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Algorithms for Reasoning with graphical models Slides Set 9: AND/OR search for Probabilistic Networks Rina Dechter (Dechter1 chapter 6 and 7 ) slides9 828X 2019 Algorithms for Reasoning with graphical models Class6: AND/OR search for

  1. Arc Weights for AND/OR Trees ( | , ) ( | ) ( | ) ( A ) A A P E A B P B A P C A P A B=0 B=1 A C=0 C=1 A P(A) A B E=0 E=1 B 0 .4 .6 0 .2 .8 0 .6 0 0 .4 .6 B C 1 .1 .9 1 .7 .3 1 .4 0 1 .5 .5 E C 1 0 .7 .3 1 1 .2 .8 D E D Evidence: E=0 OR A .6 .4 AND 0 1 OR B B .4 .6 .1 .9 AND 0 1 0 1 OR E C E C E C E C .4 .2 .8 .5 .2 .8 .7 .1 .9 .2 .1 .9 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D D D D D D D D .8 .9 .5 .8 .9 .7 .5 .7 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ( | , ) P D B C OR to AND arc weight <X,x> is the B C D=0 D=1 0 0 .2 .8 product of factors that all their 0 1 .1 .9 arguments are just assigned at AND 1 0 .3 .7 1 1 .5 .5 node X=x but not before Evidence: D=1 slides9 828X 2019

  2. Cost of a Solution Tree ( | , ) ( | ) ( | ) ( A ) A A P E A B P B A P C A P A B=0 B=1 A C=0 C=1 A P(A) A B E=0 E=1 B 0 .4 .6 0 .2 .8 0 .6 0 0 .4 .6 B C 1 .1 .9 1 .7 .3 1 .4 0 1 .5 .5 E C 1 0 .7 .3 1 1 .2 .8 D E D Evidence: E=0 OR A .6 .4 AND 0 1 OR B B .4 .6 .1 .9 AND 0 1 0 1 OR E C E C E C E C .4 .2 .8 .5 .2 .8 .7 .1 .9 .2 .1 .9 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D D D D D D D D .8 .9 .5 .8 .9 .7 .5 .7 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 A solution tree includes the root and has a single child for ( | , ) P D B C any OR node, and all children of any of its AND nodes B C D=0 D=1 0 0 .2 .8 0 1 .1 .9 Cost of the solution tree: the product of weights on its arcs 1 0 .3 .7 1 1 .5 .5 Cost of (A=0,B=1,C=1,D=1,E=0) = 0.6 ∙ 0.6 ∙ 0.5 ∙ 0.8 ∙ 0.5 = 0.0720 Evidence: D=1

  3. Arc Weights for AND/OR Trees OR to AND arc weight (X,x) is the product of factors f that are instantiated at the AND node X=x but not before A P(A=0) 0 B P(B=0|A=0) P(B=1|A=0) 0 1 E D E D P(E=0|A=0,B=0) P(E=1|A=0,B=0) P(E=0|A=0,B=1) P(E=1|A=0,B=1) 0 1 0 1 0 1 0 1 C C C C P(D=0|B=0,C=0)× P(D=0|B=0,C=1)× P(D=1|B=0,C=0)× P(D=1|B=0,C=1)× P(D=0|B=1,C=0)× P(D=0|B=1,C=1)× P(D=1|B=1,C=1)× P(D=1|B=1,C=0)× P(C=0|A=0) P(C=1|A=0) P(C=0|A=0) P(C=1|A=0) P(C=0|A=0) P(C=1|A=0) P(C=1|A=0) P(C=0|A=0) 0 1 0 1 0 1 0 1 slides9 828X 2019

  4. The Value Function for (Probability of Evidence) ( | , ) ( | ) ( | ) ( A ) A A P E A B P B A P C A P A B=0 B=1 A C=0 C=1 A P(A) A B E=0 E=1 B 0 .4 .6 0 .2 .8 0 .6 0 0 .4 .6 B C 1 .1 .9 1 .7 .3 1 .4 0 1 .5 .5 E C 1 0 .7 .3 1 1 .2 .8 P(D=1,E=0)=? D E D Evidence: E=0 OR A .6 .4 AND 0 1 OR B B .4 .6 .1 .9 OR AND 0 1 0 1 OR E C E .54 C E C E C .4 .2 .8 .5 .2 .8 .7 .1 .9 .2 .1 .9 AND .7 .5 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D D .7 .5 D D . D D D D .8 .9 .5 .8 .9 .7 .5 .7 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ( | , ) P D B C Value of node = updated belief for sub‐problem below B C D=0 D=1 0 0 .2 .8 𝑤�𝑜 � � � AND node: product 0 1 .1 .9 � � ∈����������� 1 0 .3 .7 𝑥 𝑜, 𝑜 � 𝑤�𝑜 � � � 1 1 .5 .5 OR node: Marginalization by summation ��∈����������� Evidence: D=1,E=0 slides9 828X 2019

  5. The Value Function (Probability of Evidence) ( | , ) ( | ) ( | ) ( A ) A A P E A B P B A P C A P A B=0 B=1 A C=0 C=1 A P(A) A B E=0 E=1 B 0 .4 .6 0 .2 .8 0 .6 0 0 .4 .6 B C 1 .1 .9 1 .7 .3 1 .4 0 1 .5 .5 E C 1 0 .7 .3 1 1 .2 .8 P(D=1,E=0)=? D E D Evidence: E=0 OR .24408 A .6 .4 AND AND .3028 0 .1559 1 OR .3028 .1559 B B .4 .6 .1 .9 OR .352 (=.4 · . 88) AND .623 0 .27 1 0 .104 1 OR .5 .7 .4 E .88 C E .54 C E .89 C .2 E .52 C .4 .2 .8 .5 .2 .8 .7 .1 .9 .2 .1 .9 AND .8 .9 .7 .5 .8 .9 .7 .5 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR .8 D D .9 .7 .5 .8 D D .9 .7 .5 D D D D .8 .9 .5 .8 .9 .7 .5 .7 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ( | , ) P D B C Value of node = updated belief for sub‐problem below B C D=0 D=1 0 0 .2 .8 𝑤�𝑜 � � � AND node: product 0 1 .1 .9 � � ∈����������� 1 0 .3 .7 𝑥 𝑜, 𝑜 � 𝑤�𝑜 � � � 1 1 .5 .5 OR node: Marginalization by summation ��∈����������� Evidence: D=1 slides9 828X 2019

  6. The Value Function ( | , ) ( | ) ( | ) ( A ) A A P E A B P B A P C A P A B=0 B=1 A C=0 C=1 A P(A) A B E=0 E=1 B 0 .4 .6 0 .2 .8 0 .6 0 0 .4 .6 B C 1 .1 .9 1 .7 .3 1 .4 0 1 .5 .5 E C 1 0 .7 .3 1 1 .2 .8 D E D Evidence: E=0 OR A .6 .4 AND 0 1 OR B B .4 .6 .1 .9 AND 0 1 0 1 OR E C E C E C E C .4 .2 .8 .5 .2 .8 .7 .1 .9 .2 .1 .9 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D D D D D D D D .8 .9 .5 .8 .9 .7 .5 .7 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ( | , ) P D B C • B C D=0 D=1 V(n) is dictated by the query of interest 0 0 .2 .8 • V(n) the value of the sub‐problem represented by T(n) 0 1 .1 .9 1 0 .3 .7 • For sum‐inference it is the probability mess below n 1 1 .5 .5 • Can be computed recursively based on child values. Evidence: D=1,E=0 slides9 828X 2019

