Local Search and Constraint Reasoning. CS 510 Lecture 3 October 6, 2009

Transcription

1 Warm Up A* Exercise In groups, draw the tree that A* builds to find a path from Oradea to Bucharest marking g, h, and f for each node. Use straight-line distance as h(n) In what order are the nodes expanded? What is the path found?

2 Local Search and Constraint Reasoning CS 510 Lecture 3 October 6, 2009

3 Reminder Project proposal : due next week before class (mail me a pdf) send draft of project proposal after class to peer review partners send feedback to partners on proposal by thursday night

4 Reminder Midterm : 2 weeks from today (Oct 20) Will cover book, lectures, readings Ch 1-3, , 5-6, 17.6 Open book/note (not open device) Practice [2.2, 2.5, 3.1, 3.8, 4.1, 4.2, 4.4, 5.5b, 5.6, 6.2, 6.3abc, 17.9, 17.13]

5 Overview Local Search (Hill climbing, simulated annealing, genetic algorithms) Constraint Reasoning (Constraint Satisfaction problems (CSP), constraint optimization (COP), Distributed COP Project groups Paper discussion

6 Local Search In many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution State space = set of "complete" configurations Find configuration satisfying constraints, e.g., n- queens In such cases, we can use local search algorithms keep a single "current" state, try to improve it

7 Example: n-queens Put n queens on an n n board with no two queens on the same row, column, or diagonal

8 Hill-Climbing Search "Like climbing Everest in thick fog with amnesia"

9 Hill-climbing search Problem: depending on initial state, can get stuck in local maxima

10 Hill-climbing search 8-queens problem h = number of pairs of queens that are attacking each other, either directly or indirectly h = 17 for the above state

11 Hill-climbing search 8-queens problem A local minimum with h=1

12 Annealing in Metallurgy Remove crystal dislocations in metal Heat the material - cause atoms to become unstuck from position Then slowly cool, atoms wander try to find configurations with lower energy

13 Simulate this process to escape local minima Replace solution with nearby solution How nearby depends on temperature Heat - nearly random solution Slowly cool - gradually improve

14 Simulated annealing search Idea: escape local maxima by allowing some "bad" moves but gradually decrease their frequency

15 Properties of Simulated annealing search One can prove: If T decreases slowly enough, then simulated annealing search will find a global optimum with probability approaching 1 Widely used in VLSI layout, airline scheduling, etc

16 Local beam search Keep track of k states rather than just one Start with k randomly generated states At each iteration, all the successors of all k states are generated If any one is a goal state, stop; else select the k best successors from the complete list and repeat.

17 Genetic Algorithms A successor state is generated by combining two parent states Start with k randomly generated states (population) A state is represented as a string over a finite alphabet (often a string of 0s and 1s) Evaluation function (fitness function). Higher values for better states. Produce the next generation of states by selection, crossover, and mutation

18 Genetic algorithms Fitness function: number of non-attacking pairs of queens (min = 0, max = 8 7/2 = 28) 24/( ) = 31% 23/( ) = 29% etc

19 Genetic algorithms

20 Constraint Reasoning

21 Constraint Satisfaction Problem (CSP) Variables: V1,V2,V3...with domains D1, D2, D3 Constraints: Set of allowed value pairs V1 not equal V2 = {(red,blue),(green, blue,(green, red)} Solution: Assign values to variables that satisfy all constraints V1=red, V2=blue, V3=green V1 = = D1={red, green} V2 D2={blue,red} V3 D3={green}

22 3SAT as CSP xi are variables values are true and false Constraints expressed in conjunctive normal form ex 1: (x1 v x2 v x3) ^ (~x1 v ~x2 v ~x3) ^ (~x1 v x2 v ~x3) ex 2: (x1 v ~x2) ^ (~x1 v ~x2) ^ (~x1 v ~x2)

23 Class Exercise Formulate the 4 queens problem as a CSP variables values constraints

24 What if no solution? Or multiple solutions? If no solution, minimize broken constraints? If multiple solutions, some solutions may be preferred Gives rise to the Constraint Optimization Problem

25 Constraint Optimization Constraints have degrees of violation (or degrees of satisfaction) Goal is to minimize violation (or maximize satisfaction) Significantly generalizes CSPs

26 Constraint Optimization Problem (COP) Given: Variables (x1,x2,...,xn) Finite, discrete domains (D1,D2,...,Dn) Foreach xi, xj: Valued constraint function fij: Di x Dj N

27 Constraint Optimization(COP) Goal: Find complete solution: an assignment A that minimizes F(A) where F(A) = Sum(fij(di,dj), xi di, xj dj in A Example: di dj f(di,dj) F(A)= F(A)=6 1 2 F(A)=

28 Example: Map Coloring Variables WA, NT, Q, NSW, V, SA, T Domains Di = {red,green,blue} Constraints: adjacent regions must have different colors e.g., WA NT, or (WA,NT) in {(red,green), (red,blue),(green,red), (green,blue), (blue,red),(blue,green)}

29 Examples: Map Coloring Solutions are complete and consistent assignments, e.g., WA = red, NT = green,q = red,nsw = green,v = red,sa = blue,t = green

30 Constraint Graph Binary CSP: each constraint relates two variables Constraint graph: nodes are variables, arcs are constraints

31 CSP as search Initial State: Empty assignment {} Successor Function: Assign value to an unassigned variable Goal Test: Complete assignment that does not violate any constraints Path Cost: Constant for each step

32 Backtracking Search Basically DFS Map coloring (no adjacent regions the same color)

33 Improving efficiency General-purpose heuristics can give huge gains in speed: Which variable should be assigned next? In what order should its values be tried? Can we detect inevitable failure early?

