Artificial Intelligence Search

8/13/2019 Artificial Intelligence Search

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Artificial Intelligence Search Algorithms

Dr Rong QuSchool of Computer Science

University of NottinghamNottingham, NG8 1BB, UK

[email protected]

Local Search Algorithms


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Konstanz, May 2012 AI Search Algorithms Local Search 2

Optimisation ProblemsFor most of real world optimisation problems

An exact model cannot be built easily;

Number of feasible solutions grow exponentiallywith growth in the size of the problem.Optimisation algorithms

Mathematical programmingTree searchHeuristic algorithms


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Optimisation Problems : MethodsMeta-heuristics

Guide an underlying heuristic/search to escapefrom being trapped in a local optima and toexplore better areas of the solution spaceExamples:

Single solution approaches: Simulated Annealing, Tabu Search, etc;Population based approaches: Geneticalgorithm, Memetic algorithm, Ant Algorithms,etc;


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Optimisation Problems : MethodsMeta-heuristics

+ : Able to cope with inaccuracies of data andmodel, large sizes of the problem and real-timeproblem solving;+ : mechanisms to escape from local optima+ : ease of implementation;+ : no need for exact model of the problem;-: usually no guarantee of optimality.


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LOCAL SEARCH



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Starts from some initial solution , moves to a betterneighbour solution until it arrives at a local optimum (does not have a better neighbour )

Examples: k-opt algorithm for TSP, etc;+: ease of implementation;+: guarantee of local optimality usually in smallcomputational time;+: no need for exact model of the problem;

-: poor quality of solution due to getting stuck in poorlocal optima;

Local Search: Method



f(X)

X

local maximum

solution

global maximum

solution

( )

Neighbourhood ofsolution

globalmaximum value

Y

Local Search: terminology



Neighbourhood function : using the concept of amove, which changes one or more attributes of agiven solution to generate another solution

Local optimum : a solution x with respect to theneighbourhood function N , if f(x) < f(y) for every y in N(x)

Local Search: terminology


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Local Search: elements

Representation of the solution

Evaluation function

Neighbourhood functionSolutions which are close to a given solutionOptimisation of real-valued functions, for a current solutionx0, neighbourhood is defined as an interval (x0 r, x0 +r)

Acceptance criterionFirst improvement, best improvement, best of non-improvingsolutions, random criteria


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1. Pick a random point in the search space2. Consider all the neighbours of the current state

3. Choose the neighbour with the best quality andmove to that state4. Repeat 2 thru 4 until all the neighbouring states are

of lower quality

5. Return the current state as the solution state

Local Search: Hill Climbing


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Local Search: Hill Climbing

Possible solutions- Try several runs, starting atdifferent positions- Increase the size of the

neighborhood (e.g. 3-opt inTSP)


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How can bad local optima be avoided?


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SIMULATED ANNEALING


Motivated by the physical annealing processMaterial is heated and slowly cooled into a uniform structure


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The SA algorithmThe first SA algorithm was developed in 1953(Metropolis)Kirkpatrick (1982)* applied SA to optimisationproblems

Compared to hill climbingSA allows downwards steps

A SA move is selected at random and then decides whetherto accept it

Better moves are always acceptedWorse moves may be accepted, depends on aprobability

*Kirkpatrick, S , Gelatt, C.D., Vecchi, M.P. 1983. Optimization bySimulated Annealing. Science, vol 220, No. 4598, pp 671-680


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To accept or not to accept?

The law of thermodynamics states that attemperature t , the probability of an increase inenergy of magnitude, E , is given by

P (E ) = exp(- E / kt )

k is a constant known as Boltzmanns constant


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To accept or not to accept?

P = exp(- c / t ) > r

c is change in the evaluation function

t the current temperaturer is a random number between 0 and 1

Change inEvaluationFunction

Temperatureof System exp(-C/T)



10 100 0.904837418 10 10 0.36787944120 100 0.818730753 20 10 0.13533528330 100 0.740818221 30 10 0.04978706840 100 0.670320046 40 10 0.01831563950 100 0.60653066 50 10 0.00673794760 100 0.548811636 60 10 0.00247875270 100 0.496585304 70 10 0.00091188280 100 0.449328964 80 10 0.00033546390 100 0.40656966 90 10 0.00012341