  7. The Value Function ( | , ) ( | ) ( | ) ( A ) A A P E A B P B A P C A P A B=0 B=1 A C=0 C=1 A P(A) A B E=0 E=1 B 0 .4 .6 0 .2 .8 0 .6 0 0 .4 .6 B C 1 .1 .9 1 .7 .3 1 .4 0 1 .5 .5 E C 1 0 .7 .3 1 1 .2 .8 D E D Evidence: E=0 OR A .6 .4 AND 0 1 OR B B .4 .6 .1 .9 AND 0 1 0 1 OR E C E C E C E C .4 .2 .8 .5 .2 .8 .7 .1 .9 .2 .1 .9 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D D D D D D D D .8 .9 .5 .8 .9 .7 .5 .7 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ( | , ) P D B C • B C D=0 D=1 V(n) is dictated by the query of interest 0 0 .2 .8 • V(n) the value of the sub‐problem represented by T(n) 0 1 .1 .9 1 0 .3 .7 • For sum‐inference it is the probability mess below n 1 1 .5 .5 • Can be computed recursively based on child values. Evidence: D=1 slides9 828X 2019

  8. The Value Function for Optimization A A B f 1 A C f 2 A E f 3 A F f 4 B C f 5 B D f 6 B E f 7 C D f 8 E F f 9 0 0 2 0 0 3 0 0 0 0 0 2 0 0 0 0 0 4 0 0 3 0 0 1 0 0 1 C A F 0 1 0 0 1 0 0 1 3 0 1 0 0 1 1 0 1 2 0 1 2 0 1 4 0 1 0 1 0 1 1 0 0 1 0 2 1 0 0 1 0 2 1 0 1 1 0 1 1 0 0 1 0 0 B 1 1 4 1 1 1 1 1 0 1 1 2 1 1 4 1 1 0 1 1 0 1 1 0 1 1 2 D B E C E 𝐺 ∗ � min � � 𝑔 � �𝑦 � � Objective function: � D F 5 OR w(A,0) = 0 w(A,1) = 0 A AND 5 0 1 OR 5 B 2 0 Node Value AND 6 5 0 1 (bottom‐up evaluation) 3 4 C C OR 3 1 E E 3 1 3 5 5 4 1 3 OR – minimization 0 1 AND 5 0 2 1 0 2 2 0 0 2 0 1 0 1 AND – summation 5 2 0 2 2 0 0 2 OR D D D D F F F F 5 6 4 2 3 0 2 2 2 4 1 0 3 0 2 2 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 slides9 828X 2019

  9. The Value Function for Optimization A A B f 1 A C f 2 A E f 3 A F f 4 B C f 5 B D f 6 B E f 7 C D f 8 E F f 9 0 0 2 0 0 3 0 0 0 0 0 2 0 0 0 0 0 4 0 0 3 0 0 1 0 0 1 C A F 0 1 0 0 1 0 0 1 3 0 1 0 0 1 1 0 1 2 0 1 2 0 1 4 0 1 0 1 0 1 1 0 0 1 0 2 1 0 0 1 0 2 1 0 1 1 0 1 1 0 0 1 0 0 B 1 1 4 1 1 1 1 1 0 1 1 2 1 1 4 1 1 0 1 1 0 1 1 0 1 1 2 D B E C E 𝐺 ∗ � min � � 𝑔 � �𝑦 � � Objective function: � D F 5 OR A 0 0 5 AND 7 0 1 OR 5 7 B B 2 0 1 4 6 5 6 4 AND 0 1 0 1 3 4 C 4 4 C C 3 C OR 1 2 E E E 0 E 3 1 3 5 5 4 1 3 0 2 5 2 2 5 3 0 5 2 0 2 2 0 0 1 0 2 5 2 1 0 2 0 0 1 1 0 AND 0 1 0 1 0 1 0 1 0 1 0 1 5 2 0 2 2 0 0 2 5 2 1 0 2 0 1 0 D D D D D D OR F F D D F F F F F F AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 5 6 4 2 3 0 2 2 2 4 1 0 3 0 2 2 5 6 4 2 1 2 0 4 2 4 1 0 1 2 0 4 AND node = Combination operator (summation) OR node = Marginalization operator (minimization)

  10. The AND/OR Counting Value(#CSP) A A F B B C solution E C E D D F OR 11 A AND 5 6 0 1 OR 5 B B 6 OR node: Marginalization operator (summation) AND 4 1 0 0 4 2 1 1 OR 2 2 1 E C 1 E C 4 2 E C 1 E 1 C AND node: Combination operator (product) AND 2 0 0 0 1 0 1 1 0 1 0 1 2 1 0 1 0 1 2 0 1 0 1 1 1 1 1 0 1 0 0 1 OR 2 1 2 0 1 1 1 D F D F 0 D 2 F 1 D 2 F 1 1 1 D F D F 1 D F D F 1 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 0 1 1 0 0 1 0 1 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 0 Value of node = number of solutions below it slides9 828X 2019

  11. Summary: AND/OR Search Tree for GMs • The AND/OR search tree of R relative to a pseudo‐tree, T, has: – Alternating levels of: OR nodes (variables) and AND nodes (values) • Successor function: – The successors of OR nodes X are all its consistent values along its path A – The successors of AND <X,v> are all X child variables in T F – Arc‐weight are assigned from the model factors B C • E D A solution is a consistent subtree. Its cost, the product of the weights. • Query: compute the value of the root node A B E C OR A D F AND 0 1 OR B B AND 0 1 0 1 OR E C E C E C E C AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D F D F D F D F D F D F D F D F AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 slides9 828X 2019

  12. Size and Traversal of AND/OR Search Tree AND/OR tree OR tree Space O(n) O(n) O(n k h ) O(k n ) Size= O(n k w* log n ) Time (Freuder & Quinn85), (Collin, Dechter & Katz91), (Bayardo & Miranker95), (Darwiche01) k = domain size h= height of pseudo‐tree ∗ h n = number of variables w*= treewidth slides9 828X 2019

  13. AND/OR vs. OR Spaces OR space AND/OR space width height Time (sec.) Nodes Time (sec.) AND nodes OR nodes 5 10 3.15 2,097,150 0.03 10,494 5,247 4 9 3.13 2,097,150 0.01 5,102 2,551 5 10 3.12 2,097,150 0.03 8,926 4,463 4 10 3.12 2,097,150 0.02 7,806 3,903 5 13 3.11 2,097,150 0.10 36,510 18,255 Random graphs with 20 nodes, 20 edges and 2 values per node slides9 828X 2019

  14. Pseudo‐trees slides9 828X 2019

  15. PseudoTree Definition Given undirected graph G = (V, E), a directed rooted tree T = (V, E′) defined on all its nodes is a pseudo tree if any arc of G which is not included in E′ is a back-arc in T , namely it connects a node in T to an ancestor in T . The arcs in E′ may not all be included in E. Given a pseudo tree T of G, the extended graph of G relative to T includes also the arcs in E′ that are not in E: as GT = (V, E ∪ E′). slides9 828X 2019

