Advances in Bayesian Network Learning using Integer Programming

Size: px

Start display at page:

Download "Advances in Bayesian Network Learning using Integer Programming"

Douglas Paul
6 years ago
Views:

1 Advances in Bayesian Network Learning using Integer Programming Mark Bartlett and James Cussens UAI-13, Supported by the UK Medical Research Council (Project Grant G ) Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

2 Outline Linear programming Integer linear programming BN learning with IP Improving efficiency Results Current and future work Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

3 Linear programming The diet problem Fat Sugar Salt Cost Chocolate Chips Needs y Minimise 5x + 4y, subject to: x, y 0 4x + 6y 12 5x + y 8 x + 8y 4 x, y R x Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

4 Linear programming Solving an LP using SCIP presolved problem has 2 variables (0 bin, 0 int, 0 impl, 2 cont) and 3 constraints LP iter cols rows dualbound primalbound gap e+01 Inf e e % e e % chocolate chips 1.25 Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

5 Integer linear programming Discrete dieting Fat Sugar Salt Cost Chocolate Chips Needs y Minimise 5x + 4y, subject to: x, y 0 4x + 6y 12 5x + y 8 x + 8y 4 x, y Z x Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

6 Integer linear programming Solving an IP using SCIP presolved problem has 2 variables (0 bin, 2 int, 0 impl, 0 cont) and 3 constraints LP iters cols rows cuts dualbound p bnd gap e+01 Inf e+01 Inf e e % e e % e e % e e % e e % Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

7 Integer linear programming Separating the LP solution with a cutting plane Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

8 Integer linear programming Separating the LP solution with a cutting plane Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

9 Integer linear programming Branching on a fractional variable Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

10 Integer linear programming Branching on a fractional variable Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

11 Integer linear programming Branch and cut 1. Let x* be the LP solution. 2. If x* worse than incumbent then exit. 3. If there are valid inequalities not satisfied by x* add them and go to 1. Else if x* is integer-valued then the current problem is solved Else branch on a variable with non-integer value in x* to create two new sub-problems (propagate if possible) Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

12 BN learning with IP Encoding graphs with binary IP variables Suppose there are p random variables V in some dataset. Want to learn an optimal BN (with p vertices) for some decomposable score. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

13 BN learning with IP Encoding graphs with binary IP variables Suppose there are p random variables V in some dataset. Want to learn an optimal BN (with p vertices) for some decomposable score. Can encode any graph by creating a binary IP variable I (u W ) for each BN variable u V and each candidate parent set W I (0 ) = 1 I (1 {0}) = 1 I (2 {0, 1}) = 1 All other IP variables zero. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

14 BN learning with IP Encoding graphs with binary IP variables Suppose there are p random variables V in some dataset. Want to learn an optimal BN (with p vertices) for some decomposable score. Can encode any graph by creating a binary IP variable I (u W ) for each BN variable u V and each candidate parent set W. With no restrictions on candidate parent sets that could be a lot of variables! And computing objective coefficients for each of them (from the data) could take a while. More on this problem later. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

15 BN learning with IP Encoding graphs with binary IP variables Suppose there are p random variables V in some dataset. Want to learn an optimal BN (with p vertices) for some decomposable score. Can encode any graph by creating a binary IP variable I (u W ) for each BN variable u V and each candidate parent set W. Assume known parameters (pedigrees) or Dirichlet parameter priors (general BN) and a uniform (or at least decomposable ) structural prior. Each I (u W ) has an (assumed precomputed) local score c(u, W ). Instantiate the I (u W ) to maximise: u,w c(u, W )I (u W ) subject to the I (u W ) representing a DAG. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

16 BN learning with IP Ruling out non-dags with linear constraints u V : W I (u W ) = 1 Where C V : u C W :W C= I (u W ) 1 (1) Let x be the solution to the LP relaxation. We search for a cluster C such that x violates (1) and then add (1) to get a new LP. Repeat as long as a cluster cut can be found. Cluster constraints introduced by Jaakkola, Sontag, Globerson and Meila ( AISTATS2010 ) [Jaakkola et al., 2010]. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

17 BN learning with IP Solving in the root node Eskimo pedigree BN variables. At most 2 parents. Simulated genotypes IP variables. time frac cuts dualbound primalbound gap 1110s e e % 1139s e e % 1171s e e % 1209s e e % 1228s e e % 1264s e e % *1266s e e % SCIP Status : problem is solved [optimal solution found Solving Time (sec) : Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

18 BN learning with IP Solving after branching Alarm. 37 BN variables. At most 3 parents datapoints IP variables. time node left frac strbr gap 19.5s % 20.1s % 20.4s % 20.5s % 21.2s % R21.3s % s % 63.3s % SCIP Status : problem is solved [optimal solution found Solving Time (sec) : Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