34 Most Constrained Variable Most constrained variable: choose the variable with the fewest legal values a.k.a. minimum remaining values (MRV) heuristic

35 Most Constraining Variable Tie-breaker among most constrained variables Most constraining variable: Choose the variable with the most constraints on remaining variables

36 Least constraining variable assignment Given a variable, choose the least constraining value: the one that rules out the fewest values in the remaining variables Combining these heuristics makes 1000 queens possible (as opposed to about 25)

37 Forward Checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values

38 Forward Checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values

39 Constraint Propagation Forward checking propagates information from assigned to unassigned variables, but doesn't provide early detection for all failures: NT and SA cannot both be blue! Constraint propagation repeatedly enforces constraints locally

40 Arc Consistency Simplest form of propagation makes each arc consistent X Y is consistent iff for every value x of X there is some allowed y

41 Arc Consistency Simplest form of propagation makes each arc consistent X Y is consistent iff for every value x of X there is some allowed y If X loses a value, neighbors of X must be rechecked

42 Arc Consistency Simplest form of propagation makes each arc consistent X Y is consistent iff for every value x of X there is some allowed y If X loses a value, neighbors of X must be rechecked Detects failures quicker than forward checking

43 Local Search for CSPs Hill-climbing, simulated annealing typically work with "complete" states, i.e., all variables assigned To apply to CSPs: allow states with unsatisfied constraints operators reassign variable values Variable selection: randomly select any conflicted variable Value selection by min-conflicts heuristic: choose value that violates the fewest constraints i.e., hill-climb with h(n) = total number of violated constraints

44 Example: 4 Queens States: 4 queens in 4 columns (4 4 = 256 states) Actions: move queen in column Goal test: no attacks Evaluation: h(n) = number of attacks Given random initial state, can solve n-queens in almost constant time for arbitrary n with high probability (e.g., n = 10,000,000)

45 Distributed Optimization Problem How do a set of agents optimize over a set of alternatives that have varying degrees of global quality? Examples allocating resources constructing schedules planning activities Difficulties No global control/knowledge Localized communication Quality guarantees required Limited Time 45

46 Interesting Research Questions How to evaluate performance? Cycles, CBR, CCC, etc? How much distribution counts? OptAPO? Local approximations of optimization How to apply DCR to new domains? More flexible DCOP? multiply constrained, resource constraints? Privacy? 46

47 Privacy in Constraint Optimization Common model for automated coordination Assign values to variables, subject to constraints Problems Constraints may be personal or proprietary Current algorithms not designed for privacy NP-hard! 47

48 Eucalyptus Grower Supply Chain Bundled leaves Transport Company Gas Packed leaves Distributor Corn Farmer Bundled corn Transport Company Packed leaves Alcohol Shipping Company Bundled corn Need to coordinate when to send and how much Wholesaler Tanked alcohol Retailer 48

49 Algs: Asynchronous Backtracking (ABT) Not DCOP, constraint satisfaction Agents form communication/priority chain Pass down tentative assignments (ok? messages) If value inconsistent with higher agents, change Pass up violations (nogood messages) nogoods causes higher agents to change value All happens concurrently 49

50 Synchronous Branch and Bound (SynchBB) Simulate branch and bound in distributed environment Messages sent in predefined, sequential order First agent sends a partial solution (one variable) Next agent adds variable, evaluates If < bound (best complete solution cost), send on If > bound, try new values If all values tried, backtrack 50

51 Synchronous Iterative Deepening (SynchID) Simulates iterative deepening First agent picks value, sends it and cost limit (initially 0) to next agent Next agent tries to find value under limit If succeed, pass on values, limit to next agent If fail, backtrack up and increase the limit 51

52 Adopt (Asynchronous Optimization) Agents organized in tree, not chain, constraints between ancestors/descendants not siblings When agents receive messages choose value with min cost send VALUE message to descendants send COST message to parents send THRESHOLD message to child 52

53 DPOP (Dynamic Pseudotree) Same tree as Adopt, but sequential Dynamic programming/variable elimination approach Each agent sends only one message up (with constraint information) and then one message down (with value information) 53

54 Project Groups (Stacey) and (Dan and Brian) (Balaji and Huiqing) and (Melissa, Anya, and Yiannis) (Raffi, Lenrik, and Andrew) and (Kevin) Alexis, Olufunke, and Rich 54

55 Discussion: PageRank and Eigenvalues Create matrix whose values aij are the number of backlinks into page i from page j divided by number of forward links of page j. a14 = 1 link from page 4/2 total links from page 4 = 1/2 To get page rank find eigenvector Ax=lambda(x) where lambda =1 (eigenvalue) 55

56 Discussion: PageRank and Eigenvalues To get page rank find eigenvector Ax=lambda(x) where lambda =1 (eigenvalue) A [ ] = [ ] All multiples of eigenvectors are eigenvectors (normalize to fraction) [12/31 4/31 9/31 6/31] 56

57 Page Rank Have to add in E to fix link problems Vector [ ] for democratic E or [x 0 0 0] for specific E rank now c(a + E x 1)R = R still looking for an eigenvector just more complex one 57

58 Iterating Rank R0 = [ ] R1 = df R0 R1 = [ ] d = R1 = R1 + [ ] v = sqrt(v1 2 + v vn 2 ) delta = [ ]

59 Is PageRank Admissible? How does the type of search talked about here connect with the other search algorithms we ve studied?

60 Customized PageRank Where does google customize pagerank today? What are the pitfalls of this? 60