Lectur ture 1 e 10 Stochastic Local Search (4. (4.8) Slide 1 - - PowerPoint PPT Presentation

Computer Science CPSC 322

Lectur ture 1 e 10 Stochastic Local Search

(4. (4.8)

Slide 1

Lect cture re O Overvi rview

• Recap
• Domain Splitting for Arc Consistency
• Local Search
• Stochastic Local Search (SLS)
• Comparing SLS

2

Arc C Consi sist stency A cy Algori rithm

• Go through all the arcs in the network
• Make each arc consistent by pruning the appropriate

domain, when needed

• Reconsider arcs that could be turned back to being

inconsistent by this pruning

• Eventually reach a ‘fixed point’: all arcs consistent

3

• When AC reduces the domain of a variable X to make an arc

〈X,c〉 arc consistent, which arcs does it need to reconsider? AC does not need to reconsider other arcs

• If arc 〈Y,c〉 was arc consistent before, it will still be arc consistent.

“Consistent before” means each element yi in Y must have an element xi in X that satisfies the constraint. Those xi would not be pruned from Dom(X), so arc 〈Y,c〉 stays consistent

• If an arc 〈X,ci〉 was arc consistent before, it will still be arc consistent

The domains of Zi have not been touched

• Nothing changes for arcs of constraints not involving X

Whi hich arc arcs nee need t to

• be rec

be reconsidered?

Z1 c1 Z2 c2 Z3 c3 Y c T H E S E X A c4

4

Arc C Consi sist stency A cy Algori rithm: m: C Comp mplexi xity

• Let’s determine Worst-case complexity of this

procedure (compare with DFS O(dn))

• let the max size of a variable domain be d
• let the number of variables be n
• The max number of binary constraints is ?

(n (n * (n (n-1)) )) / 2

• How many times, at worst, the same arc

can be inserted in the ToDoArc list? O(d) d)

• How many steps are involved in checking

the consistency of an arc? O(d2)

• Overall

ll complexit ity: O(n2d3)

• Compare to O(dN) of DFS. Arc consistency is MUCH faster

5

So did we find a polynomial algorithm to solve CPSs? No, AC does not always solve the CPS. It is a way to possibly simplify the original CSP and make it easier to solve

Overall complexity: O(n2d3)

Ar Arc Consi sist stency ency Algorithm hm: Interp erpret reting ng Outco comes es

• Three possible outcomes (when all arcs are arc

consistent):

• Each domain has a single value,

 e.g. built-in AISpace example “Scheduling problem 1”  We have: a (unique) solution.

• At least one domain is empty,

 We have: No solution! All values are ruled out for this variable.  e.g. try this graph (can easily generate it by modifying Simple Problem 2)

• Some domains have more than one value,

 There may be: one solution, multiple ones, or none  Need to solve this new CSP (usually simpler) problem: – same constraints, domains have been reduced

6

Lect cture re O Overvi rview

• Recap
• Domain Splitting for Arc Consistency
• Local Search
• Stochastic Local Search (SLS)
• Comparing SLS

7

Sear arch v h vs. Domai ain S n Splitting ng

• Arc consistency ends: Some domains have more than
• ne value → may or may not have a solution
• A. Apply Depth-First Search with Pruning or
• B. Split the problem in a number of disjoint cases CSPi: for

instance CSP with dom(X) = {x1, x2, x3, x4} becomes CSP1 with dom(X) = {x1, x2} and CSP2 with dom(X) = {x3, x4}

• Solution to CSP is the union of solutions to CSPi

8

Exa Example

Run “Scheduling Problem 2” in AIspace

• Try spitting on E (select 4 first, then 2 then 3)

9

Anot

• ther

her E Exampl ple

“Crossword 1” in Aispace,

• try splitting on D3 and then A3 (always “select half”)

10

Domain in s splitti tting

• For each subCSP, which arcs have to be on the ToDoArcs

list when we get the subCSP by splitting the domain of X?

