Iterative Join Graph Propagation

Transcription

1 Iterative Join Graph Propagation Vibhav Gogate Stat methods class dapted from Robert Mateescu s slides The University of Texas at Dallas

2 What is IJGP? IJGP is an approximate algorithm for belief updating in Bayesian networks IJGP is a version of join-tree clustering which is both anytime and iterative IJGP applies message passing along a join-graph, rather than a join-tree Empirical evaluation shows that IJGP is almost always superior to other approximate schemes (IBP, M)

3 Outline IBP - Iterative Belief Propagation M - Mini lustering IJGP - Iterative Join Graph Propagation Empirical evaluation onclusion

4 Iterative Belief Propagation - IBP Belief propagation is exact for poly-trees IBP - applying BP iteratively to cyclic networks U 1 One step: update BEL(U 1 ) X1 u ) ( 1 U2 ( x 1 ) U 2 U 3 U3 ( x 1 ) X 1 X 2 ( u 1 ) X 2 No guarantees for convergence Works well for many coding networks

5 Mini-lustering - M luster Tree Elimination Mini-lustering, i=3 h ( b, c) p( a) p( b a) p( c a, ( 1,2) b a 2 1 ) B p(a), p(b a), p(c a,b) B B B D D D 2 p(d b), p(d b), h h (1,2) (1,2) (b,c), p(f c,d) p(f c,d) h ( b, c) p( a) p( b a) p( c a, 1 ( 1,2) b a ) h ( b, f ) p( d b) h ( b, c) p( f c, d 1 ( 2,3) (1,2) ) c, d B D E 3 B sep(2,3) = {B,} elim(2,3) = {,D} B E p(e b,f) E h h 1 ( b) p( d b) h(1,2) ( b, c, d ( f ) p( f c, d 1 ( 2,3) c 2 ( 2,3) ) c, d ) G 4 E G p(g e,f)

6 TE vs. M 1 B B h( 1,2) ( b, c) 1 B B H (1,2) 1 h( 1,2) ( b, c) h 1 (2,1) ( b) 2 BD B h ( b, ( 2,1) c ) h( 2,3) ( b, f ) 2 BD B H (2,1) H (2,3) h h h 2 (2,1) 1 (2,3) 2 (2,3) ( c) ( b) ( f ) 3 BE E 4 EG h h ( b, ( 3,2) f ( e, ( 3,4) f ) ) h( 4,3) ( e, f ) 3 BE E 4 EG H (3,2) H (3,4) H (4,3) h ( b, 1 ( 3,2) f 1 h( 3,4) ( e, f 1 h( 4,3) ( e, f ) ) ) luster Tree Elimination Mini-lustering, i=3

7 IJGP - Motivation IBP is applied to a loopy network iteratively not an anytime algorithm when it converges, it converges very fast M applies bounded inference along a tree decomposition M is an anytime algorithm controlled by i-bound IJGP combines: the iterative feature of IBP the anytime feature of M

8 IJGP - The basic idea pply luster Tree Elimination to any join-graph We commit to graphs that are minimal I-maps void cycles as long as I-mapness is not violated Result: use minimal arc-labeled join-graphs

9 IJGP - Example B B B B D E BDE BE BE DE E G H DE GH H H G GH H I J GI GI GHIJ a) Belief network a) The graph IBP works on

10 rc-minimal join-graph B B B B B B BDE BE BE BDE BE DE E DE E DE GH H H DE GH H H G GH H G GH GI GI GHIJ GI GI GHIJ

11 Minimal arc-labeled join-graph B B B B B B BDE BE BDE BE DE E DE E DE GH H H DE GH H H G GH GH GI GI GHIJ GI GI GHIJ

12 Join-graph decompositions B B B BDE BE BDE B BE BDE B BE DE E DE E DE E DE GH H H DE GH DE GH GH GH GH GI GI GHIJ GI GI GHIJ GI GI GHIJ a) Minimal arc-labeled join graph b) Join-graph obtained by collapsing nodes of graph a) c) Minimal arc-labeled join graph

13 Tree decomposition BDE B BE BDE DE E DE DE DE GH GH GI GI GHIJ GHI GHI GHIJ a) Minimal arc-labeled join graph a) Tree decomposition

14 Join-graphs BDE GI B BE GHIJ DE GH H B B BE DE E H G GH H GI BDE GI B BE GHIJ DE GH H B B DE E H GH GI BDE GI BE GHIJ DE GH B DE E GH GI BDE GHI GHIJ DE DE GHI more accuracy less complexity

15 Minimal arc-labeled decomposition BDE B BE BDE B BE DE E DE E DE DE a) ragment of an arc-labeled join-graph a) Shrinking labels to make it a minimal arc-labeled join-graph Use a DS algorithm to eliminate cycles relative to each variable

16 Message propagation BDE B BE DE DE E BDE h (3,1) (bc) p(a), p(c), p(b ac), BD 1 p(d abe),p(e b,c) B 3 h(3,1)(bc) GH GH h (1,2) DE E GI GI GHIJ 2 DE Minimal arc-labeled: sep(1,2)={d,e} elim(1,2)={,b,} Non-minimal arc-labeled: sep(1,2)={,d,e} elim(1,2)={,b} h h ( de) p( a) p( c) p( b ac) p( d abe) p( e bc) h ( bc ( 1,2) (3,1) ) a, b, c ( 1,2) ( cde) p( a) p( c) p( b ac) p( d abe) p( e bc) h(3,1) ( bc) a, b