  16. Pseudo Trees A psedo‐tree of a graph is a tree spanning its nodes, where all arcs in the graph not in the tree are back‐arcs 1 4 1 6 2 3 2 7 5 3 ∗ h (a) Graph 4 7 1 1 2 7 5 3 5 3 5 4 2 7 6 6 4 6 (b) DFS tree (c) Pseudo tree (d) Chain height=3 height=2 height=6 slides9 828X 2019

  17. Extended Graphs Question: given, n,k,w,h,b develop and expression that study the size of the AND/OR search tree As a function of these parameters, which are not independent of each other slides9 828X 2019

  18. From DFS‐Trees to Pseudo‐Trees 4 1 6 6 7 2 4 6 4 5 3 3 5 2 7 1 1 3 2 7 5 (c) (b) (a) 237 AND nodes 1 OR a b c AND 2 7 2 7 2 7 OR b c b c a c a c a b a b AND 3 3 5 5 3 3 5 5 3 3 5 5 OR a c a b a c a b b c a b b c a b b c a c b c a c AND OR 4 4 4 4 6 6 6 6 4 4 4 4 6 6 6 6 4 4 4 4 6 6 6 6 b c b b c c b c b b c c a c a c a c a c a c a c a a b b a b a a b b a b AND 1 OR 108 AND nodes a b c AND 3 5 3 5 3 5 OR a b c a b c a b c a b c a b c a b c AND 4 2 4 2 4 2 7 6 7 6 7 6 4 2 4 2 4 2 7 6 7 6 7 6 4 2 4 2 4 2 7 6 7 6 7 6 OR AND b c b c c c b b b c b c c c b b c c a c a c a a c c a c a c a a b b a a a b a b b b a a a b a b slides9 828X 2019

  19. From DFS to Pseudo Trees OR 1 AND a b c OR 2 7 AND a b c a b c DFS tree OR 3 3 3 5 5 5 AND a b c a b c a b c a b c a b c a b c depth = 3 OR 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 AND a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c OR 1 AND a b c OR 3 5 AND a b c a b c pseudo- tree OR 4 2 4 2 4 2 7 6 7 6 7 6 depth = 2 AND a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c slides9 828X 2019

  20. Finding Min‐Depth Pseudo‐Trees • Finding min depth DFS, or pseudo tree is NP‐ complete, but: • Given a tree‐decomposition whose treewidth is w*, there exists a pseudo ‐tree T of G whose depth, satisfies h <= w* log n, slides9 828X 2019

  21. Finding Min‐height Pseudo‐Trees • Finding a min height pseudo‐tree is NP‐complete, but: • Given a tree‐decomposition with treewidth w*, there exists a pseudo‐tree whose height satisfies – h <= w* log n • Optimality of h and w* cannot be achieved at once. 4 1 2 2 6 3 3 7 5 1 W*=1 4 h=7 5 6 w*=2 h=3 7 slides9 828X 2019

  22. AND/OR Search‐tree properties (k = domain size, h = pseudo‐tree height. n = number of variables) • Theorem: Any AND/OR search tree based on a pseudo‐tree is sound and complete (expresses all and only solutions) • Theorem: Size of AND/OR search tree is O(n k h ) Size of OR search tree is O(k n ) • Theorem: Size of AND/OR search tree can be bounded by O(exp(w* log n)) • When the pseudo‐tree is a chain we get an OR space slides9 828X 2019

  23. Summary: Queries and Value of Nodes • V( n) is the value of the tree T(n) for the task: – Counting: v(n) is number of solutions in T(n) – Consistency: v(n) is 0 if T(n) inconsistent, 1 othewise. – Max‐Inference: v(n) is the optimal solution in T(n) – Sum‐Inference: v(n) is probability of evidence in T(n). – Mixed‐Inference: v(n) is the marginal map in T(n). • Goal : compute the value of the root node recursively traversing the AND/OR tree. Complexity of searching depth‐first is – Space: O(n) O(n 𝑙 � ) – Time: – Time: 𝑃�𝑙 �∗���� � slides9 828X 2019

  24. Outline: Search • Review Graphical Modes OR A • AND/OR search spaces, pseudo‐trees AND 0 1 OR B B – AND/OR search trees AND 0 1 0 1 OR C C C C E E E E AND 0 1 0 1 0 1 0 1 – AND/OR search graphs OR D D F F D D F F AND 01 01 – Generating good pseudo‐trees Context minimal AND/OR search graph – Brute‐force AND/OR 18 AND nodes • Heuristic search (HS) for AND/OR spaces – Basic Heuristic search (Depth and Best) – AND/OR Depth‐first HS (branch and bound) – AND/OR Best‐first heuristic search – The Guiding MBE heuristic – Marginal Map (max‐sum‐product) A=yellow A=green • Hybrids of search and Inference E D M B=blue B=green B=red B=blue L C A E D M E D M E D M E D M • Summary and Class 2 L L L L B K C C C C H F K K K K G J H H H H F F F F G G G G J J J J slides9 828X 2019

  25. From Search Trees to Search Graphs • Any two nodes that root identical subtrees or subgraphs can be merged slides9 828X 2019

  26. From Search Trees to Search Graphs • Any two nodes that root identical subtrees or subgraphs can be merged slides9 828X 2019

  27. J A A AND/OR Tree F B B C K E C E D D F G J G H OR A H K AND 0 1 OR B B AND 0 1 0 1 OR E C E C E C E C AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D F D F D F D F D F D F D F D F AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 G G J J G G J J J J G G J J G G J J G G J J G G J J G G J J OR G G AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K H H H H K K K K AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 slides9 828X 2019

  28. J A A AND/OR Graph F B B C K E C E D D F G J G H OR A H K AND 0 1 OR B B AND 0 1 0 1 OR E C E C E C E C AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D F D F D F D F D F D F D F D F AND 0 1 0 1 G G J J OR AND 0 1 0 1 0 1 0 1 OR H H H H K K K K AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 slides9 828X 2019

  29. Merging Based on Context • context (X) = ancestors of X in pseudo tree, connected to X, or to descendants of X C C A A F F • context (X) = parents in the D D B B E E induced graph • max |context| = induced width [ ] A = treewidth B [A ] [ AB ] [ A B ] C E pseudo tree [AE ] D F [ BC ] context( ) = [ ] slides9 828X 2019

  30. Context‐Based Minimal AND/OR Search Graph Definition 7.2.13 (context minimal AND/OR search graph) The AND/OR search graph of M guided by a pseudo-tree T that is closed under context-based merge operator, is called the context minimal AND/OR search graph and is denoted by C T ( R ) . slides9 828X 2019

  31. AND/OR Tree DFS Algorithm (Value=Sum‐Product) ( | , ) ( | ) ( | ) ( A ) A A P E A B P B A P C A P A B=0 B=1 A C=0 C=1 A P(A) A B E=0 E=1 B 0 .4 .6 0 .2 .8 0 .6 0 0 .4 .6 B C 1 .1 .9 1 .7 .3 1 .4 0 1 .5 .5 E C 1 0 .7 .3 1 1 .2 .8 Result: P(D=1,E=0) D E D Evidence: E=0 OR .24408 A .6 .4 AND .3028 0 .1559 1 OR .3028 .1559 B B .4 .6 .1 .9 AND .352 .623 0 .27 1 0 .104 1 OR .5 .7 .4 E .88 C E .54 C E .89 C .2 E .52 C .4 .2 .8 .5 .2 .8 .7 .1 .9 .2 .1 .9 AND .8 .9 .7 .5 .8 .9 .7 .5 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR .8 D D .9 .7 .5 .8 D D .9 .7 .5 D D D D .8 .9 .5 .8 .9 .7 .5 .7 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ( | , ) P D B C B C D=0 D=1 0 0 .2 .8 0 1 .1 .9 1 0 .3 .7 1 1 .5 .5 Evidence: D=1,E=0 slides9 828X 2019

