1. Local Search Algorithms
CPS 4801

2. Outline Hill-Climbing Search Simulated Annealing
Local Beam Search (briefly)

3. Local search algorithms
In many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution.
Find the final 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
generally move to neighbors
The path are not retained

4. Local search algorithms
uses very little memory
useful for solving pure optimization problems
can often find reasonable solutions in large state spaces.

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

6. Hill-climbing search (steepest-ascent version)
A simple loop that continuously moves in the direction of increasing value – uphill
Terminates when reaches a “peak”
does not look ahead beyond the immediate neighbors, does not maintain a search tree

7. 8-queens problem Each state has 8*7 = 56 successors.
complete-state formulation
vs.
incremental formulation

8. 8-queens problem
h = number of pairs of queens that are attacking each other, either directly or indirectly (h=0 solution)
h = 17 for the above state

9. Hill-climbing search “Greedy local search” makes rapid progress
grabs a good neighbor state without thinking ahead about where to go next
makes rapid progress

10. Hill-climbing search: 8-queens problem
5 steps from the state in the previous slide
A local minimum with h = 1

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

12. Hill-climbing search
 Starting from a randomly generated 8-queen state, steepest-ascent hill climbing gets stuck 86% of the time.
 It takes 4 steps on average when it succeeds and 3 when it gets stuck.
The steepest ascent version halts if the best successor has the same value as the current.

13. Hill-climbing search allow a sideways move
shoulder
flat local maximum, that is not a shoulder
Solution: a limit on the number of consecutive sideway moves
E.g., 100 consecutive sideways movies in the 8-queens problem
successful rate: raises from14% to 94%
cost: 21 steps on average for each successful instance, 64 for each failure

14. Variants of hill climbing
Stochastic hill climbing
chooses at random from among the uphill moves
converge more slowly, but finds better solutions
First-choice hill climbing
generates successors randomly until one is better than the current state
good when with many (thousands) of successors

15. Variants of hill climbing
Random-restart hill climbing
“If you don’t succeed, try, try again.”
conducts a series of hill-climbing searches from randomly generated initial states, until a goal is found.

16. Simulated Annealing
A hill-climbing algorithm that never makes “downhill” moves is guaranteed to be incomplete.
Idea: escape local maxima by allowing some “bad” moves

17. Simulated Annealing Picks a random move (instead of the best)
If “good move”
accepted;
else
accepted with some probability
The probability decreases exponentially with the “badness” of the move

18. Simulated Annealing

19. Simulated Annealing
Simulated annealing was first used extensively to solve VLSI (Very-Large-Scale Integration) layout problems.
It has been applied widely to factory scheduling and other large-scale optimization tasks.

20. Local Beam Search
Idea: keep k states instead of 1; choose top k of all their successors
Not the same as k searches run in parallel!
Searches that find good states recruit other searches to join them
moves the resources to where the most progress is being made

21. Local Beam Search
Problem: quite often, all k states end up on same local hill (concentrated in a small region)
Idea: choose k successors randomly (stochastic beam search)