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Matrix Factorization

Matrix Factorization. Recovering latent factors in a matrix. m movies. n users. V[ i,j ] = user i’s rating of movie j. Recovering latent factors in a matrix. m movies. m movies. ~. n users. V[ i,j ] = user i’s rating of movie j. KDD 2011. talk pilfered from  ….

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Matrix Factorization

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  1. Matrix Factorization

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

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

  4. KDD 2011 talk pilfered from  …..

  5. 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

  6. for image denoising

  7. Matrix factorization as SGD step size

  8. Matrix factorization as SGD - why does this work? step size

  9. Matrix factorization as SGD - why does this work? Here’s the key claim:

  10. Checking the claim • Think for SGD for logistic regression • LR loss = compare y and ŷ= dot(w,x) • similar but now update w (user weights) and x (movie weight)

  11. What loss functions are possible? N1, N2 - diagonal matrixes, sort of like IDF factors for the users/movies “generalized” KL-divergence

  12. What loss functions are possible?

  13. What loss functions are possible?

  14. ALS = alternating least squares

  15. KDD 2011 talk pilfered from  …..

  16. Similar to McDonnell et al with perceptron learning

  17. Slow convergence…..

  18. More detail…. • Randomly permute rows/cols of matrix • Chop V,W,H into blocks of size d x d • m/d blocks in W, n/d blocks in H • Group the data: • Pick a set of blocks with no overlapping rows or columns (a stratum) • Repeat until all blocks in V are covered • Train the SGD • Process strata in series • Process blocks within a stratum in parallel

  19. More detail…. Z was V

  20. 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

  21. Wall Clock Time8 nodes, 64 cores, R/snow

  22. Number of Epochs

  23. Varying rank100 epochs for all

  24. Hadoop scalability Hadoop process setup time starts to dominate

  25. Hadoop scalability

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