Advanced Search Techniques

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• Advanced Search Techniques
• CS570 Lecture Notes
• by Jin Hyung Kim
• Computer Science Department
• KAIST

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Optimization and Search

• Search for maximal value of Objective function
• find i MAX Obj(pi)
• where pi is parameter
• Traveling Sales Person problem
• find minimal distance path
• Best-first search keeping track of maximal object
  function value

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Search for Optimal Parameter

• Finding Optimal Weight matrix Neural Network
  training
• Deterministic Method
• Step-by-step procedure
• Hill-Climbing search
• gradient search error back propagation algrthm
• Statistical Method
• Pseudo-random change
• retain if it improve

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Hill Climbing Search

• 1. Set n to be the initial node
• 2. If obj(n) gt max obj(childi (n)) then exit
• 3. Set n to be the highest-value child of n
• 4. Return to step 2
• Terminate at Local maxima
• no guarantee on Global optimal
• Gradient Search
• Hill climbing for continuous, differentiable
  functions
• step width ?
• Slow in near optimal

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Avoiding Local Optimal Problem

• Many starting points
• Momentum factor
• Add Stochastic flavor
• Simulated Annealing

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Simulated Annealing

• Solves combinatorial optimization
• variant of Metropolis algorithm
• by S. Kirkpatric (83)
• finding minimum-energy solution of a neural
  network finding low temperature state of
  physical system
• To overcome local minimum problem
• Instead always going downhill, try to go downhill
  most of the time

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Iterative Statistical

• Simple Iterative Algorithm (TSP)
• 1. find a path p
• 2. make p, a variation of p
• 3. if p is better than p, keep p as p
• 4. goto 2
• Metropolis Algorithm
• 3 if (p is better than p) or (random lt Prob),
  then keep p as p
• a kind of Monte Carlo method
• Simulated Annealing
• T is reduced as time passes

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About T

• Metropolis Algorithm
• Prob p(DE) exp ( DE / T)
• Simulated Annealing
• Prob pi(DE) exp ( DE / Ti)
• if Ti is reduced too fast, poor quality
• if Tt gt T(0) / log(1t) - Geman
• System will converge to minimun configuration
• Tt k/1t - Szu
• Tt a T(t-1) where a is in between 0.8 and 0.99

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Function Simulated Annealing

• current ? select a node (initialize)
• for t ? 1 to ? do
• T ? schedulet
• if T0 then return current
• next ? a random selected successor of current
• ?E ? valuenext - valuecurrent
• if ?E gt 0 then current ? next
• else current ? next only with probability e?E /T

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Genetic Algorithm / Evolutionary Strategy

• Motivated as natural selection and genetics
• mechanisms of evolution
• J. Holland, Adaptation in Natural and Artificial
  Systems, U of Michigan Press, 1975
• one compete for opportunity of reproduction
• survival of fittest individuals
• a.k.o. Probabilistic Search Algorithm
• cost minimization
• high probability of locating global optimal
  solution
• Intl conference on Genetic Algorithms

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Natural Selection

• Genetic contents survival capacity features
• each feature gene
• set of gene chromosome
• Generate-and-Test type algorithm
• Evolution change of species feature
• Survival of the fittest
• Gene of the fittest survive, gene of weaker die
  out
• Reproduction
• new combinations of genes are generated from
  parents - crossover

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Genetic Algorithm

• Encoded representation of solution as bit strings
• reproduction
• selected for next generation
• crossover
• generic material is exchanged
• mutation
• random alteration of bits of strings
• each solution is associated with fitness value

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GA Structure
Initialize population
Termination Cond ?
Select solutions for next population
Perform Crossover and Mutation
Evaluate population
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GA Component

• Population of binary strings
• Control parameters
• Fitness function
• Genetic operators
• crossover
• mutation
• Selection Mechanism
• Mechanism to encode solution to binary string

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Encoding Mechanism

• Encode each solution to unique binary string
• TSP problem
• path representation
• 5 4 3 1 2 - city 5 first, then city 4 , .
• Order representation
• order in the remaining city list
• 5 4 3 1 2 (5 4 3 1 1)
• 1 2 3 4 5 (1 1 1 1 1)
• The representation want to be valid after
  crossover
• Partially matched crossover (p 159)

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Fitness Function

• Objective function that optimized
• fitness of a string
• normalize to range of 0 to 1, if possible
• Selection is based on the evaluation

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Selection

• Models natures survival-of-the-fittest
• fitter string has more offspring
• high chance of surviving
• Proportionate selection scheme
• fi / aver(f)
• Monte Calro method to handle fractions
• Ranking Scheme - predefined surv. Prob.
• Elite Saving Scheme - local minima
• Tournament Selection
• etc.

