Junction trees Trees where each node is a set of variables

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Efficient Principled Learning of Junction Trees A A AB AC AD 0.3 0.4 0.2 Anton Chechetka and Carlos Guestrin Carnegie Mellon University Motivation Constructing a junction tree Using Alg.1 for every SV, obtain a list L of pairs (S,Q) s.t I(Q,V\SQ|S)<|V|(+) Example: Experimental results Theoretical guarantees Intuition: if the intra-clique dependencies are strong enough, guaranteed to find a well- approximating JT in polynomial time. • Finding almost independent subsets • Question: if S is a separator of an -JT, which variables are on the same side of S? • More than one correct answer possible: • We will settle for finding one • Drop the complexity to polynomial from exponential • Junction trees • Trees where each node is a set of variables • Running intersection property: every clique between Ci and Cj contains Ci Cj • Ci and Cj are neighbors  Sij=Ci Cj is called a separator • Example: • Notation: Vij is a set of all variables on the same side of edge i-j as clique Cj: • V34={GF}, V31={A}, V43={AD} • Encoded independencies: (Vij Vji | Sij) • Constraint-based learning • Naively • for every candidate sep. Sof sizek • for every XV\S • if I(X, V\SX | S) <  • add (S,X) to the “list of useful components”L • find a JT consistent with L • Probabilistic graphical models are everywhere • Medical diagnosis, datacenter performance monitoring, sensor nets, … Model quality (log-likelihood on test set) • Compare this work with • ordering-based search (OBS) [Teyssier+Koller:UAI05] • Chow-Liu alg. [Chow+Liu:IEEE68] • Karger-Srebro alg.[Karger+Srebro:SODA01] • local search • this work + local searchcombination (using our algorithm to initialize local search) Complexity: B B B C BE Theorem: Suppose a maximal -JT tree of treewidth k exists for P(V) s.t. for every clique C and separator S of tree it holds that minX(C\S)I(X,C\SX|S) > (k+3)(+) then our algorithm will find a k|V|(+)-JT for P(V) with probability at least (1-) using AB , , , , S A CD EF D : • Main advantages • Compact representation of probability distributions • Exploit structure to speed up inference BC Q C E E S={A}: {B}, {C,D} OR {B,C}, {D} EF CD , , Separators ABCD ABEF F 1 V34 EG AB B E 5 • Problem: From L, reconstruct a junction tree. • This is non-trivial. Complications: • L may encode more independencies than a single JT encodes • Several different JTs may be consistent with independencies in L B BC BE Cliques samples and E 3 4 C • But also problems • Compact representation≠ tractable inference. • Exact inference#P-complete in general • Often still need exponential time even for compact models • Example: • Often do not even have structure, only data • Best structure is NP-complete to find • Most structure learning algorithms return complex models, where inference is intractable • Very few structure learning algorithms have global quality guarantees • We address both of these issues! We provide • efficient structure learning algorithm • guaranteed to learn tractable models • with global guarantees on the results quality CD Intuition: Consider set of variables Q={BCD}. Suppose an -JT (e.g. above) with separator S={A} exists s.t. some of the variables in Q ({B}) are on the left of S and the remaining ones ({CD}) on the right.then a partitioning of Q into X and Y exists s.t. I(X,Y|S)< EF 2 Key theoretical result Efficient upper bound for I(,|) 6 time Key insight [Arnborg+al:,SIAM-JADM1987, Narasimhan+Bilmes: UAI05]: In a junction tree, components (S,Q) have recursive decomposition: Data: Beinlich+al:ECAIM1988 37 variables, treewidth 4, learned treewidth 3 Intuition: Suppose a distribution P(V) can be well approximated by a junction tree with clique size k. Then for every set SV of size k, A,BV of arbitrary size, to check that I(A,B | S) is small, it is enoughto check for all subsets XA, YB of size at most k that I(X,Y|S) is small. possible partitionings