100 100 0.367879441 100 10 4.53999E-05110 100 0.332871084 110 10 1.67017E-05120 100 0.301194212 120 10 6.14421E-06130 100 0.272531793 130 10 2.26033E-06140 100 0.246596964 140 10 8.31529E-07150 100 0.22313016 150 10 3.05902E-07160 100 0.201896518 160 10 1.12535E-07170 100 0.182683524 170 10 4.13994E-08180 100 0.165298888 180 10 1.523E-08190 100 0.149568619 190 10 5.6028E-09200 100 0.135335283 200 10 2.06115E-09

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SImulated Annealing Acceptance Probability

Temp = 100

Tem = 10

* Need to use a scientific calculator to calculate exp()


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The SA algorithm

For t = 1 to do T = Schedule [t]If T = 0 then return Current Next = a randomly selected neighbour of Current

E = VALUE[Next ] VALUE[Current ]if E > 0 then Current = Next

else Current = Next with probability exp(- E/T)


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The SA algorithm

To implement an SA algorithm: implement hillclimbing with an accept function and modify theacceptance criteria

The cooling schedule is hidden in this algorithm -but it is important (more later)

The algorithm assumes that annealing will continueuntil temperature is zero - this is not necessarily thecase


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SA - Cooling Schedule

Starting Temperature

Final Temperature

Temperature Decrement

Iterations at each temperature


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Starting TemperatureMust be hot enough to allow moves to almost neighbourhood state (else we are in danger ofimplementing hill climbing)Must not be so hot that we conduct a randomsearch for a period of timeProblem is finding a suitable startingtemperature


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Starting TemperatureIf we know the maximum change in the costfunction we can use this to estimateStart high, reduce quickly until about 60% ofworse moves are accepted. Use this as thestarting temperatureHeat rapidly until a certain percentage areaccepted the start cooling


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Final TemperatureUsual decrease temperature to 0However, the algorithm runs for a lot longerIn practise, it is not necessary to decrease thetemperature to 0Chances of accepting a worse move are almostthe same as the temperature being equal to 0Therefore, the stopping criteria can either be asuitably low temperature or when the system is

frozen at the current temperature (i.e. nobetter or worse moves are being accepted)


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Temperature DecrementLinear

temp = temp x

Geometrictemp = temp * aExperience has shown that should be between 0.8and 0.99. Of course, the higher the value of , thelonger it will take


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Iterations at each temperature A constant number of iterations at eachtemperature

Another method, first suggested by (Lundy,1986) is to only do one iteration at eachtemperature, but to decrease the temperaturevery slowly

t = t/(1 + t)

where is a suitably small value




Temperatureof System e xp( -C/T)

10 100 0.904837418 10 10 0.36787944120 100 0.818730753 20 10 0.135335283

30 100 0.740818221 30 10 0.04978706840 100 0.670320046 40 10 0.01831563950 100 0.60653066 50 10 0.00673794760 100 0.548811636 60 10 0.00247875270 100 0.496585304 70 10 0.00091188280 100 0.449328964 80 10 0.00033546390 100 0.40656966 90 10 0.00012341

100 100 0.367879441 100 10 4.53999E-05110 100 0.332871084 110 10 1.67017E-05120 100 0.301194212 120 10 6.14421E-06130 100 0.272531793 130 10 2.26033E-06140 100 0.246596964 140 10 8.31529E-07150 100 0.22313016 150 10 3.05902E-07160 100 0.201896518 160 10 1.12535E-07170 100 0.182683524 170 10 4.13994E-08180 100 0.165298888 180 10 1.523E-08190 100 0.149568619 190 10 5.6028E-09200 100 0.135335283 200 10 2.06115E-09

.0.5

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SImulated Annealing Acceptance Proba bility

Temp = 100

Tem = 10


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Iterations at each temperature An alternative: dynamically change the numberof iterations as the algorithm progresses

At lower temperatures: a large number of iterationsare done so that the local optimum can be fully explore At higher temperatures: the number of iterations canbe less


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TABU SEARCH


a meta -heuristic superimposed on another heuristic. The overallapproach is to avoid entrapment in cycles by forbidding or penalizingmoves which take the solution, in the next iteration, to points in thesolution space previously visited (hence tabu ).