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Crossover

• Pairs of strings are picked up at random
• Single-point crossover
• randomly cut into two parts
• exchange to form a new string
• Multiple-point crossover
• a variant is Uniform Crossover - using bit mask
• (Alternative)
• if random gt Pc (Xover-rate) Crossover
• otherwise remain unaltered

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Mutation

• (independently) Flipping bits with random
• In order to get diversity
• controlled by Mutation rate, Pm
• if no mutation, only space governed by initial
  population is searched
• If Pm is too large, a random search
• Dynamic change of Pm ge
• Adaptive mutation
• depending on distance of crossover result
• near -gtmore mutation

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Generation Model

• Discrete generation model
• Continuous generation model
• asynchronous operation
• one-at-the-time model
• Typical parameter values
• population size 30 -200
• crossover rate 0.5 -1
• mutation rate 0.001 - 0.05

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Generation n
11
13
2
9
selection
crossover
mutation
8
15
1
13
Generation n1
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Genetic Algorithm

• (defun distribution (population)
• (let ((genotypes (noduplicates population))
• (sum (apply (mapcar fitness
  genotypes))))
• (mapcar (lambda (x)
• (cons (/ (fitness x) sum) x))
• genotypes)))
• (defun reproduce (population)
• (let ((offstring nil)(d (distribution
  population)))
• (dotimes (I (/ (length population) 2))
• (let ((x (select d))(y (select d)))
• (crossover x y)
• (setq offstring (nconc (list x y)
  offstring))))
• offstring))

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(defun select (distribution) (let ((random
(random 1.0))(prob 0) genotype) (some (lambda
(pair) (setq prob ( prob (first
pair))) (if (gt random prob) nil
(setq genotype (rest pair)))) distribution) (
mutate genotype))) (defun mutate
(genotype) (mapcar (lambda (x) (if (gt (random
1.0) 0.03) x (if ( x 1) 0
1))) genotype))) (defun crossover (x y) (if (gt
(random 1.0) 0.6) (list x y) (let ((site
(random (length x))) (swap (rest (nthcdr site
x)))) (setf (rest (nthcdr site x))(rest
(nthcdr site y)) (rest (nthcdr site y))
swap))))
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Artificial Life

• "The study of man-made systems that exhibit
  behaviors characteristic of natural living
  systems."
• Artificial Life is a field of study devoted to
  understanding life by attempting to abstract the
  fundamental dynamical principles underlying
  biological phenomena, and recreating these
  dynamics in other physical media -- such as
  computers -- making them accessible to new kinds
  of experimental manipulation and testing. - C. G.
  Langton
• "Artificial Life is the enterprise of
  understanding biology by constructing biological
  phenomena out of artificial components, rather
  than breaking natural life forms down into their
  component parts. It is the synthetic rather than
  the reductionist approach." - T. S. Ray

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Artificial Life related topics

• Autonomous Agents
• Cellular Automata
• Genetic Algorithms / Programming
• Neural Networks
• Simulators

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Complex Adaptive Systems

• Within science, complexity is a watchword for a
  new way of thinking about the collective behavior
  of many basic but interacting units, be they
  atoms, molecules, neurons, or bits within a
  computer. To be more precise, our definition is
  that complexity is the study of the behavior of
  macroscopic collections of such units that are
  endowed with the potential to evolve in time.
  Their interactions lead to coherent collective
  phenomena, so-called emergent properties that can
  be described only at higher levels than those of
  the individual units. In this sense, the whole is
  more than the sum of its components...
• Peter Coveney and Roger Highfield, Frontiers of
  Complexity The Search for Order in a Chaotic
  World, 1995.

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John Conway's Game of Life

• played on a field of cells, each of which has
  eight neighbors (adjacent cells). A cell is
  either occupied (by an organism) or not. The
  rules for deriving a generation from the previous
  one are these
• Death If an occupied cell has 0, 1, 4, 5, 6, 7,
  or 8 occupied neighbors, the organism dies (0, 1
  neighbors of loneliness 4 thru 8 of
  overcrowding).
• Survival If an occupied cell has two or three
  neighbors, the organism survives to the next
  generation.
• Birth If an unoccupied cell has three occupied
  neighbors, it becomes occupied.
• http//www.tech.org/stuart/life/life.html