L5 Problem Solving n Search p4

8/3/2019 L5 Problem Solving n Search p4

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241-320 Design Architecture and Engineeringfor Intelligent System

Suntorn Witosurapot

Contact Address:Phone: 074 287369 or

Email: [email protected]

November 2009


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Lecture 5:

Problem Solving and Search

part 4

Informed Search


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241-320 Design Architecture &Engineering for Intelligent System

Problem Solving and Search part 4 3

Overview

Heuristics

Informed Search Methods

Greedy Best-First Search

A Search

Iterative Deepening A Search

Local Search

Conclusion


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Local search algorithms

In many optimization problems, the path to the goal isirrelevant; 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

(Ex:as in our part-2 slide, the Evaluation function should be

lower and lower, when proceeding in each step)


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Example: n-queens

Put nqueens on an n nboard with no two queens onthe same row, column, or diagonal

Note: The eight queens puzzle has 92 distinct solutions

[Ref] http://en.wikipedia.org/wiki/Eight_queens_puzzle


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Hill-climbing with Steepest Descent

It is a kind of local search algorithm.

Analogy: Imagine you have climbed a hill, but it hasgot dark before you are back down. Your goal is thebottom of the hill. Your problem is finding it. Youhave no GPS, so greedy search is not an option.

You would make sure that every step you made tookyou down hill and, ignoring cliffs, as steeply aspossible


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Hill-climbing with Steepest Descent

That is the fastest way to the bottom, unless: You are at the bottom of a dip (M point), and every step you

might make takes you up hill

You are on a plateau (B region) and every step makes nodifference

Objective function

Current stateState space

A

MB

Descenddirection


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Steepest Descent Search

Steepest descent searches are designed to take(you guessed it) the steepest path down theevaluation function

The evaluation function equals zero at the goal andis positive elsewhere

You actually dont need a goal, you could just betrying to minimise the cost of something

Or reverse it all and try to maximise something


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Steepest Descent

1) S initial state

2) Repeat:

a) S arg minS

SUCCESSORS(S)

{h(S)}

b) if GOAL?(S) return S

c) if h(S) < h(S) then S S else return failure


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Application: 8-QueenRepeat n times:

1) Pick an initial state S at random with one queen in each column2) Repeat k times:

a) If GOAL?(S) then return S

b) Pick an attacked queen Q at random

c) Move Q in its column to minimize the number of attackingqueens new S [min-conflicts heuristic]

3) Return failure

1

2

3

3

2

2

3

2

2

2

2

2

0

2


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Application: 8-QueenRepeat n times:

1) Pick an initial state S at random with one queen in each column2) Repeat k times:


b) Pick an attacked queen Q at random

c) Move Q in its column to minimize the number of attackingqueens new S [min-conflicts heuristic]

3) Return failure

1

2

3

3

2

2

3

2

2

2

2

2

0

2

Why does it work ???1) There are many goal states that are

well-distributed over the state space2) If no solution has been found after a few

steps, its better to start it all over again.Building a search tree would be much lessefficient because of the high branchingfactor

3) Running time almost independent of thenumber of queens


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Steepest Descent Search(cont.)

Problem: depending on initial state, can get stuck inlocal minima

Objective function

Current state Local minima global minima

State space

A

MB

Suppose we are at point A,and would like to be at point B,the goal

Everything goes fine until weget to m, a local minimum.Then we are stuck.

Descenddirection


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Simulated Annealing Search

Note: If youre curious, annealing refers to the process used toharden metals by heating them to a high temperature (hence,mealting) and then gradually cooling them

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

A

MB

Current stateBad moveGood move


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Simulated Annealing Search: No pain, no gain

x0 x1 x2

z(x)Allow non-improving moves sothat it is possible to go down

x11x

4x5x6x7 x

8x9x10

x12x

13

x

in order to rise again

to reach global optimum


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Simulated Annealing

Improving moves always accepted Non-improving moves may be accepted probabilistically

and in a manner depending on the temperature parameterT. loosely

the worse the move the lesslikely it is to be accepted a worsening move is less likely to be accepted, the cooler the

temperature

The temperature T starts high and is gradually cooled asthe search progresses.

