Chapter 7- Local Search part 2

1. Chapter 7- Local Search part 2 Ryan Kinworthy CSCE 990-06 Advanced Constraint Processing

2. Outline • Chapter Introduction • Greedy Local Search (SLS) • Random Walk • Properties of Local Search • Empirical Evaluation • Hybrids of Local Search and Inference • Effects of Constraint Propagation on SLS • Local Search on Cycle-Cutset • Chapter Summary

3. Example(2): Simulated Annealing • Uses a noise model from statistical mechanics • At each step, the algorithm computes the change in the cost function () when the value of the variable is changed to the value picked. • If the change improves or doesn’t affect the cost function the change is made • Otherwise the change is made with probability e-t where T is Temperature • Temp can be held constant, or slowly reduced from a high temp to a low temp according to some schedule. This algorithm converges to the exact solution if the temperature T is reduced gradually.

4. Empirical Evaluation • Evaluating Local Search Algorithms • Empirically • Use either benchmarks or randomly generated problems • Recent trends are to generate hard random problems from phase transition • Empirical evaluation of GSAT algorithms • As the number of variables and clauses increases, the running time of the algorithms increases • SLS algorithms listed in order of efficiency (most to least) • Simulated Annealing • With Random Walk with Noise • with Random Walk • Basic GSAT

5. Hybrids of Local Search & Inference • Inference with general search is very effective, how well will inference work with local search? • Effects of constraint propagation on SLS • Certain classes of problems, easy for BT, are very hard for SLS • Example: certain variations of 3SAT are extremely hard for SLS. • However, when inference methods are combined with SLS, these 3SAT problems become trivial

6. Why Is Inference so Effective for Local Search? • Not definitively explained • Conjectures given • Enforcing local consistency eliminates many “near” solutions and thusly reduces the search space. • That is, assignments that satisfy almost all clauses whose cost function would normally be near zero become high cost due to consistency enforcing. • Doesn’t always hold • Problems with uniform structure perform worse with inference • Application of combining inference and SLS • Cycle-cutset

7. What is Cycle-Cutset? • First description in Chapter 5 (pp. 146) • Definition: Given an undirected graph, a subset of nodes in the graph is a cycle-cutset if its removal results in a graph with no cycles. • The Cycle-Cutset scheme alternates between two algorithms • BT search on the cutset portion • Tree inference on the rest • Key benefits • Once a variable is instantiated, it can be removed from the constraint graph. • If the set of instantiated variables forms a cycle-cutset, then we know that the remaining nodes in the graph form a tree and we can use directional rather than full consistency algorithms to solve it. • Applicable to Local Search/Inference hybrids • If we can guarantee that the constraint graph is a tree, we can then use directional arc-consistency as an inference method for solving the constraint graph. • Example on page 206

8. Local Search on Cycle-Cutset • Instantiated variables cut the flow of information on any path they are on. • In other words, the network is equivalent to one in which the instantiated variable is deleted from the network and the influence of its value is propagated to all neighboring nodes. • So when the group of instantiated variables removes all cycles in the graph, the remaining network can be viewed as a tree and consequently can be solved by a tree-inference algorithm (e.g., arc-consistency). • Complexity • Can be bounded exponentially in the size of the cutset. • Analysis not specifically given, but I assume it to be NP-Hard

9. Hybrid Local Search on Cycle-Cutset • Where does SLS fit in? • Since SLS approximates search, it replaces BT search • A mechanism for collaboration in hybrids • Tree Algorithm works for networks with cycles • Any assignment it produces will minimize the number of violated constraints across all its subnetworks

10. Tree Algorithm • Input: • An arc consistent network R • Variables X partitioned, X = Z U Y • into cycle-cutset Y and • tree variables Z • An assignment Y = y. • Output: • An assignment Z = z that minimizes the #violated constraints of the entire network when Y = y.

11. Tree Algorithm (cont.) • Initialization • For any value y[i] of any cutset variable yi, the cost Cyi (y[i],y) is 0. • Algorithm body • 1) Going from leaves to root in the tree • For every variable zi, and any value aiin Dzi, compute the cost of each assignment • 2) Compute, from root to leaves new assignments for every tree variable zi • For a tree variable zi, let Dzi be its consistent values with vpithe value assigned to its parent pi, assign each variable a value based on the results of the previous calculation

12. Tree Algorithm (AAAI’96) • More clearly explained in Dechter and Kask’s paper (today’s handout) • A Graph-Based Method for Improving GSAT • Appeared in the AAAI’96 conference proceedings • Tree Algorithm • Generalization of Mackworth & Freuder ’85 • Generalized to work with cyclic networks • In acyclic (tree) networks functions the same as M&F • In cyclic networks • Finds an assignment that minimizes the sum of unsatisfied constraints over all its tree subnetworks

13. Tree Algorithm (AAAI’96) cont. • Example on board • Still unclear on how the weight of a constraint is calculated • Discussion

14. SLS With Cycle-Cutset • Benefit of using tree algorithm • Minimizes the cost of tree subnetworks given a cycle-cutset assignment • This means we can replace BT search with an SLS search • Need to combine the tree algorithm with SLS • Results in a concrete algorithm: • SLS + CC

15. Overview of SLS + CC • Algorithm executes a certain #tries • For each try • Start from a random initialization • Alternate between SLS and tree algorithm • SLS chooses initial assignment for Y variables • TA find min cost of assignment to Z variables • SLS fix Z, choose best y, fix y • TA find best assignment for Z, fix z • SLS on Y… • TA on Z… • Note: • Only adjacent tree variables affect the behavior of SLS • The algorithm must enforce this property • Otherwise the performance of SLS + CC deteriorates by several orders of magnitude • Cycle-Cutset idea can be generalized • SLS + CC is an algorithm specific to the case when w* = 1(tree) • In cases where w* >1, SLS cannot be used and a general backtracking search must be used instead

16. SLS + CC • Input: • An arc consistent network R • Variables X partitioned, X = Z  Y • into cycle cutset Y and • tree variables Z • Output: • An assignment Z = z, Y = y that is a local minimum of the #violated constraints C(z,y)

17. SLS + CC (cont.) • Repeat MAX_TRIES times • Algorithm body • 1) Random initial assignment for all variables • 2) Alternate between these steps until problem is solved, or the TA doesn’t change the values of the variables or no progress is made • a) When the values of cycle-cutset variables are fixed, run TA on the Z variables • b) When the values of tree variables are fixed, run SLS on the Y variables

18. SLS + CC Performance • SLS + CC vs. SLS • Empirical evaluation (pp. 210-211) • For problems where the Cycle-Cutset < 30% of the variables: • SLS + CC can solve 3-4 times more problems then SLS alone given equal CPU time • When Cycle-Cutset  30% • SLS + CC performs about the same as plain SLS • For problems where the Cycle-Cutset > 30% of the variables: • SLS is better than SLS + CC

19. Summary of Local Search • The good • Significantly faster in some problem domains than BT • Can solve previously unsolvable problems • Does more with less CPU time • Hybrid algorithms even more efficient • The bad • Not complete or sound (doesn’t guarantee a solution) • Not applicable to all domains • Can get stuck in local minima • The ugly • If applied to the wrong domain, can be a waste of time • Code carefully or performance will deteriorate rapidly!!

20. Discussion • Questions? • Thoughts? • Opinions?