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LOCAL SEARCH AND CONTINUOUS SEARCH

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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

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

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

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

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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

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A different view of search

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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 ”

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Hill - climbing search

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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

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Hill - climbing search : 8- queens problem 5 steps later… A local minimum with h = 1 (a common problem with hill climbing)

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Drawbacks of hill climbing Problem : depending on initial state, can get stuck in local maxima

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Approaches to local minima Try again Sideways moves

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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

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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 ?

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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

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Comparison of approaches for 8- queens problem TechniqueSuccess rateAverage number of moves Hill Climbing14%3.9 Hill 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

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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…

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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

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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

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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 )

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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

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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.

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Genetic algorithms Inspired by nature New states generated from two parent states. Throw some randomness into the mix as well…

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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

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Genetic algorithms 32752411

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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.

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Genetic algorithms Probability of selection is weighted by the normalized fitness function.

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Genetic algorithms Probability of selection is weighted by the normalized fitness function. Crossover from the top two parents.

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Genetic algorithms

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Genetic Algorithms 1. Initialize population ( k random states ) 2. Calculate fitness function 3. Select pairs for crossover 4. Apply mutation 5. Evaluate fitness of children 6. From the resulting population of 2* k individuals, probabilistically pick k of the best. 7. Repeat.

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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

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Example

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What can we do to solve this problem ?

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Searching Continuous Space Discretize the state space Turn it into a grid and do what we ’ ve always done.

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

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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

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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

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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

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Questions ?

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“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 Link to the Article

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Local Search Algorithms and Optimization Problems

Local Search Algorithms and Optimization Problems

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