  32. AND/OR Search Graph (Value=Sum‐Product) Context ( | , ) ( | ) ( | ) ( A ) P E A B P B A P C A P A [ ] A A B=0 B=1 A C=0 C=1 A P(A) A B E=0 E=1 0 .4 .6 0 .2 .8 0 .6 0 0 .4 .6 [A] 1 .1 .9 1 .7 .3 1 .4 B 0 1 .5 .5 B C 1 0 .7 .3 [AB] 1 1 .2 .8 E C [AB] Result: P(D=1,E=0) Evidence: E=0 E D D [BC] .24408 A .6 .4 .3028 .1559 0 1 .3028 .1559 B B .4 .6 .1 .9 .352 .623 .27 .104 0 1 0 1 .5 .7 .4 .88 .54 .89 .2 .52 E C E C E C E C .4 .2 .8 .5 .2 .8 .7 .1 .9 .2 .1 .9 .8 .9 .7 .5 .8 .9 .7 .5 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 .8 .9 .7 .5 D D D D .8 .9 .5 .7 ( | , ) P D B C B C Value 0 1 0 1 0 1 0 1 0 0 .8 B C D=0 D=1 0 1 .9 0 0 .2 .8 1 0 .7 0 1 .1 .9 1 1 .1 1 0 .3 .7 1 1 .5 .5 Cache table for D Evidence: D=1,E=0 slides9 828X 2019

  33. AND/OR Search Graph (Optimization) A A B f 1 A C f 2 A E f 3 A F f 4 B C f 5 B D f 6 B E f 7 C D f 8 E F f 9 0 0 2 0 0 3 0 0 0 0 0 2 0 0 0 0 0 4 0 0 3 0 0 1 0 0 1 C A F 0 1 0 0 1 0 0 1 3 0 1 0 0 1 1 0 1 2 0 1 2 0 1 4 0 1 0 1 0 1 1 0 0 1 0 2 1 0 0 1 0 2 1 0 1 1 0 1 1 0 0 1 0 0 B 1 1 4 1 1 1 1 1 0 1 1 2 1 1 4 1 1 0 1 1 0 1 1 0 1 1 2 D B E C E 𝐺 ∗ � min � � 𝑔 � �𝑦 � � Objective function: � D F OR A AND 0 1 OR B B AND 0 1 0 1 C C C C OR E E E E 0 1 0 1 0 1 AND 0 1 0 1 0 1 0 1 0 1 D D OR D D F F F F B C Value AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1 1 0 Context minimal AND/OR search graph 1 1 Cache table for D slides9 828X 2019

  34. Merging Based on Context • context (X) = ancestors of X in pseudo tree, connected to X, or to descendants of X C C A A F F • context (X) = parents in the D D B B E E induced graph • max |context| = induced width [ ] A = treewidth B [A ] [ AB ] [ A B ] C E pseudo tree [AE ] D F [ BC ] context( ) = [ ] slides9 828X 2019

  35. How Big Is The Context? • Theorem: The maximum context‐size of a pseudo‐tree equals the treewidth along the pseudo tree. C [ ] F G K H [C] [C] B A max context size = treewidth L A [CK] [CH] J N B [CKL] [CHA] H E O E [CKLN] [CHAB] F [AB] C D P J [CKO] [CHAE] G [AF] L D [CEJ] K M N M [CD] P O (C K H A B E J L N O D P M F G) slides9 828X 2019

  36. Treewidth vs. Pathwidth G ABC BDFG D A B BDEF F C treewidth = 3 EFH E M K = (max cluster size) ‐ 1 H L FHK J HJ KLM G ABC D A B BDEFG F C E EFH pathwidth = 4 M K H = (max cluster size) ‐ 1 FHKJ L J KLM slides9 828X 2019

  37. A All Four Search Spaces B E C D F A 0 1 A 0 1 B 0 1 0 1 B 0 1 0 1 C 0 1 0 1 0 1 0 1 C 0 1 0 1 0 1 0 1 D D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 E E 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F 0 1 0 1 0 1 0 10 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 1 0 1 0 1 F 0 1 Full OR search tree Context minimal OR search graph 126 nodes 28 nodes OR A OR A AND 0 1 AND 0 1 OR B B OR B B AND 0 1 0 1 AND 0 1 0 1 C C OR C E C OR C C C C E E E E E E E AND 0 1 0 1 0 1 0 1 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D D D D F F F F OR D D D D D D F F F F D D F F F F AND 0 1 0 1 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Context minimal AND/OR search graph Full AND/OR search tree 18 AND nodes 54 AND nodes k = domain size n = number of variables Any query is best computed w*= treewidth over the context‐minimal AND/OR space pw*= pathwidth slides9 828X 2019

  38. A All Four Search Spaces B E C D F A 0 1 A 0 1 B AND/OR graph OR graph 0 1 0 1 B 0 1 0 1 C 0 1 0 1 0 1 0 1 C 0 1 0 1 0 1 0 1 D D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 O(n k w* ) O(n k pw* ) Space E E 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F 0 1 0 1 0 1 0 10 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 1 0 1 0 1 F 0 1 Full OR search tree Time Context minimal OR search graph O(n k w*+1 ) O(n k pw*+1 ) 126 nodes 28 nodes size OR A OR A AND 0 1 AND 0 1 OR B B OR B Computes any query: B AND 0 1 0 1 AND 0 1 0 1 • Constraint satisfaction C C OR C E C OR C C C C E E E E E E E • Max‐Inference: Optimization AND 0 1 0 1 0 1 0 1 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 • Sum‐Inference: Weighted counting OR D D D D F F F F OR D D D D D D F F F F D D F F F F AND • Mixed‐Inference: Marginal Map, 0 1 0 1 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 • Maximum expected utility Context minimal AND/OR search graph Full AND/OR search tree 18 AND nodes 54 AND nodes k = domain size n = number of variables Any query is best computed w*= treewidth over the context‐minimal AND/OR space pw*= pathwidth slides9 828X 2019

  39. AND/OR Search and Variable Elimination AND/OR Search C F G 0 1 B A K K H H J 0 1 0 1 0 1 0 1 H L L L L E A A A A 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 C D L N N N N N N N N B B B B B B B B K M 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 N P O O O O O O O O O O O O O O O O O E E E E E E E E E E E E E E E E 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 C [ ] P P P P P P P P J J J J J J J J J J J J J J J J K [C] H [C] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 L A [CK] [CH] N B [CKL] [CHA] D D D D D D D D F F F F 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 O E [CKLN] [CHAB] F [AB] J P [CKO] [CHAE] M M M M G G G G G [AF] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 D [CEJ] M [CD] (C K H A B E J L N O D P M F G) slides9 828X 2019

