Extrapolation
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The document proposes methods to accelerate PageRank computations using extrapolation techniques. It discusses how PageRank works and is typically computed using an iterative power method. The authors' approach is to use successive PageRank vectors to estimate the components in the directions of the first few eigenvectors, subtracting them to remove their influence and speed convergence. Empirical results show quadratic extrapolation can significantly speed up PageRank convergence, though not enough for truly personalized computations. The extrapolation techniques may help accelerate other similar iterative methods.
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NFSFIXES
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D026017036
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Reference. Machine Learning: The art and science of algorithms that make sense of data
Paper Study: Melding the data decision pipeline
Melding the data decision pipeline: Decision-Focused Learning for Combinatorial Optimization from AAAI2019.
Derive the math equation from myself and match the same result as two mentioned CMU papers [Donti et. al. 2017, Amos et. al. 2017] while applying the same derivation procedure.
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New approach for wolfe’s modified simplex method to solve quadratic programmi...
Abstract
In this paper, an alternative method for Wolfe’s modified simplex method is introduced. This method is easy to solve quadratic programming problem (QPP) concern with non-linear programming problem (NLPP). In linear programming models, the characteristic assumption is the linearity of the objective function and constraints. Although this assumption holds in numerous practical situations, yet we come across many situations where the objective function and some or all of the constraints are non-linear functions. The non-linearity of the functions makes the solution of the problem much more involved as compared to LPPs and there is no single algorithm like the simplex method, which can be employed to solve efficiently all NPPs.
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Data Analysis Homework Help
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A talk I gave at the SIAM Annual Meeting Mini-symposium on the mathematics of the power grid organized by Mahantesh Halappanavar. I discuss a few ideas on how our dynamic centrality could help analyze such situations.
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Extrapolation
- 1. Extrapolation Methods for Accelerating PageRank Computations Sepandar D. Kamvar Taher H. Haveliwala Christopher D. Manning Gene H. Golub Stanford University
- 4. Link Counts Linked by 2 Important Pages Linked by 2 Unimportant pages Sep’s Home Page Taher’s Home Page Yahoo! CNN DB Pub Server CS361
- 6. Definition of PageRank Yahoo! CNN DB Pub Server Taher Sep 1/2 1/2 1 1 0.1 0.1 0.1 0.05 0.25
- 7. PageRank Diagram Initialize all nodes to rank 0.333 0.333 0.333
- 8. PageRank Diagram Propagate ranks across links (multiplying by link weights) 0.167 0.167 0.333 0.333
- 9. PageRank Diagram 0.333 0.5 0.167
- 10. PageRank Diagram 0.167 0.167 0.5 0.167
- 11. PageRank Diagram 0.5 0.333 0.167
- 12. PageRank Diagram After a while… 0.4 0.4 0.2
- 14. Matrix Notation 0 .2 0 .3 0 0 .1 .4 0 .1 = .1 .3 .2 .3 .1 .1 .2 . 1 .3 .2 .3 .1 .1
- 15. Matrix Notation Find x that satisfies: . 1 .3 .2 .3 .1 .1 0 .2 0 .3 0 0 .1 .4 0 .1 = .1 .3 .2 .3 .1 .1 .2
- 20. Power Method u 1 1 u 2 2 u 3 3 u 4 4 u 5 5 Express x (0) in terms of eigenvectors of A
- 21. Power Method u 1 1 u 2 2 2 u 3 3 3 u 4 4 4 u 5 5 5
- 22. Power Method u 1 1 u 2 2 2 2 u 3 3 3 2 u 4 4 4 2 u 5 5 5 2
- 23. Power Method u 1 1 u 2 2 2 k u 3 3 3 k u 4 4 4 k u 5 5 5 k
- 24. Power Method u 1 1 u 2 u 3 u 4 u 5
- 27. Power Method u 1 1 u 2 2 u 3 3 u 4 4 u 5 5 u 1 1 u 2 2 2 u 3 3 3 u 4 4 4 u 5 5 5 u 1 1 u 2 2 2 2 u 3 3 3 2 u 4 4 4 2 u 5 5 5 2
- 29. Our Approach u 1 u 2 u 3 u 4 u 5 Estimate components of current iterate in the directions of second two eigenvectors, and eliminate them.
- 31. Using Successive Iterates u 1 x (0) u 1 u 2 u 3 u 4 u 5
- 32. Using Successive Iterates u 1 x (1) x (0) u 1 u 2 u 3 u 4 u 5
- 33. Using Successive Iterates u 1 x (1) x (0) x (2) u 1 u 2 u 3 u 4 u 5
- 34. Using Successive Iterates x (0) u 1 x (1) x (2) u 1 u 2 u 3 u 4 u 5
- 35. Using Successive Iterates x (0) x’ = u 1 x (1) u 1 u 2 u 3 u 4 u 5
- 42. Results Quadratic Extrapolation speeds up convergence. Extrapolation was only used 5 times!
- 43. Results Extrapolation dramatically speeds up convergence, for high values of c (c=.99)
Editor's Notes
- Why does the Power Method Work?
- Assume that lambda 1 is less than 1 and all other eigenvalues are strictly less than 1.
- Here, talk about in the past, how lambda 2 is often close to 1, so the power method is not useful. However, in our case,
- Note : derivation given here is slightly different from what’s in the paper the one here is perhaps more intuitive the one in the paper is more compact