Proposed independently by Glover (1986) and Hansen (1986)


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The TS algorithm Accepts non-improving solutions deterministically inorder to escape from local optima by guiding asteepest descent local search (or steepest ascent hillclimbing) algorithm

After evaluating a number of neighbourhoods, weaccept the best one, even if it is low quality on costfunction.

Accept worse move


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The TS algorithmUses of past experiences (memory) to improvecurrent decision making in two ways

prevent the search from revisiting previouslyvisited solutionsexplore the unvisited areas of the solution space

By using memory (a tabu list) to prohibit certainmoves

makes tabu search a global optimizer rather thana local optimizer


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Tabu Search - uses of memoryTabu move what does it mean?

Not allowed to re-visit exactly the same state thatweve been before

Discouraging some patterns in solution: e.g. inTSP problem, tabu a state that has the townslisted in the same order that weve seen before. If the size of problem is large, lot of time justchecking if weve been to certain state before.


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Tabu Search algorithmCurrent = initial solutionWhile not terminate

Next = a highest-valued neighbour of Current

If(not Move_Tabu( H, Next ) or Aspiration( Next )) then Current = Next Update BestSolutionSeen H = Recency( H + Current)

Endif

End-While Return BestSolutionSeen


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Elements of Tabu Search

Memory related - recency (How recent thesolution has been reached)

Tabu List (short term memory): to record a limitednumber of attributes of solutions (moves,selections, assignments, etc) to be discouraged inorder to prevent revisiting a visited solution;Tabu tenure (length of tabu list): number ofiterations a tabu move is considered to remain

tabu;


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Memory related - recency (How recent thesolution has been reached)

Tabu tenureList of moves does not grow forever restrictthe search too muchRestrict the size of listFIFOOther ways: dynamic


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Memory related frequencyLong term memory: to record attributes of elitesolutions to be used in:

Diversification: Discouraging attributes of elitesolutions in selection functions in order todiversify the search to other areas of solutionspace;Intensification: giving priority to attributes of a

set of elite solutions (usually in weightedprobability manner)


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If a move is good, but its tabu -ed, do we still rejectit?

Aspiration criteria: accepting an improving solutioneven if generated by a tabu move

Similar to SA in always accepting improvingsolutions, but accepting non-improving ones whenthere is no improving solution in theneighbourhood;


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Example: TSP using Tabu Search

Find the list of towns to be visited so that thetravelling salesman will have the shortestroute

Short term memory:Maintain a list of t towns and prevent them frombeing selected for consideration of moves for anumber of iterations;

After a number of iterations, release those townsby FIFO


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Long term memory:Maintain a list of t towns which have beenconsidered in the last k best (worst) solutions

encourage (or discourage) their selections infuture solutionsusing their frequency of appearance in the set ofelite solutions and the quality of solutions whichthey have appeared in our selection function


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Aspiration:If the next moves consider those moves in thetabu list but generate better solution than thecurrent one

Accept that solution anywayPut it into tabu list


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Tabu Search Pros & ConsPros

Generated generally good solutions foroptimisation problems compared with other AI

methodsConsTabu list construction is problem specificNo guarantee of global optimal solutions


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Other Local Search

Variable Neighbourhood SearchIterative Local SearchGuided Local SearchGRASP (Greedy Random Adaptive Search Procedure)

Talbi, Metaheuristics From design toimplementation , Wiley, 2009


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Cost Function

The evaluation function is calculated at everyiteration

Often the cost function is the most expensive partof the algorithm

Therefore

We need to evaluate the cost function as efficiently aspossibleUse Delta EvaluationUse Partial Evaluation


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Cost Function

If possible, the cost function should also bedesigned so that it can lead the search

Avoid cost functions where many states returnthe same valueThis can be seen as a plateau in the searchspace, the search has no knowledge where itshould proceedBin Packing


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Cost Function

Bin Packing A number of items, a number of binsObjective

As many items as possible As less bins as possible Other objectives depending on the problems

Cost function? a) number of bins

b) number of items c) both a) and b)How about there are weights for the items?


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Cost Function

Graph Colouring A undirected graph G =(V, E), V: vertices; E:edges connecting vertices

Objective colouring the graph with the minimal number ofcolours so that no adjacent vertices are of the same colour

Cost function? a) number of colours

How about different colourings (during the search)of the same number of colours?