Initially (when things are hot) virtually anything is accepted,at the end (when things are nearly frozen) only improving moves

are allowed (and the search effectively reduces to hill-climbing)


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Simulated Annealing Search

1) S initial state

2) Repeat:

a) S arg minS

SUCCESSORS(S)

{h(S)}

b) if GOAL?(S) return S

c) if h(S) < h(S) then S S

d) else with probability p

S S

This part is differentfrom hill-climbing

Q: How can we calculate p?


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241-320 Design Architecture &Engineering for Intelligent System Problem Solving and Search part 4 18

Setting p

What if p is too low?

We dont make many downhill moves and wemight not get out of many local maxima

What if p is too high?

We may be making too many suboptimal moves

Should p be constant?

We might be making too many random moveswhen we are near the global maximum


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Setting p (cont.)

Decrease p as iterations progress

Accept more uphill moves early, accept fewer assearch goes on

Intuition: as search progresses, we are movingtowards more promising areas and quite likelytoward a global minimum

Decrease p as h(s) - h(s) increases

Accept fewer uphill moves if slope is high

See next slide for intutition


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Decreasing p as h(S)-h(S) increases

h(S)-h(S) is large: weare likely moving towards

a sharp (interesting)minimum so dont moveuphill too much

h(S)-h(S) is small: we arelikely moving towards a

smooth (uninteresting)minimum so we want toescape this local minimum

h(s)h(s)h(s)

h(s)


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Complete Simulated Annealing SearchAlgorithm

1) S initial state

2) Iterate:

Repeat k times:


b) S successor of S picked at random

c) if h(S)

h(S) then S

Sd) else

- h = h(S)-h(S)

- with probability ~ exp(h/T), where T is calledthe temperature, do: S S

3) T = T

definitely accept the change

Else, accept the change with probability

When enough iterations have

passed without improvement,terminate.

Simulated annealing lowers T over the k iterations.It starts with a large T and slowly decreases T


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Simulated Annealing SearchAlgorithm (cont.)

Probability of moving downhillfor negative h values atdifferent temperature ranges:

High temp: accept all moves

(Random Walk) Low temp: Stochastic Hill-Climbing


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Convergence

If the schedule lowers T slowly enough, thealgorithm will find a global optimum withprobability approaching 1

In practice, reaching the global optimumcould take an enormous number of iterations


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Very Basic Simulated Annealing Example

1 Do 400 trial moves




m Do 400 trial moves

100=T

95.0=TT

95.0=TT

95.0=TT

95.0=TT

00001.0=Tn Do 400 trial moves

95.0=TT

Iteration


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Conclusion: Simulated annealing

Design of neighborhood is critical

Lots of parameters to tweak eg. , K, initial

temperature

Simulated annealing is usually better thanhillclimbing if you can find the right

parameters


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Parallel Local Search Techniques

They perform several local searchesconcurrently, but not independently:

Beam search

Genetic algorithms(will be studied later)


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Conclusion

Walking downhill is not as easy as youd think

Informed search algorithms try to move quicklytowards a goal based on the distance metric from

their current point

Greedy search algorithms only follow paths searchspace that bring them closest to the goal

Local search algorithms have no memory to storetree structures, but work by intelligently coveringselected parts of search space


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Search problems

Blind search

Heuristic search:best-first and A*

Construction of heuristics Local searchVariants of A*


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See Visualization of N-Queen Solutions

For viewing an live demo of N-Queens solutionalgorithms with different algorithms, visit

http://yuval.bar-or.org/index.php?item=9
http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9http://yuval.bar-or.org/index.php?item=9


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L5 Problem Solving n Search p4