Extrapolation

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

Motivation Problem: Speed up PageRank Motivation: Personalization “ Freshness” 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 Definition of PageRank Computation of PageRank Convergence Properties Outline of Our Approach Empirical Results 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 The importance of a page is given by the importance of the pages that link to it. 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 Initialize: Repeat until convergence: 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 Initialize: Repeat until convergence:

PageRank doesn’t actually use P T . Instead, it uses A=cP T + (1-c)E T . So the PageRank problem is really: not: A side note Find x that satisfies: Find x that satisfies:

Power Method And the algorithm is really . . . Initialize: Repeat until convergence:

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? Imagine our n x n matrix A has n distinct eigenvectors u i . u 1 1 u 2  2 u 3  3 u 4  4 u 5  5 Then, you can write any n -dimensional vector as a linear combination of the eigenvectors of A .

Why does it work? From the last slide: To get the first iterate, multiply x (0) by A . First eigenvalue is 1. Therefore: 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

The smaller  2 , the faster the convergence of the Power Method. 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? For traditional problems: A is smaller, often dense.  2 often close to   , making the power method slow. In our problem, A is huge and sparse More importantly,  2 is small 1 . Therefore, Power method is actually much faster than other methods. 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? Assume x (k) can be written as a linear combination of the first three eigenvectors ( u 1 , u 2 , u 3 ) of A. Compute approximation to { u 2 , u 3 }, and subtract it from x (k) to get x (k) ’

Assume Assume the x (k) can be represented by first 3 eigenvectors of A

Linear Combination Let’s take some linear combination of these 3 iterates.

Rearranging Terms We can rearrange the terms to get: Goal: Find  1 ,  2 ,  3 so that coefficients of u 2 and u 3 are 0, and coefficient of u 1 is 1.

Summary We make an assumption about the current iterate. Solve for dominant eigenvector as a linear combination of the next three iterates. We use a few iterations of the Power Method to “clean it up”.

Outline Definition of PageRank Computation of PageRank Convergence Properties Outline of Our Approach Empirical Results 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 Speeds up PageRank by a fair amount, but not by enough for true Personalized PageRank. Ideas are useful for further speedup algorithms. Quadratic Extrapolation can be used for a whole class of problems.

The End Paper available at http://dbpubs.stanford.edu/pub/2003-16

Extrapolation

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