  40. AND/OR Search and Variable Elimination C F G 0 1 B A K K H H J 0 1 0 1 0 1 0 1 H L L L L E A A A A 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 C D L N N N N N N N N B B B B B B B B K M 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 N P O O O O O O O O O O O O O O O O O E E E E E E E E E E E E E E E E 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 C [ ] P P P P P P P P J J J J J J J J J J J J J J J J K [C] H [C] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 L A [CK] [CH] N B [CKL] [CHA] D D D D D D D D F F F F 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 O E [CKLN] [CHAB] F [AB] J P [CKO] [CHAE] M M M M G G G G G [AF] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 D [CEJ] M Variable Elimination [CD] (C K H A B E J L N O D P M F G) slides9 828X 2019

  41. AND/OR Search and Variable Elimination C AND/OR Search F G 0 1 B A K K H H J 0 1 0 1 0 1 0 1 H L L L L E A A A A 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 C D L N N N N N N N N B B B B B B B B K M 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 N P O O O O O O O O O O O O O O O O O E E E E E E E E E E E E E E E E 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 C [ ] P P P P P P P P J J J J J J J J J J J J J J J J K [C] H [C] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 L A [CK] [CH] N B [CKL] [CHA] D D D D D D D D F F F F 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 O E [CKLN] [CHAB] F [AB] J P [CKO] [CHAE] M M M M G G G G G [AF] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 D [CEJ] M Variable Elimination [CD] (C K H A B E J L N O D P M F G) slides9 828X 2019

  42. Road Map: Search • Review Graphical Modes OR A • AND/OR search spaces, pseudo‐trees AND 0 1 OR B B – AND/OR search trees AND 0 1 0 1 A OR C C C C E E E E AND 0 1 0 1 0 1 0 1 – AND/OR search graphs B OR D D F F D D F F AND 01 01 E C – Generating good pseudo‐trees Context minimal AND/OR search graph D F – Brute‐force AND/OR 18 AND nodes • Heuristic search for AND/OR spaces – Basic Heuristic search (Depth and Best) – Depth‐first AND/OR branch and bound – Best‐first AND/OR search – The Guiding MBE heuristic – Marginal Map (max‐sum‐product) A=yellow A=green E D M B=blue B=green B=red B=blue L C A E D M E D M E D M E D M • Hybrids of search and Inference L L L L B K C C C C H F K K K K G J • H H H H Summary and Class 2 F F F F G G G G J J J J slides9 828X 2019

  43. Finding Good Pseudo-Trees slides9 828X 2019

  44. Finding Min‐Height Pseudo‐Trees • Finding min height pseudo tree is NP‐complete, but: • Given a tree‐decomposition whose tree‐width is w*, there exists a pseudo‐tree T of G whose depth, satisfies h <= w* log n, slides9 828X 2019

  45. Constructing Pseudo‐Trees • Min‐Fill [Kjaerulff, 1990] – Depth‐first traversal of the induced graph obtained along the min‐fill elimination order – Variables ordered according to the smallest “fill‐set” • Hypergraph Partitioning [Karypis and Kumar, 2000] – Functions are vertices in the hypergraph and variables are hyperedges – Recursive decomposition of the hypergraph while minimizing the separator size at each step – Using state‐of‐the‐art software package hMeTiS slides9 828X 2019

  46. Variable Orderings and Pseudo‐Trees Note: we order from top to bottom here A (A) A Bucket‐tree = pseudo‐tree bucket-A A B (AB) B C (AC) (BC) C AB bucket-B A F C E (AE) (BE) (AE) E bucket-C ABC ABE bucket-E E D (BD) (DE) D B D F (AF) (EF) F bucket-D BDE AEF bucket-F d : A B C E D F Bucket-tree based on d Induced graph Bucket-tree A A OR 0 1 AND Finding small height or small width pseudo‐trees is NP‐hard B OR B B So, which orderings would give good pseudo‐trees? AND 0 1 0 1 C E OR C E C E C E C E AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 D F OR D F D F D F D F D F D F D F D F AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Bucket-tree used as AND/OR search tree pseudo-tree slides9 828X 2019

  47. Quality of Pseudo‐Trees Network hypergraph min‐fill Network hypergraph min‐fill width depth width depth width depth width depth barley 7 13 7 23 spot5 47 152 39 204 diabetes 7 16 4 77 spot28 108 138 79 199 link 21 40 15 53 spot29 16 23 14 42 mildew 5 9 4 13 spot42 36 48 33 87 munin1 12 17 12 29 spot54 12 16 11 33 munin2 9 16 9 32 spot404 19 26 19 42 munin3 9 15 9 30 spot408 47 52 35 97 munin4 9 18 9 30 spot503 11 20 9 39 water 11 16 10 15 spot505 29 42 23 74 pigs 11 20 11 26 spot507 70 122 59 160 Bayesian Networks Repository SPOT5 Benchmarks For more see [Dechter 2013 ] slides9 828X 2019

  48. Finding Min‐height Pseudo‐Trees • Finding a min height pseudo‐tree is NP‐complete, but: • Given a tree‐decomposition with treewidth w*, there exists a pseudo‐tree whose height satisfies – h <= w* log n • Optimality of h and w* cannot be achieved at once. 4 1 2 2 6 3 3 7 5 1 W*=1 4 h=7 5 6 w*=2 h=3 7 slides9 828X 2019

  49. Hypergraphs Decomposition slides9 828X 2019

  50. The Impact of the Pseudo‐Tree F G W=4,h=8 Min‐Fill C C [ ] (Kjaerulff90) 0 1 B A K K H H K [C] H [C] 0 1 0 1 0 1 0 1 J L L L L H A A A A L A 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 [CK] [CH] E N N N N N N N N B B B B B B B B C 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 D N [CKL] B [CHA] O O O O O O O O O O O O O O O O E E E E E E E E E E E E E E E E L 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 O E 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 [CKLN] [CHAB] K M F [AB] P P P P P P P P J J J J J J J J J J J J J J J J N J P [CKO] [CHAE] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 P G [AF] O What is a good D D D D D D D D F F F F D [CEJ] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 pseudo‐tree? Hypergraph M [CD] M M M M G G G G How to find a 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Partitioning (C K H A B E J L N O D P M F G) • Choose pseudo-tree with a minimal search graph good one? (h‐Metis) C • But determinism and pruning for optimization is 1 0 C [ ] D D K K unpredictable 0 1 0 1 0 1 0 1 W=5,h=6 B B B B M M M M O O O O K D [C] [C] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 A A A A A A A A L L L P P P P P P P P L L L L L 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 B M [CD] O 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 [CD] [CK] 0 1 0 1 0 1 0 1 0 1 0 1 F F F F 0 1 0 1 0 1 0 1 A [BCD] L N N N N N N N N P [CKO] [CKO] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 G G G G J J J J J J J J J J J J J J J J 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F J [AB] [ABCD] N [KLO] E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 G [AF] E H [ABCDJ] [ABCJ] H H H H H H H H H H H H H H H H 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 slides9 828X 2019 (C D K B A O M L N P J H E F G)

