaran, Sanghavi, Parrilo, and Willsky, 2009; Candes, Li, Ma, and Wright, 2011; Zhou, Li, Wright, Candes, and Ma, 2010] Matrix   Comple5on = In Principal   Component   Analysis + + In Low-rank = In ‘noise’ Low-rank = Robust  Matrix   Factoriza5on ‘noise’ + Low-rank + Sparse Outliers ‘noise’
  57. 57. Video  Surveillance ✦ Goal:  separate  foreground  from  background   ✦ ✦ ✦ Store  video  as  matrix Low-rank  =  background Outliers  =  movement
  58. 58. Video  Surveillance ✦ Goal:  separate  foreground  from  background   ✦ ✦ ✦ Store  video  as  matrix Low-rank  =  background Outliers  =  movement Original  Frame
  59. 59. Video  Surveillance ✦ Goal:  separate  foreground  from  background   ✦ ✦ ✦ Store  video  as  matrix Low-rank  =  background Outliers  =  movement Original  Frame Nuclear  Norm (342.5s)
  60. 60. Video  Surveillance ✦ Goal:  separate  foreground  from  background   ✦ ✦ ✦ Store  video  as  matrix Low-rank  =  background Outliers  =  movement Original  Frame Nuclear  Norm (342.5s) DFC-­‐5% (24.2s) DFC-­‐0.5% (5.2s)
  61. 61. Subspace  SegmentaOon [Liu, Lin, and Yu, 2010] Matrix   Comple5on = In + Low-rank ‘noise’
  62. 62. Subspace  SegmentaOon [Liu, Lin, and Yu, 2010] Matrix   Comple5on = In Principal   Component   Analysis + + ‘noise’ Low-rank = In ‘noise’ Low-rank
  63. 63. Subspace  SegmentaOon [Liu, Lin, and Yu, 2010] Matrix   Comple5on = In Principal   Component   Analysis + + ‘noise’ Low-rank = In Subspace   Segmenta5on ‘noise’ Low-rank = In + Low-rank ‘noise’
  64. 64. MoOvaOon:  Face  images
  65. 65. MoOvaOon:  Face  images ... Principal   Component   Analysis ... In
  66. 66. MoOvaOon:  Face  images ... Principal   Component   Analysis ... In = + ‘noise’ Low-rank ✦  Model  images  of  one  person  via  one  low-­‐dimensional  subspace
  67. 67. MoOvaOon:  Face  images
  68. 68. MoOvaOon:  Face  images Subspace   Segmenta5on In
  69. 69. MoOvaOon:  Face  images Subspace   Segmenta5on In
  70. 70. MoOvaOon:  Face  images Subspace   Segmenta5on In
  71. 71. MoOvaOon:  Face  images Subspace   Segmenta5on In
  72. 72. MoOvaOon:  Face  images Subspace   Segmenta5on In
  73. 73. MoOvaOon:  Face  images Subspace   Segmenta5on In
  74. 74. MoOvaOon:  Face  images Subspace   Segmenta5on = In + ‘noise’ Low-rank ✦  Model  images  of  five  people  via  five  low-­‐dimensional  subspaces
  75. 75. MoOvaOon:  Face  images Subspace   Segmenta5on = In + ‘noise’ Low-rank ✦  Model  images  of  five  people  via  five  low-­‐dimensional  subspaces ✦  Recover  subspaces                        cluster  images
  76. 76. MoOvaOon:  Face  images Subspace   Segmenta5on = In ✦ + ‘noise’ Low-rank Nuclear  norm  heurisOc  to  provably  recovers  subspaces ✦ Guarantees  are  preserved  with  DFC [TMMFJ, ICCV13]
  77. 77. MoOvaOon:  Face  images Subspace   Segmenta5on = In + ‘noise’ Low-rank ✦ Toy  Experiment:  IdenOfy  images  corresponding  to  same  person   (10  people,  640  images) ✦ DFC  Results:  Linear  speedup,  State-­‐of-­‐the-­‐art  accuracy  
  78. 78. Video  Event  DetecOon
  79. 79. Video  Event  DetecOon ✦ ✦ Input:  videos,  some  of  which  are  associated  with  events Goal:  predict  events  for  unlabeled  videos
  80. 80. Video  Event  DetecOon ✦ ✦ ✦ Input:  videos,  some  of  which  are  associated  with  events Goal:  predict  events  for  unlabeled  videos Idea: ✦ Featurize  each  video
  81. 81. Video  Event  DetecOon ✦ ✦ ✦ Input:  videos,  some  of  which  are  associated  with  events Goal:  predict  events  for  unlabeled  videos Idea: ✦ ✦ Featurize  each  video Learn  video  clusters  via  nuclear  norm  heurisOc
  82. 82. Video  Event  DetecOon ✦ ✦ ✦ Input:  videos,  some  of  which  are  associated  with  events Goal:  predict  events  for  unlabeled  videos Idea: ✦ ✦ ✦ Featurize  each  video Learn  video  clusters  via  nuclear  norm  heurisOc Given  labeled  nodes  and  cluster  structure,  make  predicOons
  83. 83. Video  Event  DetecOon ✦ ✦ ✦ Input:  videos,  some  of  which  are  associated  with  events Goal:  predict  events  for  unlabeled  videos Idea: ✦ ✦ ✦ Featurize  each  video Learn  video  clusters  via  nuclear  norm  heurisOc Given  labeled  nodes  and  cluster  structure,  make  predicOons                                            Can  do  this  at  scale  with  DFC!
  84. 84. DFC  Summary ✦ DFC:  distributed  framework  for  matrix  factorizaOon ✦ Similar  recovery  guarantees ✦ Significant  speedups   ✦ DFC  applied  to  3  classes  of  problems: ✦ Matrix  compleOon ✦ Robust  matrix  factorizaOon ✦ Subspace  recovery ✦ Extend  DFC  to  other  MF  methods,  e.g.,  ALS,  SGD?
  85. 85. Big  Data  and  Distributed  CompuOng   are  valuable  resources,  but  ...
  86. 86. Big  Data  and  Distributed  CompuOng   are  valuable  resources,  but  ... ✦ Challenge  1:  ML  not  developed  with  scalability  in  mind
  87. 87. Big  Data  and  Distributed  CompuOng   are  valuable  resources,  but  ... ✦ Challenge  1:  ML  not  developed  with  scalability  in  mind Divide-­‐and-­‐Conquer  (e.g.,  DFC)
  88. 88. Big  Data  and  Distributed  CompuOng   are  valuable  resources,  but  ... ✦ Challenge  1:  ML  not  developed  with  scalability  in  mind Divide-­‐and-­‐Conquer  (e.g.,  DFC) ✦ Challenge  2:  ML  not  developed  with  ease-­‐of-­‐use  in  mind
  89. 89. Big  Data  and  Distributed  CompuOng   are  valuable  resources,  but  ... ✦ Challenge  1:  ML  not  developed  with  scalability  in  mind ML base ML base Divide-­‐and-­‐Conquer  (e.g.,  DFC) ML base ML base ✦ Challenge  2:  ML  not  developed  with  ease-­‐of-­‐use  in  mind ML base ML base www.mlbase.org ML base

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