WA

NT

SA

Q

NSW

V

T


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Cost Function

WeightingsHard constraints: a large weighting. Thesolutions which violate those constraints have a

high cost functionSoft constraints: weighted depending ontheir importanceCan be dynamically changed as the algorithmprogresses. This allows hard constraints to beaccepted at the start of the algorithm butrejected later


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Neighbourhood

How do you move from one state to another?When you are in a certain state, what other statesare reachable?

Examples: bin packing, timetablingSome researchthe neighbourhood structure should be symmetric. i.e. ifmove from state i to state j , then possible to move fromstate j to state i

However, a weaker condition can hold in order toensure convergenceEvery state must be reachable from every other.

Important: when thinking about your problem, ensurethat this condition is met


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Neighbourhood

The smaller the search space, the easier the searchwill beIf we define cost function such that infeasible

solutions are accepted, the search space will beincreased As well as keeping the search space small, alsokeep the neighbourhood small


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Performance

What is performance? Quality of the solution returnedTime taken by the algorithm

We already have the problem of findingsuitable SA parameters (cooling schedule)


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Performance

Improving Performance - Initialisation Start with a random solution and let theannealing process improve on that.Might be better to start with a solution thathas been heuristically built (e.g. for the TSPproblem, start with a greedy search)


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APPENDIX


SA modifications in the literature


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Acceptance Probability

The probability of accepting a worse movein SA is normally based on the physicalanalogy (based on the Boltzmann

distribution)

But is there any reason why a differentfunction will not perform better for all, or at

least certain, problems?

Konstanz, May 2012 58 AI Search Algorithms Local Search


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Why should we use a different acceptancecriteria?

The one proposed does not work. Or we suspect

we might be able to produce better solutionsThe exponential calculation is computationallyexpensive.Johnson (1991) found that the acceptancecalculation took about one third of the

computation time



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Johnson experimented withP() = 1 / t

This approximates the exponentialPlease read in conjunction with the simulated annealing handout

Set these parametersClassic Acceptance Criteria Approximate Acceptance Criteria C ha nge i n E va lu at io n F unc ti on , c 2 0

exp(-c/t) 1 - c / t Temperature, t 1000.818730753 0.8



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A better approach was found by building alook-up table of a set of values over therange / t

During the course of the algorithm / t wasrounded to the nearest integer and thisvalue was used to access the look-up tableThis method was found to speed up the

algorithm by about a third with nosignificant effect on solution quality



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The Cooling Schedule

If you plot a typical cooling schedule you are likelyto find that at high temperatures many solutionsare acceptedIf you start at too high a temperature a randomsearch is emulated and until the temperature coolssufficiently any solution can be reached and couldhave been used as a starting position



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At lower temperatures, a plot of the coolingschedule, is likely to show that very few worsemoves are accepted; almost making simulated

annealing emulate hill climbingTaking this one stage further, we can say thatsimulated annealing does most of its work duringthe middle stages of the cooling scheduleConnolly (1990) suggested annealing at a constanttemperature



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But what temperature?It must be high enough to allow movement but notso low that the system is frozen

But, the optimum temperature will vary from onetype of problem to another and also from oneinstance of a problem to another instance of thesame problem



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One solution to this problem is to spend some timesearching for the optimum temperature and thenstay at that temperature for the remainder of thealgorithmThe final temperature is chosen as the temperaturethat returns the best cost function during thesearch phase



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Neighbourhood

The neighbourhood of any move is normallythe same throughout the algorithm but

The neighbourhood could be changed asthe algorithm progresses

For example, a different neighbourhood can beused to helping jumping from local optimal



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Cost Function

The cost function is calculated at every iteration ofthe algorithm

Various researchers (e.g. Burke,1999) have shownthat the cost function can be responsible for a largeproportion of the execution time of the algorithmSome techniques have been suggested which aimto alleviate this problem



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Cost Function

Rana (1996) - Coors BreweryGA but could be applied to SAThe evaluation function is approximated (one

tenth of a second)Potentially good solution are fully evaluated(three minutes)



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Cost Function

Ross (1994) uses delta evaluation on thetimetabling problem

Instead of evaluating every timetable as only

small changes are being made between onetimetable and the next, it is possible to evaluate just the changes and update the previous costfunction using the result of that calculation



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Cost Function

Burke (1999) uses a cacheThe cache stores cost functions (partial andcomplete) that have already been evaluatedThey can be retrieved from the cache ratherthan having to go through the evaluationfunction again

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