• A. arcs <Zi, r(Zi,X)>
• B. arcs < Zi, r(Zi,X)> and <X, r(Zi,X)>

C. arcs < Zi, r(Z,X)> and <Y, r(X,Y)>

Z1 c1 Z2 c2 Z3 c3 Y c X A c4

• D. All arcs in the figure
• E. All 8 arcs related to constraints involving X: (c1,c2,c3,c)

11

Domain in s splitti tting

• For each subCSP, which arcs have to be on the ToDoArcs

list when we get the subCSP by splitting the domain of X?

C. arcs < Zi, r(Z,X)> and <Y, r(X,Y)>

Z1 c1 Z2 c2 Z3 c3 Y c X A c4 T H E S E T H I S

12

If domains with multiple values Split on one

Sear archi hing b ng by domai ain s n splitting ng

CSP, apply AC

CSP1, apply AC CSP2, apply AC

If domains with multiple values Split on one If domains with multiple values…..Split on one

13

Anot nother er form

• rmulation

n of

• f CSP as

as sear search

Arc consistency with domain splitting

• States: vector (D(V1), …, D(Vn)) of remaining domains,

with D(Vi) ⊆ dom(Vi) for each Vi

• Start state: vector of original domains (dom(V1), …, dom(Vn))
• Successor function:
• reduce one of the domains + run arc consistency
• Goal state: vector of unary domains that satisfies all

constraints

• That is, only one value left for each variable
• The assignment of each variable to its single value is a model
• Solution: that assignment

14

Arc consistency + domain splitting: example

({1,2,3,4}, {1,2,3,4}, {1,2,3,4}) ({1,3}, {1,2,3,4}, {1,2,3,4}) ({1,3}, {1,3}, {1,3}) ({2,4}, {1,2,3,4}, {1,2,3,4}) ({1,3}, {1}, {1,3}) AC ({}, {}, {}) ({1,3}, {3}, {1,3}) ({}, {}, {}) AC AC ({2,4}, {2,4}, {2,4}) ({2,4}, {2}, {2,4}) AC ({}, {}, {}) ({2,4}, {4}, {2,4}) ({}, {}, {}) AC AC A ϵ {1,3} A ϵ {2,4} B ϵ {1} B ϵ {3} B ϵ {2} B ϵ {4} 3 variables: A, B, C Domains: all {1,2,3,4} A=B, B=C, A≠C No solution No solution No solution ({1,2,3,4}, {1,2,3,4}, {1,2,3,4}) AC (arc consistency)

15

Arc consistency + domain splitting: another example

({1,2,3,4}, {1,2,3,4}, {1,2,3,4}) ({1,3}, {1,2,3,4}, {1,2,3,4}) ({1,3}, {1,3}, {1,3}) ({2,4}, {1,2,3,4}, {1,2,3,4}) ({1,3}, {1}, {1,3}) AC ({1}, {1}, {1}) ({1,3}, {3}, {1,3}) ({3}, {3}, {3}) AC AC ({2,4}, {2,4}, {2,4}) ({2,4}, {2}, {2,4}) AC ({2}, {2}, {2}) ({2,4}, {4}, {2,4}) ({4}, {4}, {4}) AC AC A ϵ {1,3} A ϵ {2,4} B ϵ {1} B ϵ {3} B ϵ {2} B ϵ {4} 3 variables: A, B, C Domains: all {1,2,3,4} A=B, B=C, A=C Solution Solution Solution Solution ({1,2,3,4}, {1,2,3,4}, {1,2,3,4}) AC (arc consistency)

16

If domains with multiple values Split on one

Sear archi hing b ng by domai ain s n splitting ng

How many CSPs do we need to keep around at a time? Assume solution at depth m and b children at each split

CSP, apply AC

CSP1, apply AC CSP2, apply AC

If domains with multiple values Split on one If domains with multiple values…..Split on one

• A. O(bm)
• B. O(bm)
• B. O(mb)
• B. O(b2)

17

If domains with multiple values Split on one

Sear archi hing b ng by domai ain s n splitting ng

How many CSPs do we need to keep around at a time? Assume solution at depth m and b children at each split O(bm): It is DFS

