Matrix Factorization via SGD

1. Matrix Factorizationvia SGD

2. Background

4. Recovering latent factors in a matrix r m movies m movies ~ H W V n users V[i,j] = user i’s rating of movie j

7. MF VIA SGD

8. Matrix factorization as SGD local gradient …scaled up by N to approximate gradient step size

9. Key claim:

10. What loss functions are possible?

12. ALS = alternating least squares

13. Distributed MF Via SGD

14. KDD 2011 talk pilfered from  …..

15. NAACL 2010

16. Parallel Perceptrons • Simplest idea: • Split data into S “shards” • Train a perceptron on each shard independently • weight vectors are w(1) , w(2) , … • Produce some weighted average of the w(i)‘s as the final result

17. Parallelizing perceptrons Instances/labels Split into example subsets Instances/labels – 1 Instances/labels – 2 Instances/labels – 3 Compute vk’s on subsets vk -1 vk- 2 vk-3 Combine by some sort of weighted averaging vk

18. Parallel Perceptrons – take 2 • Idea: do the simplest possible thing iteratively. • Split the data into shards • Let w = 0 • For n=1,… • Train a perceptron on each shard with one passstarting with w • Average the weight vectors (somehow) and let wbe that average • Extra communication cost: • redistributing the weight vectors • done less frequently than if fully synchronized, more frequently than if fully parallelized All-Reduce

19. Parallelizing perceptrons – take 2 Instances/labels Split into example subsets w (previous) Instances/labels – 1 Instances/labels – 2 Instances/labels – 3 Compute local vk’s w -1 w- 2 w-3 Combine by some sort of weighted averaging w

23. Similar to McDonnell et al with perceptron learning

24. Slow convergence…..

31. More detail…. M= • Initialize W,H randomly • not at zero  • Choose a random ordering (random sort) of the points in a stratum in each “sub-epoch” • Pick strata sequence by permuting rows and columns of M, and using M’[k,i] as column index of row i in subepoch k • Use “bold driver” to set step size: • increase step size when loss decreases (in an epoch) • decrease step size when loss increases • Implemented in Hadoop and R/Snowfall

33. Wall Clock Time8 nodes, 64 cores, R/snow In-memory implementation

38. Number of Epochs

43. Varying rank100 epochs for all

44. Wall Clock TimeHadoop One map-reduce job per epoch

45. Hadoop scalability Hadoop process setup time starts to dominate

46. Hadoop scalability