19 Improving efficiency Enough background! The advances in this paper are all about improving efficiency. But this required a better understanding of the geometry of the relevant polytopes. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

20 Improving efficiency Enough background! The advances in this paper are all about improving efficiency. But this required a better understanding of the geometry of the relevant polytopes. GOBNILP 1.3 is available It requires SCIP ( Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

21 Improving efficiency Enough background! The advances in this paper are all about improving efficiency. But this required a better understanding of the geometry of the relevant polytopes. GOBNILP 1.3 is available It requires SCIP ( Just need to write the manual for GOBNILP 1.4! Results in paper with GOBNILP 1.3, those on the slides from GOBNILP 1.4. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

22 Improving efficiency Propagation Propagation If, say, I (1 {2, 3}) and I (4 {1}) set to 1 in some subproblem then immediately fix e.g. I (2 {4}) to 0 in that subproblem. Efficient implementation important of course. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

23 Improving efficiency Primal heuristic Primal heuristics in IP A good early (typically suboptimal) solution helps prune the search tree. Can also help in the root search node due to reduced cost strengthening. If we fail to solve to optimality (or want good anytime behaviour) even more important to have a reasonable solution. SCIP has 35 built-in primal heuristics but we turn off all apart from a few fast ones based on rounding. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

24 Improving efficiency Primal heuristic Sink finding As we learn from [Silander and Myllymäki, 2006]... Every DAG has at least one sink node (node with no children). For this node we can choose the best parents without fear of creating a cycle. Once a sink v p selected from V then just worry about learning the best BN with nodes V \ {v p }. Basically the same as finding the best total order (in reverse). Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

25 Improving efficiency Primal heuristic Using the LP solution to find sinks I (1 W 1,1 ) I (1 W 1,2 )... I (1 W 1,k1 ) I (2 W 2,1 ) I (2 W 2,2 )... I (2 W 2,k2 ) I (3 W 3,1 ) I (3 W 3,2 )... I (3 W 3,k3 ) I (p W p,1 ) I (p W p,2 )... I (p W p,kp ) For each variable, order its parent set choices from best to worst. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

26 Improving efficiency Primal heuristic Using the LP solution to find sinks I (1 W 1,1 ) I (1 W 1,2 )... I (1 W 1,k1 ) I (2 W 2,1 ) I (2 W 2,2 )... I (2 W 2,k2 ) I (3 W 3,1 ) I (3 W 3,2 )... I (3 W 3,k3 ) I (p W p,1 ) I (p W p,2 )... I (p W p,kp ) (With only the acyclicity constraint) for an optimal BN at least one BN variable has its best parent set selected. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

27 Improving efficiency Primal heuristic Using the LP solution to find sinks I (1 W 1,1 ) I (1 W 1,2 )... I (1 W 1,k1 ) I(2 W 2,1 ) I (2 W 2,2 )... I (2 W 2,k2 ) I (3 W 3,1 ) I (3 W 3,2 )... I (3 W 3,k3 ) I (p W p,1 ) I (p W p,2 )... I (p W p,kp ) (With only the acyclicity constraint) for an optimal BN at least one BN variable has its best parent set selected. Let x be the LP solution and suppose x (2 W 2,1 ) is closer to 1 than the best parent set choice for any other variable. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

28 Improving efficiency Primal heuristic Using the LP solution to find sinks I (1 W 1,1 ) I (1 W 1,2 )... I (1 W 1,k1 ) I(2 W 2,1 ) I (2 W 2,2 )... I (2 W 2,k2 ) I (3 W 3,1 ) I (3 W 3,2 )... I (3 W 3,k3 ) I (p W p,1 ) I (p W p,2 )... I (p W p,kp ) (With only the acyclicity constraint) for an optimal BN at least one BN variable has its best parent set selected. Let x be the LP solution and suppose x (2 W 2,1 ) is closer to 1 than the best parent set choice for any other variable. Select it. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

29 Improving efficiency Primal heuristic Using the LP solution to find sinks I (1 W 1,1 ) I (1 W 1,2 )... I (1 W 1,k1 ) I(2 W 2,1 ) I (2 W 2,2 )... I (2 W 2,k2 ) I (3 W 3,1 ) I (3 W 3,2 )... I (3 W 3,k3 ) I (p W p,1 ) I (p W p,2 )... I (p W p,kp ) (With only the acyclicity constraint) for an optimal BN at least one BN variable has its best parent set selected. Let x be the LP solution and suppose x (2 W 2,1 ) is closer to 1 than the best parent set choice for any other variable. Select it. Suppose 2 is a member of W 1,1, W 3,2, W p,1 and W p,2 Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