CSP, apply AC

CSP1, apply AC CSP2, apply AC

If domains with multiple values Split on one If domains with multiple values…..Split on one

18

Syst Systematically so solvi ving C CSPs SPs: Su Summary

• Build Constraint Network
• Apply Arc Consistency
• One domain is empty →
• Each domain has a single value →
• Some domains have more than one value →
• Apply Depth-First Search with Pruning OR
• Split the problem in a number of disjoint cases
• Apply Arc Consistency to each case, and repeat

19

Limitat ation o

• n of System

emat atic A Appr proac

• aches

es

• Many CSPs (scheduling, DNA computing, etc.) are

simply too big for systematic approaches

• If you have 105 vars with dom(vari) = 104

Systematic Search Branching factor b = Solution depth d = Complexity = Constraint Network Size = Complexity of AC =

20

Limitat ation o

• n of System

emat atic A Appr proac

• aches

es

• Many CSPs are simply too big for systematic approaches
• If you have 105 vars with dom(vari) = 104

Systematic Search Branching factor b = 104 Solution depth d = 105 Time Complexity = O((104 ) 105) Constraint Network Size = O(105 + 105 *105) Time Complexity of AC = O((105*2 * 104*3)

21

Lea Learning G Goal

• als for C
• r CSP
• Define possible worlds in term of variables and their domains
• Compute number of possible worlds on real examples
• Specify constraints to represent real world problems differentiating

between:

• Unary and k-ary constraints
• List vs. function format
• Verify whether a possible world satisfies a set of constraints (i.e.,

whether it is a model, a solution)

• Implement the Generate-and-Test Algorithm. Explain its

disadvantages.

• Solve a CSP by search (specify neighbors, states, start state, goal

state). Compare strategies for CSP search. Implement pruning for DFS search in a CSP.

• Define/read/write/trace/debug the arc consistency algorithm. Compute

its complexity and assess its possible outcomes

• Define/read/write/trace/debug domain splitting and its integration with

arc consistency

22

Lect cture re O Overvi rview

• Recap
• Domain Splitting for Arc Consistency
• Local Search
• Stochastic Local Search (SLS)
• Comparing SLS

23

• Solving CSPs is NP-hard
• Search space for many CSPs is huge
• Exponential in the number of variables
• Even arc consistency with domain splitting is often not enough
• Alternative: local search
• use algorithms that search the space locally, rather than

systematically

• Often finds a solution quickly, but are not guaranteed to find a

solution if one exists (thus, cannot prove that there is no solution)

Loc

• cal Sear

arch: Mo Motivation

• n

24

Loca cal S Search rch

• Idea:
• Consider the space of complete assignments of values to variables

(all possible worlds)

• Neighbours of a current node are similar variable assignments
• Move from one node to another according to a function that scores

how good each assignment is

• Useful method in practice

– Best available method for many constraint satisfaction and constraint optimization problems

25

Local al S Sear arch P h Probl blem em: D Defini nition

• n

Definition: A local search problem consists of a: CSP: a set of variables, domains for these variables, and constraints on their joint values. A node in the search space will be a complete assignment to all

• f the variables.

Neighbour relation: an edge in the search space will exist when the neighbour relation holds between a pair of nodes. Scoring function: h(n), judges cost of a node (want to minimize)

• E.g. the number of constraints violated in node n.
• E.g. the cost of a state in an optimization context.

26

Local al S Sear arch P h Probl blem em: E Exampl ple

Definition: A local search problem consists of a: CSP: a set of variables, {V1 ….,Vn }, each with domain Dom(Vi), and constraints on their joint values. A node in the search space will be a complete assignment to all

• f the variables.

Nneighbour relation: The neighbors of node with assignment A= {V1 / v1,…,Vn / vn }

are nodes with assignments that differ from A for one value only

Scoring function: h(n), judges cost of a node (want to minimize)

• E.g. the number of constraints violated in node n.
• E.g. the cost of a state in an optimization context.

