Introduction to Artificial Intelligence

Local Search(updated 4/30/2006)

Henry Kautz

Local Search in Continuous Spaces

initial state vector

( ) quantity to optimized

step size

until Goal_Test( ) do

( )

S

f S

S

fS S S

S

negative step to minimize fpositive step to maximize f

grad

ient

Local Search in Discrete State Spaces

state = choose_start_state();while ! GoalTest(state) do

state := arg min { h(s) | s in Neighbors(state) }endreturn state;

• Terminology: – “neighbors” instead of “children”– heuristic h(s) is the “objective function”, no need to be

admissible• No guarantee of finding a solution

– sometimes: probabilistic guarantee• Best goal-finding, not path-finding• Many variations

Local Search versus Systematic Search

• Systematic Search– BFS, DFS, IDS, Best-First, A*– Keeps some history of visited nodes– Always complete for finite search spaces,

some versions complete for infinite spaces– Good for building up solutions incrementally

• State = partial solution• Action = extend partial solution

Local Search versus Systematic Search

• Local Search– Gradient descent, Greedy local search,

Simulated Annealing, Genetic Algorithms– Does not keep history of visited nodes– Not complete. May be able to argue will

terminate with “high probability”– Good for “fixing up” candidate solutions

• State = complete candidate solution that may not satisfy all constraints

• Action = make a small change in the candidate solution

N-Queens Problem

N-Queens Systematic Searchstate = choose_start_state();add state to Fringe;while ! GoalTest(state) do

choose state from Fringe according to h(state);Fringe = Fringe U { Children(state) }

endreturn state;

• start = empty board• GoalTest = N queens are on the board• h = (N – number of queens on the board) • children = all ways of adding one queen without creating

any attacks

N-Queens Local Search, V1state = choose_start_state();while ! GoalTest(state) do

state := arg min { h(s) | s in Neighbors(state) }endreturn state;

• start = put down N queens randomly• GoalTest = Board has no attacking pairs• h = number of attacking pairs • neighbors = move one queen to a different square on the

board

N-Queens Local Search, V2state = choose_start_state();while ! GoalTest(state) do

state := arg min { h(s) | s in Neighbors(state) }endreturn state;

• start = put a queen on each square with 50% probability• GoalTest = Board has N queens, no attacking pairs• h = (number of attacking pairs + max(0, N - # queens))• neighbors = add or delete one queen

N Queens Demo

States Where Greedy Search Must Succeed

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States Where Greedy Search Might Succeed

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Local Search Landscapeo

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Local Minimum

Plateau

Variations of Greedy Search

• Where to start?– RANDOM STATE– PRETTY GOOD STATE

• What to do when a local minimum is reached?– STOP– KEEP GOING

• Which neighbor to move to?– BEST neighbor– Any BETTER neighbor (Hill Climbing)

• How to make local search more robust?

Restarts

for run = 1 to max_runs dostate = choose_start_state();flip = 0;while ! GoalTest(state) && flip++ < max_flips do

state := arg min { h(s) | s in Neighbors(state) }endif GoalTest(state) return state;

endreturn FAIL

Uphill Moves: Random Noise

state = choose_start_state();while ! GoalTest(state) do

with probability noise dostate = random member Neighbors(state)

else state := arg min { h(s) | s in

Neighbors(state) }end

endreturn state;

Uphill Moves: Simulated Annealing (Constant Temperature)

state = start;while ! GoalTest(state) do

next = random member Neighbors(state);deltaE = h(next) – h(state);if deltaE 0 then

state := next;else

with probability e-deltaE/temperature dostate := next;

endendif

endreturn state;

Book reverses, because is looking

for max h state

Uphill Moves: Simulated Annealing (Geometric Cooling Schedule)

temperature := start_temperature;state = choose_start_state();while ! GoalTest(state) do

next = random member Neighbors(state);deltaE = h(next) – h(state);if deltaE 0 then

state := next;else

with probability e-deltaE/temperature dostate := next;

endtemperature := cooling_rate * temperature;

endreturn state;

Simulated Annealing

• For any finite problem with a fully-connected state space, will provably converge to optimum as length of schedule increases:

• But: fomal bound requires exponential search time

• In many practical applications, can solve problems with a faster, non-guaranteed schedule

cooling_rate 1lim Pr(optimum) 1

Other Local Search Strategies

• Tabu Search– Keep a history of the last K visited states– Revisiting a state on the history list is “tabu”

• Genetic algorithms– Population = set of K multiple search points– Neighborhood = population U mutations U crossovers

• Mutation = random change in a state• Crossovers = random mix of assignments from two states• Typically only a portion of neighbor is generated

– Search step: new population = K best members of neighborhood