  51. Road Map: Search • Review Graphical Modes OR A • AND/OR search spaces, pseudo‐trees AND 0 1 OR B B – AND/OR search trees AND 0 1 0 1 OR C C C C E E E E AND 0 1 0 1 0 1 0 1 – AND/OR search graphs OR D D F F D D F F AND 01 01 – Generating good pseudo‐trees Context minimal AND/OR search graph – Brute‐force AND/OR 18 AND nodes • Heuristic search (HS) for AND/OR spaces – Basic Heuristic search (Depth and Best) – AND/OR Depth‐first HS (branch and bound) – AND/OR Best‐first heuristic search – The Guiding MBE heuristic – Marginal Map (max‐sum‐product) • Hybrids of search and Inference • Summary and Class 2 slides9 828X 2019

  52. Types of queries  Max-Inference Harder  Sum-Inference  Mixed-Inference • • NP-hard : exponentially many terms All solved by AND/OR Depth‐first search, • Linear memory, exp(h) time or • We will focus on approximation algorithms • exp(w*) memory and time • But, we can do better by: – Anytime : very fast & very approximate ! Slower & more accurate • Pruning while searching • Generating upper and lower bounds anytime slides9 828X 2019

  53. Search Spaces for a Tree GM slides9 828X 2019

  54. AND/OR Tree DFS Algorithm (Belief Updating) ( | , ) ( | ) ( | ) ( A ) A A P E A B P B A P C A P A B=0 B=1 A C=0 C=1 A P(A) A B E=0 E=1 B 0 .4 .6 0 .2 .8 0 .6 0 0 .4 .6 B C 1 .1 .9 1 .7 .3 1 .4 0 1 .5 .5 E C 1 0 .7 .3 1 1 .2 .8 Result: P(D=1,E=0) D E D Evidence: E=0 OR .24408 A .6 .4 AND .3028 0 .1559 1 Searching the AND/OR tree Dfs is straightforward OR .3028 .1559 B B .4 .6 .1 .9 AND .352 .623 0 .27 1 0 .104 1 OR .5 .7 .4 E .88 C E .54 C E .89 C .2 E .52 C .4 .2 .8 .5 .2 .8 .7 .1 .9 .2 .1 .9 AND .8 .9 .7 .5 .8 .9 .7 .5 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR .8 D D .9 .7 .5 .8 D D .9 .7 .5 D D D D .8 .9 .5 .8 .9 .7 .5 .7 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ( | , ) P D B C OR node: Marginalization operator (summation) B C D=0 D=1 0 0 .2 .8 0 1 .1 .9 AND node: Combination operator (product) 1 0 .3 .7 1 1 .5 .5 Value of node = updated belief for sub‐problem below Evidence: D=1 slides9 828X 2019

  55. AND/OR Graph DFS Algorithm (Belief Updating) Context ( | , ) ( | ) ( | ) ( A ) P E A B P B A P C A P A [ ] A A B=0 B=1 A C=0 C=1 A P(A) A B E=0 E=1 0 .4 .6 0 .2 .8 0 .6 0 0 .4 .6 [A] 1 .1 .9 1 .7 .3 1 .4 B 0 1 .5 .5 B C 1 0 .7 .3 [AB] 1 1 .2 .8 E C [AB] Result: P(D=1,E=0) Evidence: E=0 E D D [BC] .24408 A .6 .4 .3028 .1559 0 1 Searching the AND/OR graph should avoid .3028 .1559 dead caches, less simple B B .4 .6 .1 .9 .352 .623 .27 .104 0 1 0 1 .5 .7 .4 .88 .54 .89 .2 .52 E C E C E C E C .4 .2 .8 .5 .2 .8 .7 .1 .9 .2 .1 .9 .8 .9 .7 .5 .8 .9 .7 .5 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 .8 .9 .7 .5 D D D D .8 .9 .5 .7 ( | , ) P D B C B C Value 0 1 0 1 0 1 0 1 0 0 .8 B C D=0 D=1 0 1 .9 0 0 .2 .8 1 0 .7 0 1 .1 .9 1 1 .1 1 0 .3 .7 1 1 .5 .5 Cache table for D Evidence: D=1 slides9 828X 2019

  56. AND/OR Search Graph (Optimization) A A B f 1 A C f 2 A E f 3 A F f 4 B C f 5 B D f 6 B E f 7 C D f 8 E F f 9 0 0 2 0 0 3 0 0 0 0 0 2 0 0 0 0 0 4 0 0 3 0 0 1 0 0 1 C A F 0 1 0 0 1 0 0 1 3 0 1 0 0 1 1 0 1 2 0 1 2 0 1 4 0 1 0 1 0 1 1 0 0 1 0 2 1 0 0 1 0 2 1 0 1 1 0 1 1 0 0 1 0 0 B 1 1 4 1 1 1 1 1 0 1 1 2 1 1 4 1 1 0 1 1 0 1 1 0 1 1 2 D B E C E 𝐺 ∗ � min � � 𝑔 � �𝑦 � � Objective function: � D F OR A AND 0 1 OR B B AND 0 1 0 1 C C C C OR E E E E 0 1 0 1 0 1 AND 0 1 0 1 0 1 0 1 0 1 D D OR D D F F F F B C Value AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1 1 0 Context minimal AND/OR search graph 1 1 Cache table for D slides9 828X 2019

  57. Dead Caches F G B A J H E C C D 0 1 L K M K N K H H C [ ] P 0 1 0 1 O 0 1 0 1 K H [C] [C] L L L L A A A A 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 L A [CK] [CH] N N N N N N N N B B B B B B B B N B [CKL] [CHA] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 O E [CKLN] [CHAB] O O O O O O O O O O O O O O O O E E E E E E E E E E E E E E E E F [AB] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 P J [CKO] [CHAE] G [AF] P P P P P P P P J J J J J J J J J J J J J J J J D [CEJ] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 M [CD] D D D D D D D D F F F F (C K H A B E J L N O D P M F G) 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 M M M M G G G G 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 (Darwiche 2000) slides9 828X 2019

  58. Dead Caches F G B A J H E C C D 0 1 L K M K N K H H C [ ] P 0 1 0 1 O 0 1 0 1 K H [C] [C] L L L L A A A A 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 L A [CK] [CH] N N N N N N N N B B B B B B B B N B [CKL] [CHA] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 O E [CKLN] [CHAB] O O O O O O O O O O O O O O O O E E E E E E E E E E E E E E E E F [AB] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 P J [CKO] [CHAE] G [AF] P P P P P P P P J J J J J J J J J J J J J J J J D [CEJ] 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 M [CD] D D D D D D D D F F F F (C K H A B E J L N O D P M F G) 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 M M M M G G G G 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 (Darwiche 2000) slides9 828X 2019

  59. slides9 828X 2019

  60. Searching AND/OR Graphs • AND/OR(i): searches depth‐first, cache i‐context – i = the max size of a cache table (i.e. number of variables in a context) i=0 i=w* i Space: O(exp w*) Space: O(n) Time: O(exp w*) Time: O(exp(w* log n)) Space: O(exp(i) ) Time: O(exp(m_i+i ) m_i is related to the size of the i-cutset. slides9 828X 2019

  61. Different Levels of Caching slides9 828X 2019

  62. Search for Mixed Deterministic and Probabilistic Graphical Models slides9 828X 2019

  63. AND/OR Search for Mixed Networks . • Graph‐based No‐good and good learning are automatically performed by AND/OR (backjumping) and by caching. slides9 828X 2019