27

V1 = v1 ,V2 = v1 ,.., Vn = v1

Search Space

V1 = v2 ,V2 = v1 ,.., Vn = v1 V1 = v4 ,V2 = v1 ,.., Vn = v1 V1 = v1 ,V2 = vn ,.., Vn = v1 V1 = v4 ,V2 = v2 ,.., Vn = v1 V1 = v4 ,V2 = v3 ,.., Vn = v1 V1 = v4 ,V2 = v1 ,.., Vn = v2

• Only the current node is kept in memory at each step.
• Very different from the systematic tree search approaches we

have seen so far!

• Local search does NOT backtrack!

29

Itera rative ve B Best I Impro rove veme ment

• How to determine the neighbor node to be selected?
• Iterative Best Improvement:
• select the neighbor that optimizes some evaluation function
• Which strategy would make sense? Select neighbor with …
• D. Minimal number of constraint violations
• B. Similar number of constraint violations as current state
• A. Maximal number of constraint violations
• C. No constraint violations

30

Itera rative ve B Best I Impro rove veme ment

• How to determine the neighbor node to be selected?
• Iterative Best Improvement:
• select the neighbor that optimizes some evaluation function
• Which strategy would make sense? Select neighbour with …
• Evaluation function:

h(n): number of constraint violations in state n

• Greedy descent: evaluate h(n) for each neighbour, pick the neighbour n

with minimal h(n) – minimize the number of unsatisfied constraints

• Hill climbing: equivalent algorithm for maximization problems
• Here: Maximize number of satisfied constraints

Minimal number of constraint violations

31

Hill Climbing

NOTE: Everything that will be said for Hill Climbing is also true for Greedy Descent

evaluation/scoring function: number of satisfied contrs. Current nt s state/pos e/possibl ble w e world X 1 assume both vars have integer domain X2

32

Exa Example: N N-Quee ueens ns

• Put n queens on an n × n board with no two queens
• n the same row, column, or diagonal (i.e attacking

each other)

• Positions a queen

can attack

33

Exam ampl ple: e: N N-queen as queen as a a loc

• cal

al s sear earch pr ch probl

• blem

em

CSP: N-queen CSP

• One variable per column; domains {1,…,N} => row where the queen in

the ith column seats;

• Constraints: no two queens in the same row, column or diagonal

Neighbour relation: value of a single column differs Scoring function: number of constraint violations (i.e., number

• f attacks)

34

Exam ample: G Gree reedy des descent f for N

• r N-Quee

ueen

For each column, assign randomly each queen to a row (a number between 1 and N) Repeat

• For each column & each number: Evaluate how many constraint

violations changing the assignment to that number would yield

• Choose the column and number that leads to the fewest violated

constraints; change it Until solved

Each cell lists h (i.e. #constraints unsatisfied) if you move the queen in that column into the cell

35

h = 5 h = ? h = ?

36

h = 5 h = ? h = ?

37

Gen eneral Local al Sea earch Algo gorithm

Random initialization Local search step 1: Proced edur ure Local-Search(V,dom,C) 2: Input nputs 3: V: a set of variables 4: dom: a function such that dom(X) is the domain of variable X 5: C: set of constraints to be satisfied 6: Output put complete assignment that satisfies the constraints 7: Loc Local al 8: A[V] an array of values indexed by V 9: repea epeat 10: for

• r eac

each variable X do do 11: A[X] ←a random value in dom(X); 12: 13: wh while ile (stopping criterion not met & A is not a satisfying assignment) 14: Select a variable Y and a value V ∈dom(Y) 15: Set A[Y] ←V 16: 17: if if (A is a satisfying assignment) then hen 18: ret etur urn A 19: 20: unt until termination

38

Gen eneral Local al Sea earch Algo gorithm

Random initialization 1: Proced edur ure Local-Search(V,dom,C) 2: Input nputs 3: V: a set of variables 4: dom: a function such that dom(X) is the domain of variable X 5: C: set of constraints to be satisfied 6: Output put complete assignment that satisfies the constraints 7: Loc Local al 8: A[V] an array of values indexed by V 9: repea epeat 10: for