17 Bounded decompositions We want arc-labeled decompositions such that: the cluster size (internal width) is bounded by i (the accuracy parameter) the width of the decomposition as a graph (external width) is as small as possible Possible approaches to build decompositions: partition-based algorithms - inspired by the mini-bucket decomposition grouping-based algorithms

18 Partition-based algorithms G: (GE) E: (EB) (E) : (D) (B) D: (DB) (D) : (B) (B) B: (B) (B) (B) : () GE P(G,E) E EB P(E B,) P(,D) D B B D P(D B) DB B B P(,B) B B P(B ) B P() B D E G a) schematic mini-bucket(i), i=3 b) arc-labeled join-graph decomposition

19 IJGP properties IJGP(i) applies BP to min arc-labeled join-graph, whose cluster size is bounded by i On join-trees IJGP finds exact beliefs IJGP is a Generalized Belief Propagation algorithm (Yedidia, reeman, Weiss 2001) omplexity of one iteration: time: O(deg (n+n) d i+1 ) space: O(N d )

20 Empirical evaluation lgorithms: Exact IBP M IJGP Measures: bsolute error Relative error Kullback-Leibler (KL) distance Bit Error Rate Time Networks (all variables are binary): Random networks Grid networks (MxM) PS 54, 360, 422 oding networks

21 Random networks - KL at convergence Random networks, N=50, K=2, P=3, evid=0, w*=16, 100 instances 1e-2 Random networks, N=50, K=2, P=3, evid=5, w*=16, 100 instances 1e-2 KL distance 1e-3 1e-4 IJGP M IBP KL distance 1e-3 1e-4 IJGP M IBP 1e i-bound 1e i-bound evidence=0 evidence=5

22 Random networks - KL vs. iterations Random networks, N=50, K=2, P=3, evid=0, w*=16, 100 instances Random networks, N=50, K=2, P=3, evid=5, w*=16, 100 instances 1e-2 IJGP(2) IJGP(10) IBP 1e-1 IJGP(2) IJGP(10) IBP 1e-2 1e-3 KL distance KL distance 1e-3 1e-4 1e-4 1e e Number of iterations Number of iterations evidence=0 evidence=5

23 Random networks - Time Random networks, N=50, K=2, P=3, evid=5, w*=16, 100 instances IJGP 20 it M IBP 10 it Time (seconds) i-bound

24 Grid 81 KL distance at convergence 1e-2 Grid network, N=81, K=2, evid=5, w*=12, 100 instances KL distance 1e-3 1e-4 IJGP M IBP 1e-5 1e i-bound

25 Grid 81 KL distance vs. iterations 1e-1 Grid network, N=81, K=2, evid=5, w*=12, 100 instances 1e-2 IJGP(2) IJGP(8) IBP KL distance 1e-3 1e-4 1e Number of iterations

26 Grid 81 Time vs. i-bound 0.6 Grid network, N=81, K=2, evid=5, w*=12, 100 instances IJGP 10 it (at convergence) M IBP 10 it (at convergence) Time (seconds) i-bound

27 oding networks N=400, P=4, 500 instances, 30 iterations, w*=43

28 oding networks - BER oding, N=400, 1000 instances, 30 it, w*=43, sigma=.22 oding, N=400, 500 instances, 30 it, w*=43, sigma=.32 1e e-2 IJGP M IBP IBP IJGP BER 1e-3 BER e e i-bound i-bound sigma=.22 sigma=.32 oding, N=400, 500 instances, 30 it, w*=43, sigma=.51 oding, N=400, 500 instances, 30 it, w*=43, sigma= IBP IJGP IBP IJGP BER BER i-bound i-bound sigma=.51 sigma=.65

29 oding networks - Time 10 oding, N=400, 500 instances, 30 iterations, w*=43 Time (seconds) IJGP 30 iterations M IBP 30 iterations i-bound

30 PS 422 KL distance PS 422, evid=0, w*=23, 1instance PS 422, evid=30, w*=23, 1instance 0.1 IJGP 30 it (at convergence) M IBP 10 it (at convergence) KL distance KL distance IJGP at convergence M IBP at convergence i-bound i-bound evidence=0 evidence=30

31 PS 422 KL vs. iterations PS 422, evid=0, w*=23, 1instance PS 422, evid=30, w*=23, 1instance IJGP (3) IJGP(10) IBP 0.1 IJGP(3) IJGP(10) IBP 0.01 KL distance KL distance number of iterations number of iterations evidence=0 evidence=30

32 PS 422 Relative error 100 PS 422, evid=30, w*=23, 1instance Relative error 10 1 IJGP at convergence M IBP at convergence i-bound evidence=30

33 PS 422 Time vs. i-bound 1200 PS 422, evid=0, w*=23, 1instance IJGP 30 it (at convergence) M IBP 10 it (at convergence) Time (seconds) i-bound evidence=0

34 onclusion IJGP borrows the iterative feature from IBP and the anytime virtues of bounded inference from M Empirical evaluation showed the potential of IJGP, which improves with iteration and most of the time with i-bound, and scales up to large networks IJGP is almost always superior, often by a high margin, to IBP and M Based on all our experiments, we think that IJGP provides a practical breakthrough to the task of belief updating