Corollary: Maximal JTs of fixed treewidth s.t. for every clique C and separator S it holds that minX(C\S)I(X,C\SX|S) > for fixed >0 are efficiently PAC learnable B C ≤4 neighbors per variable (a constant!), but inference still hard a clique in the junction tree D C smaller components from L • Tractability guarantees: • Inference exponential in clique sizek • Small cliques  tractable inference  if no such splits exist, all variables of Qmust be on the same side of S B EF A I(A,B | S)=?? Only need to compute I(X,Y|S) for small X and Y! Related work A B • Alg. 1 (given candidate sep. S), threshold : • each variable of V\S starts out as a separate partition • for every QV\S of size at most k+2 • if minXQ I(X,Q\S | S) >  • merge all partitions that have variables in Q Data: Desphande+al:VLDB04 54 variables, treewidth 2 Look for such recursive decompositions in L! • JTs as approximations • Often exact conditional independence is too • strict a requirement • generalization: conditional mutual informationI(A , B | C)  H(A | B) – H(A | BC) • H() is conditional entropy • I(A , B | C) ≥0 always • I(A , B | C) = 0  (A  B | C) • intuitively: if C is already known, how much new information about A is contained in B? Y I(X,Y|S) X • DP algorithm (input list L of pairs (S,Q)): • sort L in the order of increasing|Q| • mark (S,Q)L with |Q|=1 as positive • for (S,Q)L, Q≥2,in the sorted order • if xQ,(S1,Q1), …, (Sm,Qm) Ls.t. • Si {Sx}, (Si,Qi) is positive • QiQj= • i=1:mQi=Q\x • then mark (S,Q) positive • decomposition(S,Q)=(S1,Q1),...,(Sm,Qm) • if S s.t. all (S,Qi)Lare positive • return corresponding junction tree Fixed size regardless of |Q| S Computation time is reduced from exponential in |V|to polynomial! Example: =0.25 Set S does not have to relate to the separators of the “true” JT in any way! NP-complete to decide  We use greedy heuristic Data: Krause+Guestrin:UAI05 32 variables, treewidth 3 merge; end result Pairwise I(.,.|S) Test edge, merge variables I() too low, do not merge Approximation quality guarantee: Theorem 1: Suppose an -JT of treewidth k exists for P(V). Suppose the sets SV of size k, AV\S of arbitrary size are s.t. for every XV\S of size k+1 it holds that I(XA, X(V\SA)S | S) <  then I(A, V\SA | S) < |V|(+) • This work: contributions • The first polynomial time algorithm with PAC guarantees for learning low-treewidth graphical models with • guaranteed tractable inference! • Key theoretical insight: polynomial-time upper bound on conditional mutual information for arbitrarily large sets of variables • Empirical viability demonstration Theorem [Narasimhan and Bilmes, UAI05]: If for every separator Sij in the junction tree it holds that the conditional mutual information I(Vij, Vji | Sij ) < (call it -junction tree) then KL(P||Ptree) < |V| [1] Bach+Jordan:NIPS-02 [2] Choi+al:UAI-05 [3] Chow+Liu:IEEE-1968 [4] Meila+Jordan:JMLR-01 [5] Teyssier+Koller:UAI-05 [6] Singh+Moore:CMU-CALD-05 [7] Karger+Srebro:SODA-01 [8] Abbeel+al:JMLR-06 [9] Narasimhan+Bilmes:UAI-04 • Theorem (results quality):If after invoking Alg.1(S,=) a set U is a connected component, then • For every Z s.t. I(Z, V\ZS | S)<it holds that UZ • I(U, V\US | S)<nk • Greedy heuristic for decomposition search • initialize decomposition to empty • iteratively add pairs (Si,Qi) that do not conflict with those already in the decomposition • if all variables of Q are covered, success • May fail even if a decomposition exists • But we prove that for certain distributions guaranteed to work never mistakenly put variables together • Future work • Extend to non-maximal junction trees • Heuristics to speed up performance • Using information about edges likelihood (e.g. from L1 regularized logistic regression) to cut down on computation. Incorrect splits not too bad Goal: find an –junction tree with fixed clique size k in polynomial (in |V|) time Complexity: O(nk+1). Polynomial in n,instead ofO(exp(n)) for straightforward computation Complexity: O(nk+3). Polynomial in n.