  64. AND/OR Backtrack‐Free slides9 828X 2019

  65. AND/OR CPE (Constraint Probability Evaluation) slides9 828X 2019

  66. The Effect of Constraint Propagation in AND/OR A A B C B C > D E F D E F > > G H I K A G H I K Domains are {1,2,3,4} 1 2 3 4 B C B C B B 2 3 4 2 3 4 3 4 3 4 4 CONSTRAINTS ONLY D E D D E F F F D D F F D 3 4 3 4 4 3 4 4 4 4 G H G I I G K K K G K 4 4 4 4 A 1 2 3 B C B C B 2 3 2 3 3 3 D E D F F D F FORWARD CHECKING 3 3 3 G H I K 4 4 4 4 A 1 B C 2 2 D E F 3 3 3 MAINTAINING ARC CONSISTENCY G H I K 4 4 4 4 slides9 828X 2019

  67. Available code • http://graphmod.ics.uci.edu/group/Software slides9 828X 2019

  68. Outline: Search • Review Graphical Modes OR A • AND/OR search spaces, pseudo‐trees AND 0 1 OR B B – AND/OR search trees AND 0 1 0 1 OR C C C C E E E E AND 0 1 0 1 0 1 0 1 – AND/OR search graphs OR D D F F D D F F AND 01 01 – Generating good pseudo‐trees Context minimal AND/OR search graph – Brute‐force search 18 AND nodes • Heuristic search (HS) for AND/OR spaces – Basic Heuristic search (Depth and Best) – AND/OR Depth‐first HS (branch and bound) – AND/OR Best‐first heuristic search – The Guiding MBE heuristic – Marginal Map (max‐sum‐product) • Hybrids of search and Inference • Summary and Class 2 slides9 828X 2019

  69. Basic Heuristic Search We assume min‐sum problems in the following ��𝑦 𝑔 � � ) Heuristic function computes a lower bound on the best extension of 𝑦 � � partial configuration and can be used to guide heuristic search. We focus on: 1. Branch‐and‐Bound 2. Best‐First Search ��𝑦 𝑔 � � ) Use heuristic function to Always expand the node with ��𝑦 𝑔 � � ) prune the depth‐first search tree the lowest heuristic value Linear space Needs lots of memory BnB is upper‐bound anytime MAP slides9 828X 2019

  70. Basic Heuristic Search; Best‐First Task: compute v(root): MAP, Marginal ,MMAP Each node is a sub‐problem • Best‐First Algorithms, (A*) (defined by current conditioning) – Expand nodes in OPEN list v(root) in order of min f(n) – Terminates with first full solution (for mpe) • Properties – Optimal, if h(n � � 𝒘�𝒐� v(n) g(n) – Expands least set of nodes n h(n) ; h(n) � 𝒘�𝒐� – exponential memory – Not anytime solution for MAP – Yields lower bounds on value, anytime f(n) = g(n) + h(n) � 𝑕 𝑜 � 𝑤 𝑜 =f*(n) f(n) is a lower bound on best cost through n slides9 828X 2019

  71. Basic Heuristic Search; Depth‐First • Depth‐First (B&B for MAP) – Expand in dfs order – Update UB with each solution 𝑕�𝑜� – Prunes if f(n) ≥ UB • Properties – Can use only linear memory v(n) Prunes if – Yields upper bounds anytime n h(n) ; h(n) � 𝒘�𝒐� f 𝑜 � 𝑉𝐶 ℎ�𝑜� (UB) Upper Bound = best solution so far slides9 828X 2019

  72. Partial Solution Tree for AND/OR A A A A A 0 0 0 0 B C B C B C B C B C D 0 0 0 0 1 0 1 0 Pseudo tree D D D D 0 1 0 1 (A=0, B=0, C=0, D=1) (A=0, B=1, C=0, D=0) (A=0, B=1, C=0, D=1) (A=0, B=0, C=0, D=0) Extension(T’) – solution trees that extend T’ g(T’) = conditioned value of a node V(T’) = the combined value below T’ f*(T’) = conditioned value through T’ slides9 828X 2019

  73. Exact Evaluation Function Conditioned value of a node A A A B C f 1 (ABC) A B F f 2 (ABF) B D E f 3 (BDE) 0 0 0 2 0 0 0 3 0 0 0 6 C F 0 0 1 5 0 0 1 5 0 0 1 4 B 0 1 0 3 0 1 0 1 0 1 0 8 B 0 1 1 5 0 1 1 4 0 1 1 5 C D 1 0 0 9 1 0 0 6 1 0 0 9 F 1 0 1 3 1 0 1 5 1 0 1 3 1 1 0 7 1 1 0 6 1 1 0 7 D E E 1 1 1 2 1 1 1 5 1 1 1 4 9 OR A 0 0 AND 0 1 OR 9 B 0 0 AND 9 0 1 2 4 3 3 F F v(F) OR C D C D 2 5 0 0 3 5 3 5 0 AND 0 1 4 0 5 1 0 1 0 1 0 4 5 OR E tip nodes E v(D,0) 6 4 8 5 AND 0 1 0 1 f*(T’) = w(A,0) + w(B,1) + w(C,0) + w(D,0) + v(D,0) + v(F) slides9 828X 2019

  74. Heuristic Evaluation Function A A A B C f 1 (ABC) A B F f 2 (ABF) B D E f 3 (BDE) 0 0 0 2 0 0 0 3 0 0 0 6 C F 0 0 1 5 0 0 1 5 0 0 1 4 B 0 1 0 3 0 1 0 1 0 1 0 8 B 0 1 1 5 0 1 1 4 0 1 1 5 C D 1 0 0 9 1 0 0 6 1 0 0 9 F 1 0 1 3 1 0 1 5 1 0 1 3 1 1 0 7 1 1 0 6 1 1 0 7 D E E 1 1 1 2 1 1 1 5 1 1 1 4 9 OR A 0 0 h(n) ≤ v(n) AND 0 1 OR 9 B 0 0 AND 9 0 1 2 4 3 3 F F h(F) = 5 OR C D C D 2 5 0 0 3 5 3 5 0 AND 0 1 4 0 5 1 0 1 0 1 0 4 5 OR E tip nodes E 6 4 8 5 h(D,0) = 4 AND 0 1 0 1 f(T’) = w(A,0) + w(B,1) + w(C,0) + w(D,0) + h(D,0) + h(F) = 12 ≤ f*(T’) slides9 828X 2019

  75. Depth‐First AND/OR Branch‐and‐Bound • Associate each node n with a UB (best solution so far) heuristic lower bound h(n) on v(n) 5 A 0 0 5 0 1 11 Algorithm AOBB: • 5 B 11 B EXPAND (top‐down) 0 0 0 0 – Evaluate f(T’) and prune search if f(T’) 0 1 0 1 ∞ 5 1 1 ≥ UB ∞ C 4 D 2 C 3 D – If not in cache, generate successors ∞ ∞ 1 ∞ ∞ 2 0 2 of the tip node n 0 1 0 1 0 1 0 1 3 4 3 4 3 4 3 4 E E E E f(T’) ≥ UB • PROPAGATE (bottom‐up) ∞ 3 ∞ 4 ∞ 3 ∞ 4 0 1 0 1 0 1 0 1 – Update value of the parent p of n • OR nodes: minimization • AND nodes: summation – Cache value of n based on context only if fully explored slides9 828X 2019

