Local Search Algorithms CPS 4801

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

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

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

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

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

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

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

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

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

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

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.

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

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

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.

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

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

Simulated Annealing

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.

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

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)