Informed Search Methods

1
Informed Search Methods

• Read Chapter 4
• Use text for more Examples
• work them out yourself

2
Best First

• Store is replaced by sorted data structure
• Knowledge added by the sort function
• No guarantees yet depends on qualities of the
  evaluation function
• Uniform Cost with user supplied evaluation
  function.

3
Concerns

• What knowledge is available?
• How can it be added to the search?
• What guarantees are there?
• Time
• Space

4
Greedy Search

• Adding heuristic h(n)
• h(n) estimated cost of cheapest solution from
  state n to the goal
• Require h(goal) 0.
• Complete no can be mislead.

5
Examples

• Route Finding goal from A to B
• straight-line distance from current to B
• 8-tile puzzle
• number of misplaced tiles
• number and distance of misplaced tiles

6
A

• Combines greedy and Uniform cost
• f(n) g(n)h(n) where
• g(n) path cost to node n
• h(n) estimated cost to goal
• If h(n) lt true cost to goal, then admissible.
• Best-first using admissible f is A.
• Theorem A is optimal and complete

7
A optimality Proof

• Note Along any path from root, f increases.
• Definition of monotonicity.
• Let f be cost of optimal solution.
• A expands all nodes with f(n) ltf
• A may expand nodes for which f(n) f
• Let G be optimal goal state and G2 a suboptimal
  one.

8
A Proof

• Let n be leaf node on path to G.
• h admissible gt fgt f(n)
• G2 choosen before n gt f(n)gtf(G2)
• Then G2 is not suboptimal.
• A is complete. Searches increasing contours.
• A is exponential in time and space, generally.

9
A Properties

• Dechter and Pearl A optimal among all
  algorithms using h. (Any algorithm must search at
  least as many nodes).
• If 0lth1 lt h2 and h2 is admissible, then h1 is
  admissible and h1 will search at least as many
  nodes as h2. So bigger is better.
• Sub exponential if h estimate error is within
  (approximately) log of true cost.

10
A special cases

• Suppose h(n) 0. gt Uniform Cost
• Suppose g(n) 1, h(n) 0 gt Breadth First
• If non-admissible heuristic
• g(n) 0, h(n) 1/depth gt depth first
• One code, many algorithms

11
Heuristic Generation

• Relaxation make the problem simpler
• Route-Planning
• dont worry about paths go straight
• 8-tile puzzle
• dont worry about physical constraints pick up
  tile and move to correct position
• better allow sliding over existing tiles
• Should be easy to compute

12
Iterative Deepening A

• Like iterative deepening, but
• Replaces depth limit with f-cost
• Increase f-cost by smallest operator cost.
• Complete and optimal

13
SMA

• Memory Bounded version due to authors
• Beware authors.

14
Hill-climbing

• Goal Optimizing an objective function.
• Does not require differentiable functions
• Can be applied to goal predicate type of
  problems.
• BSAT with objective function number of clauses
  satisfied.
• Intuition Always move to a better state

15
Some Hill-Climbing Algos

• Start random state or special state.
• Until (no improvement)
• Steepest Ascent find best successor
• OR (greedy) select first improving successor
• Go to that successor
• Repeat the above process some number of times
  (Restarts).
• Can be done with partial solutions or full
  solutions.

16
Hill-climbing Algorithm

• In Best-first, replace storage by single node
• Works if single hill
• Use restarts if multiple hills
• Problems
• finds local maximum, not global
• plateaux large flat regions (happens in BSAT)
• ridges fast up ridge, slow on ridge
• Not complete, not optimal
• No memory problems ?

17
Beam

• Mix of hill-climbing and best first
• Storage is a cache of best K states
• Solves storage problem, but
• Not optimal, not complete

18
Local (Iterative) Improving

• Initial state full candidate solution
• Greedy hill-climbing
• if up, do it
• if flat, probabilistically decide to accept move
• if down, dont do it
• We are gradually expanding the possible moves.

19
Local Improving Performance

• Solves 1,000,000 queen problem quickly
• Useful for scheduling
• Useful for BSAT
• solves (sometimes) large problems
• More time, better answer
• No memory problems
• No guarantees of anything

20
Simulated Annealing

• Like hill-climbing, but probabilistically allows
  down moves, controlled by current temperature and
  how bad move is.
• Let t1, t2, be a temperature schedule.
• usually t1 is high, tk 0.9tk-1.
• Let E be quality measure of state
• Goal maximize E.

21
Simulated Annealing Algorithm

• Current random state, k 1
• If Tk 0, stop.
• Next random next state
• If Next is better than start, move there.
• If Next is worse
• Let Delta E(next)-E(current)
• Move to next with probabilty e(Delta/Tk)
• k k1

22
Simulated Annealing Discussion

• No guarantees
• When T is large, edelta/t is close to e0, or 1.
  So for large T, you go anywhere.
• When T is small, edelta/t is close to e-inf, or
  0. So you avoid most bad moves.
• After T becomes 0, one often does simple
  hill-climbing.
• Execution time depends on schedule memory use is
  trivial.

23
Genetic Algorithm

• Weakly analogous to evolution
• No theoretic guarantees
• Applies to nearly any problem.
• Population set of individuals
• Fitness function on individuals
• Mutation operator new individual from old one.
• Cross-over new individuals from parents

24
GA Algorithm (a version)

• Population random set of n individuals
• Probabilistically choose n pairs of individuals
  to mate
• Probabilistically choose n descendants for next
  generation (may include parents or not)
• Probability depends on fitness function as in
  simulated annealing.
• How well does it work? Good question ?

25
Scores to Probabilities

• Suppose the scores of the n individuals are
• a1, a2,.an.
• The probability of choosing the jth individual
• prob aj/(a1a2.an).

26
GA Example

• Problem Boolean Satisfiability.
• Individual bindings for variables
• Mutation flip a variable
• Cross-over For 2 parents, randomly positions
  from 1 parent. For one son take those bindings
  and use other parent for others.
• Fitness number of clauses solved.

27
GA Example

• N-queens problem
• Individual array indicating column where ith
  queen is assigned.
• Mating Cross-over
• Fitness (minimize) number of constraint
  violations

28
GA Discussion

• Reported to work well on some problems.
• Typically not compared with other approaches,
  e.g. hill-climbing with restarts.
• Opinion Works if the mating operator captures
  good substructures.
• Any ideas for GA on TSP?