LOCAL SEARCHAND

CONTINUOUS SEARCH

Local search algorithms In many optimization problems, the path

to the goal is irrelevant; the goal state itself is the solution

In such cases, we can use local search algorithms keep a (sometimes) single "current" state, try

to improve it

Example: n-queens Put n queens on an n × n board with no

two queens on the same row, column, or diagonal

Example: n-queens Put n queens on an n × n board with no

two queens on the same row, column, or diagonal

Example: n-queens Put n queens on an n × n board with no

two queens on the same row, column, or diagonal

Local Search Operates by keeping track of only the

current node and moving only to neighbors of that node

Often used for: Optimization problems Scheduling Task assignment …many other problem where the goal is to

find the best state according to some objective function

A different view of search

Hill-climbing search Consider next possible moves (i.e.

neighbors) Pick the one that improves things the most

“Like climbing Everest in thick fog with amnesia”

Hill-climbing search

Hill-climbing search: 8-queens problem

h = number of pairs of queens that are attacking each other, either directly or indirectly

h = 17 for the above state

Hill-climbing search: 8-queens problem

• 5 steps later…• A local minimum with h = 1 (a common problem

with hill climbing)

Drawbacks of hill climbing Problem: depending on initial state, can

get stuck in local maxima

Approaches to local minima Try again Sideways moves

Try, try again Run algorithm some number of times and

return the best solution Initial start location is usually chosen randomly

If you run it “enough” times, will get answer (in the limit)

Drawback: takes lots of time

Sideways moves If stuck on a ridge, if we wait awhile and

allow flat moves, will become unstuck—maybe

Questions How long is awhile? How likely to become unstuck?

Any other extensions? First-choice hill climbing

Generate successors randomly until a good one is found

Look three moves ahead Unstuck from certain areas More inefficient Might not be any better Move quality: as good or better

Comparison of approaches for 8-queens problem

Technique Success rate Average number of moves

Hill Climbing 14% 3.9Hill Climbing + 6 restarts if needed

65% 11.5

Hill Climbing + up to 100 sideways moves if needed

94% 21

• Tradeoff between success rate and number of moves

• As success rate approaches 100% number of moves will increase rapidly

Nice properties of local search Can often get “close”

When is this useful?

Can trade off time and performance

Can be applied to continuous problems E.g. first-choice hill climbing More on this later…

Simulated annealing Insight: all of the modifications to hill

climbing are really about injecting variance Don’t want to get stuck in local maxima or

plateu

Idea: explicitly inject variability into the search process

Properties of simulated annealing

More variability at the beginning of search Since you have little confidence you’re in right place

Variability decreases over time Don’t want to move away from a good solution

Probability of picking move is related to how good it is Sideways or slight decreases are more likely than

major decreases

How simulated annealing works At each step, have temperature T

Pick next action semi-randomly Higher temperature increase randomness Select action according to goodness and

temperature Decrease temperature slightly at each time

step until it reaches 0 (no randomness)

Local Beam Search Keep track of k states rather than just one

Start with k randomly generated states

At each iteration, all the successors of all k states are generated

If any one is a goal state, stop; else select the k best successors from the complete list and repeat. Results in states getting closer together over time

Stochastic Local Beam Search

Designed to prevent all k states clustering together

Instead of choosing k best, choose k successors at random, with higher probability of choosing better states.

Terminology: stochastic means random.

Genetic algorithms Inspired by nature

New states generated from two parent states. Throw some randomness into the mix as well…

Genetic Algorithms Initialize population (k random states) Select subset of population for mating Generate children via crossover

Continuous variables: interpolate Discrete variables: replace parts of their

representing variables Mutation (add randomness to the children's

variables) Evaluate fitness of children Replace worst parents with the children

Genetic algorithms

32752411

Genetic algorithms

Fitness function: number of non-attacking pairs of queens (min = 0, max = 8 × 7/2 = 28) 24/(24+23+20+11) = 31% 23/(24+23+20+11) = 29% … etc.

Genetic algorithms

Probability of selection is weighted by the normalized fitness function.

Genetic algorithms

Probability of selection is weighted by the normalized fitness function.

Crossover from the top two parents.

Genetic algorithms

Genetic Algorithms1. Initialize population (k random states)2. Calculate fitness function3. Select pairs for crossover4. Apply mutation5. Evaluate fitness of children6. From the resulting population of 2*k

individuals, probabilistically pick k of the best.

7. Repeat.

Searching Continuous Spaces Continuous: Infinitely many values. Discrete: A limited number of

distinct, clearly defined values.

In continuous space, cannot consider all next possible moves (infinite branching factor) Makes classic hill climbing impossible

Example Want to put 3 airports in Romania, such

that the sum of squared distances from each city on the map to its closest airport is minimized.

State: coordinates of the airports

Objective function:, ¿∑𝑖=1

3

∑𝑐∈𝐶 𝑖

(𝑥𝑖−𝑥𝑐 )2+( 𝑦 𝑖+𝑦𝑐 )2

Example What can we do to solve this problem?

Searching Continuous Space

Discretize the state space Turn it into a grid and do what we’ve always

done.

Searching Continuous Space Calculate the gradient of the objective

function at the current state.

Take a step of size in the direction of the steepest slope

Problem: Can be hard or impossible to calculate. Solution: approximate the gradient through sampling.

Step size Very small takes a long time to reach

the peak Very big can overshoot the goal

What can we do…? Start high and decrease with time Make it higher for flatter parts of the space

Summary Local search often finds an approximate

solution (i.e. it end in “good” but not “best” states)

Can inject randomness to avoid getting stuck in local maxima

Can trade off time for higher likelihood of success

Real World Problems “many real world problems have a

landscape that looks more like a widely scattered family of balding porcupines on a flat floor, with miniature porcupines living on the tip of each porcupine needle, ad infinitum.”

-Russell and Norvig

Questions?

“One of the popular myths of higher education is that professors are sadists who live to inflict psychological trauma on undergraduates. …”

… “I do not “take off” points. You earn them. The difference is not merely rhetorical, nor is it trivial. In other words, you start with zero points and earn your way to a grade.”

… “this means that the burden of proof is on you to demonstrate that you have mastered the material. It is not on me to demonstrate that you have not. ”

Dear Student: I Don't Lie Awake At Night Thinking of Ways to Ruin Your Life

Art Caden, for Forbes.com

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