• r eac

each variable X do do 11: A[X] ←a random value in dom(X); 12: 13: wh while ile (stopping criterion not met & A is not a satisfying assignment) 14: Select a variable Y and a value V ∈dom(Y) 15: Set A[Y] ←V 16: 17: if if (A is a satisfying assignment) then hen 18: ret etur urn A 19: 20: unt until termination

39 Local Search Step Based on local information. E.g., for each neighbour evaluate how many constraints are unsatisfied. Greedy descent: select Y and V to minimize #unsatisfied constraints at each step

Exampl ample: N-Que ueens

h = 17 h = 1 5 steps

Each cell lists h (i.e. #constraints unsatisfied) if you move the queen from that column into the cell

40

• Which move should we

pick in this situation?

Exampl ample: N-Que ueens

41

The pro he problem of

• f loc
• cal mi

mini nima

• Which move should we

pick in this situation?

• Current cost: h=1
• No single move can

improve on this

• In fact, every single move
• nly makes things worse

(h ≥ 2)

• Locally optimal solution
• Since we are minimizing:

local minimum

42

Problems with Iterative Best Improvement

It gets misled by locally optimal points

• (Local Maxima/ Minima)

X1 X2 Eval aluat uation

• n f

func unction

• n

43 Most research in local search is about finding effective mechanisms for escaping

Lect cture re O Overvi rview

• Recap
• Domain Splitting for Arc Consistency
• Local Search
• Stochastic Local Search (SLS)
• Comparing SLS

44

Stoch chast stic Loca cal S Search rch

• We will use greedy steps to find local minima
• Move to neighbour with best evaluation function value
• We will use randomness to escape local minima

45

• Start node: random assignment
• Goal: assignment with zero unsatisfied constraints
• Heuristic function h: number of unsatisfied constraints

– Lower values of the function are better

• Stochastic local search is a mix of:

– Greedy descent: move to neighbor with lowest h – Random walk: take some random steps

– i.e., move to a neighbour with some randomness

– Random restart: reassigning values to all variables

Stochast

• chastic Loc

Local al Sear earch ch (SLS LS) f for

• r C

CSPs

46

1: Proced edur ure Local-Search(V,dom,C) 2: Input nputs 3: V: a set of variables 4: dom: a function such that dom(X) is the domain of variable X 5: C: set of constraints to be satisfied 6: Output put complete assignment that satisfies the constraints 7: Loc Local al 8: A[V] an array of values indexed by V 9: repea epeat 10: for

• r eac

each variable X do do 11: A[X] ←a random value in dom(X); 12: 13: wh while ile (stopping criterion not met & A is not a satisfying assignment) 14: Select a variable Y and a value V ∈dom(Y) 15: Set A[Y] ←V 16: 17: if if (A is a satisfying assignment) then hen 18: ret etur urn A 19: 20: unt until termination

Gen eneral SL SLS S Algo Algori rithm

Extreme case 1: random sampling. Restart at every step: Stopping criterion is “true” Random restart

47

Gen eneral SLS Algor

• rithm

1: Proced edur ure Local-Search(V,dom,C) 2: Input nputs 3: V: a set of variables 4: dom: a function such that dom(X) is the domain of variable X 5: C: set of constraints to be satisfied 6: Output put complete assignment that satisfies the constraints 7: Loc Local al 8: A[V] an array of values indexed by V 9: repea epeat 10: for

• r eac

each variable X do do 11: A[X] ←a random value in dom(X); 12: 13: wh while ile (stopping criterion not met & A is not a satisfying assignment) 14: Select a variable Y and a value V ∈dom(Y) 15: Set A[Y] ←V 16: 17: if if (A is a satisfying assignment) then hen 18: ret etur urn A 19: 20: unt until termination

Extreme case 2: greedy descent Select the neighbor with best h value (select at random among neighbors with same h)

48

Random restart

Trac racing S SLS LS al algo gorithms i in n AI AIsp space

• Let’s look at these algorithms in AIspace:
• Greedy Descent
• Random Sampling
• Simple scheduling problem 2 in AIspace:

49