Extrapolation

Extrapolation Methods for Accelerating PageRank Computations Sepandar D. Kamvar Taher H. Haveliwala Christopher D. Manning Gene H. Golub Stanford University

Motivation ,[object Object],[object Object],[object Object],[object Object],[object Object],Note: PageRank Computations don’t get faster as computers do. Results: 1. The Official Site of the San Francisco Giants Search: Giants Results: 1. The Official Site of the New York Giants

Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],0.4 0.2 0.4 Repeat: u 1 u 2 u 3 u 4 u 5 u 1 u 2 u 3 u 4 u 5

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

Definition of PageRank ,[object Object],importance of page i pages j that link to page i number of outlinks from page j importance of page j

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

PageRank Diagram Initialize all nodes to rank 0.333 0.333 0.333

PageRank Diagram Propagate ranks across links (multiplying by link weights) 0.167 0.167 0.333 0.333

PageRank Diagram 0.333 0.5 0.167

PageRank Diagram 0.167 0.167 0.5 0.167

PageRank Diagram 0.5 0.333 0.167

PageRank Diagram After a while… 0.4 0.4 0.2

Computing PageRank ,[object Object],[object Object],importance of page i pages j that link to page i number of outlinks from page j importance of page j

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

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

Power Method ,[object Object],[object Object]

[object Object],[object Object],[object Object],A side note Find x that satisfies: Find x that satisfies:

Power Method ,[object Object],[object Object],[object Object]

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

Power Method u 1 1 u 2  2  2 u 3  3  3 u 4  4  4 u 5  5  5

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

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

Power Method u 1 1 u 2  u 3  u 4  u 5 

Why does it work? ,[object Object],u 1 1 u 2  2 u 3  3 u 4  4 u 5  5 ,[object Object]

Why does it work? ,[object Object],[object Object],[object Object],[object Object],All less than 1

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

[object Object],Convergence u 1 1 u 2  2  2 k u 3  3  3 k u 4  4  4 k u 5  5  5 k

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.

Why this approach? ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],1 (“The Second Eigenvalue of the Google Matrix” dbpubs.stanford.edu/pub/2003-20.)

Using Successive Iterates u 1 x (0) u 1 u 2 u 3 u 4 u 5

Using Successive Iterates u 1 x (1) x (0) u 1 u 2 u 3 u 4 u 5

Using Successive Iterates u 1 x (1) x (0) x (2) u 1 u 2 u 3 u 4 u 5

Using Successive Iterates x (0) u 1 x (1) x (2) u 1 u 2 u 3 u 4 u 5

Using Successive Iterates x (0) x’ = u 1 x (1) u 1 u 2 u 3 u 4 u 5

How do we do this? ,[object Object],[object Object]

Linear Combination ,[object Object]

Rearranging Terms ,[object Object],Goal: Find  1 ,  2 ,  3 so that coefficients of u 2 and u 3 are 0, and coefficient of u 1 is 1.

Summary ,[object Object],[object Object],[object Object]

Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],u 1 u 2 u 3 u 4 u 5 u 1 u 2 u 3 u 4 u 5 0.4 0.2 0.4 Repeat:

Results Quadratic Extrapolation speeds up convergence. Extrapolation was only used 5 times!

Results Extrapolation dramatically speeds up convergence, for high values of c (c=.99)

Take-home message ,[object Object],[object Object],[object Object]

Extrapolation

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