  76. Anytime Performance  OR Branch‐and‐Bound is anytime  But AND/OR breaks anytime behavior of depth‐first scheme: processing  First anytime solution delayed until solved last sub‐problem starts processing optimally  Breadth‐Rotating AOBB:  Take turns processing sub‐problems  Limit number of expansions per visit  Solve each sub‐problem depth‐first  Maintain favorable complexity bounds rotate [Otten and Dechter, 2012] slides9 828X 2019

  77. Summary: AND/OR Branch‐and‐Bound Search (AOBB) • Associate each node n with a heuristic lower bound h(n) on v(n) • EXPAND (top‐down) – Evaluate f(T’) and prune search if f(T’) ≥ UB – Generate successors of the tip node n • PROPAGATE (bottom‐up) – Update value of the parent p of n • OR nodes: minimization • AND nodes: summation [Marinescu and Dechter, 2005; 2009] slides9 828X 2019

  78. Anytime Performance  OR Branch‐and‐Bound is anytime  But AND/OR breaks anytime behavior of depth‐first scheme:  First anytime solution delayed until last sub‐problem starts processing  Breadth‐Rotating AOBB:  Take turns processing sub‐problems  Limit number of expansions per visit  Solve each sub‐problem depth‐first  Maintain favorable complexity bounds rotate [Otten and Dechter, 2012] slides9 828X 2019

  79. DFS Algorithm (#CSP Example) A A F B B C solution E C E D D F OR 11 A OR node: Marginalization operator (summation ) AND 5 6 0 1 OR 5 B B 6 AND node: Combination operator (product) AND 4 1 0 0 4 2 1 1 OR 2 2 1 E C 1 E C 4 2 E C 1 E 1 C AND 2 0 0 0 1 0 1 1 0 1 0 1 2 1 0 1 0 1 2 0 1 0 1 1 1 1 1 0 1 0 0 1 OR 2 1 2 0 1 1 1 D F D F 0 D 2 F 1 D 2 F 1 1 1 D F D F 1 D F D F 1 AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 0 1 1 0 0 1 0 1 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 0 Value of node = number of solutions below it slides9 828X 2019

  80. AND/OR Tree Search for Optimization A B f 1 A C f 2 A E f 3 A F f 4 B C f 5 B D f 6 B E f 7 C D f 8 E F f 9 0 0 2 0 0 3 0 0 0 0 0 2 0 0 0 0 0 4 0 0 3 0 0 1 0 0 1 0 1 0 0 1 0 0 1 3 0 1 0 0 1 1 0 1 2 0 1 2 0 1 4 0 1 0 A 1 0 1 1 0 0 1 0 2 1 0 0 1 0 2 1 0 1 1 0 1 1 0 0 1 0 0 C A F 1 1 4 1 1 1 1 1 0 1 1 2 1 1 4 1 1 0 1 1 0 1 1 0 1 1 2 B     9 C E Goal : min D B E f X X i 5 1 i D F OR A 0 0 5 AND 7 0 1 OR 5 7 B B 2 0 1 4 6 5 6 4 AND 0 1 0 1 3 4 C 4 4 C C 3 C OR 1 2 E E E 0 E 3 1 3 5 5 4 1 3 0 2 5 2 2 5 3 0 5 2 0 2 2 0 0 1 0 2 5 2 1 0 2 0 0 1 1 0 AND 0 1 0 1 0 1 0 1 0 1 0 1 5 2 0 2 2 0 0 2 5 2 1 0 2 0 1 0 D D D D D D OR F F D D F F F F F F AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 5 6 4 2 3 0 2 2 2 4 1 0 3 0 2 2 5 6 4 2 1 2 0 4 2 4 1 0 1 2 0 4 AND node = Combination operator (summation) OR node = Marginalization operator (minimization) slides9 828X 2019

  81. AND/OR Tree Search for Optimization A B f 1 A C f 2 A E f 3 A F f 4 B C f 5 B D f 6 B E f 7 C D f 8 E F f 9 0 0 2 0 0 3 0 0 0 0 0 2 0 0 0 0 0 4 0 0 3 0 0 1 0 0 1 0 1 0 0 1 0 0 1 3 0 1 0 0 1 1 0 1 2 0 1 2 0 1 4 0 1 0 A 1 0 1 1 0 0 1 0 2 1 0 0 1 0 2 1 0 1 1 0 1 1 0 0 1 0 0 C A F 1 1 4 1 1 1 1 1 0 1 1 2 1 1 4 1 1 0 1 1 0 1 1 0 1 1 2 B     9 C E Goal : min D B E f X X i 5 1 i D F OR A 0 0 5 AND 7 0 1 OR 5 7 B B 2 0 1 4 6 5 6 4 AND 0 1 0 1 3 4 C 4 4 C C 3 C OR 1 2 E E E 0 E 3 1 3 5 5 4 1 3 0 2 5 2 2 5 3 0 5 2 0 2 2 0 0 1 0 2 5 2 1 0 2 0 0 1 1 0 AND 0 1 0 1 0 1 0 1 0 1 0 1 5 2 0 2 2 0 0 2 5 2 1 0 2 0 1 0 D D D D D D OR F F D D F F F F F F AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 5 6 4 2 3 0 2 2 2 4 1 0 3 0 2 2 5 6 4 2 1 2 0 4 2 4 1 0 1 2 0 4 AND node = Combination operator (summation) OR node = Marginalization operator (minimization) slides9 828X 2019

  82. Outline: Search • Review Graphical Modes OR A • AND/OR search spaces, pseudo‐trees AND 0 1 OR B B – AND/OR search trees AND 0 1 0 1 OR C C C C E E E E AND 0 1 0 1 0 1 0 1 – AND/OR search graphs OR D D F F D D F F AND 01 01 – Generating good pseudo‐trees Context minimal AND/OR search graph – Brute‐force search 18 AND nodes • Heuristic search (HS) for AND/OR spaces – Basic Heuristic search (Depth and Best) – AND/OR Depth‐first HS (branch and bound) – AND/OR Best‐first heuristic search – The Guiding MBE heuristic – Marginal Map (max‐sum‐product) A=yellow A=green E D M B=blue B=green B=red B=blue L C A E D M E D M E D M E D M • Hybrids of search and Inference L L L L B K C C C C H F K K K K G J H H H H F F F F • Summary and Class 2 G G G G J J J J slides9 828X 2019

  83. Conditioning versus Elimination Conditioning (search) Elimination (inference) A A D D B B C F C F E E G G The main target In conditioning is to … Trade memory for time. A=1 A=k It is not really possible to do better Then elimination D D D B B B C F C F C F E E E G G G k “sparser” problems 1 “denser” problem slides9 828X 2019

  84. The Idea of Cutset‐Conditioning We observed that when variables are assigned, connectivity reduces. The magnitude of saving is reflected through the “conditioned‐induced graph” • Cutset‐conditioning exploit this in a systematic way: • Select a subset of variables, assign them values, and • Solve the conditioned problem by bucket‐elimination. • Repeat for all assignments to the cutset. Algorithm VEC slides9 828X 2019

  85. Hybrid: (i)‐Cutset‐Conditioning Condition Until Tree‐ness Algorithm VEC(i) • Cycle‐cutset • i‐cutset • C(i)‐size of i‐cutset Space: exp(i), Time: O(exp(i+c(i))) slides9 828X 2019

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