30 Improving efficiency Primal heuristic Sink finding primal heuristic The BN returned is always best for some total ordering. Basically a greedy search for such a BN near the LP solution (L 1 ). Complications if some IP variables already fixed (due to branching). If using auxiliary variables (e.g. to encode complex user constraints) have to use SCIP s probing mode to effect the necessary propagations. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

31 Improving efficiency Tightening the LP relaxation Tightening the LP relaxation The cluster constraints of [Jaakkola et al., 2010] ensure that any integer solution is a DAG :-) But they do not define the convex hull of DAGs. :-( We want additional strong valid inequalities (preferably facets of the convex hull). Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

32 Improving efficiency Tightening the LP relaxation SCIP separators SCIP provides 12 general-purpose separators which can be used to search for strong valid inequalities. We have experimented with: Gomory Strong Chvátal-Gomory Zero-half Also some recent big wins with the Closecuts meta-separator (when used appropriately): Earlier Eskimo pedigree 1614 node BN solved in 21 mins with Closecuts (and some other improvements), took almost 32 hours before. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

33 Improving efficiency Tightening the LP relaxation Generating a close point to separate Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

34 Improving efficiency Tightening the LP relaxation Problem-specific inequalities This is a facet of the convex hull of 3-node DAGs I (1 {2, 3}) + I (2 {1, 3}) + I (3 {1, 2}) 1 Needed to separate LP solution: I (1 {2, 3}) = 1 2, I (2 {1, 3}) = 1 2, I (3 {1, 2}) = 1 2 Adding suitably generalised versions of this facet is a big win. Not too many so just add them all at the outset. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

35 Improving efficiency Tightening the LP relaxation Problem-specific inequalities We have also experimented with adding generalised versions of the facets of the convex hull of 4-node DAGs. Have found 8 types of facets. Added as cutting planes via 8 different cutting plane algorithms. Looks promising! Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

36 Results Results See paper for a bunch of GOBNILP benchmarks. Take home message 1: combined effect of various improvements has led to big performance gains. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

37 Results Results See paper for a bunch of GOBNILP benchmarks. Take home message 1: combined effect of various improvements has led to big performance gains. Take home message 2: Tightening the LP relaxation is what matters most Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

38 Results Results See paper for a bunch of GOBNILP benchmarks. Take home message 1: combined effect of various improvements has led to big performance gains. Take home message 2: Tightening the LP relaxation is what matters most The number of IP variables matters much more than the number of nodes in the BN. So with a severe limit on parent set size very large problems can be solved to optimality. For such problems the key is to find the necessary cluster constraints as quickly as possible. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

39 Results Lessons learned Benchmark/optimise/debug(!) on as wide a portfolio of problems as possible. Put in the theoretical work on tightening the LP relaxation. SCIP s default branching strategy which knows nothing about BNs is hard to beat. All BN specific searching strategies tried so far have been dramatically worse. Systematic parameter tuning (e.g. how often, how thoroughly to look for cuts) is worth the effort: programming by optimisation. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

40 Current and future work User constraints Allowing the user to add in all sorts of user-constraints, including arbitrary conditional independence constraints, is easy. But if the constraints are non-linear then need to do the work to optimise. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

41 Current and future work Column generation (what we re doing wrong) Just as one can create constraints on the fly (cutting planes) one can also create variables dynamically (column generation). Think of the not-currently-created variables as being initially fixed to zero. After solving the LP search for a parent set with a positive reduced local score. If we can t find one we have all the variables we need for an optimal solution. For the reduced local score we need (i) the (unreduced) local score (ii) dual values for all the linear constraints in which it will appear and (iii) its coefficient in each of these linear constraints. A lot of work but some hope of scaling up for more general-purpose BN structure learning. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

42 References Jaakkola, T., Sontag, D., Globerson, A., and Meila, M. (2010). Learning Bayesian network structure using LP relaxations. In Proceedings of 13th International Conference on Artificial Intelligence and Statistics (AISTATS 2010), volume 9, pages Journal of Machine Learning Research Workshop and Conference Proceedings. Silander, T. and Myllymäki, P. (2006). A simple approach for finding the globally optimal Bayesian network structure. In UAI. Mark Bartlett and James Cussens IP for BNs UAI-13, / 28

Integer Programming for Bayesian Network Structure Learning

Integer Programming for Bayesian Network Structure Learning James Cussens Prague, 2013-09-02 Supported by the UK Medical Research Council (Project Grant G1002312) James Cussens IP for BNs Prague, 2013-09-02