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Searchadditional2 Searchadditional2 Presentation Transcript

  • Chapter 4 Informed Search and Exploration
  • Outline
    • Informed (Heuristic) search strategies
      • (Greedy) Best-first search
      • A* search
    • (Admissible) Heuristic Functions
      • Relaxed problem
      • Subproblem
    • Local search algorithms
      • Hill-climbing search
      • Simulated anneal search
      • Local beam search
      • Genetic algorithms
    • Online search *
      • Online local search
      • learning in online search
  • Informed search strategies
    • Informed search
      • uses problem-specific knowledge beyond the problem definition
      • finds solution more efficiently than the uninformed search
    • Best-first search
      • uses an evaluation function f ( n ) for each node
        • e.g., Measures distance to the goal – lowest evaluation
      • Implementation :
        • fringe is a queue sorted in increasing order of f -values.
      • Can we really expand the best node first?
        • No! only the one that appears to be best based on f ( n ).
      • heuristic function h ( n )
        • estimated cost of the cheapest path from node n to a goal node
      • Specific algorithms
        • greedy best-first search
        • A* search
  • Greedy best-first search
      • expand the node that is closest to the goal
      • : Straight line distance heuristic
  • Greedy best-first search example
  • Properties of Greedy best-first search
      • Complete ?
      • Optimal ?
      • Time ?
      • Space ?
    No No – can get stuck in loops, e.g., Iasi –> Neamt – > Iasi – > Neamt Yes – complete in finite states with repeated-state checking , but a good heuristic function can give dramatic improvement – keeps all nodes in memory
  • A* search
      • evaluation function f ( n ) = g ( n ) + h ( n )
        • g ( n ) = cost to reach the node
        • h ( n ) = estimated cost to the goal from n
        • f ( n ) = estimated total cost of path through n to the goal
      • an admissible (optimistic) heuristic
        • never overestimates the cost to reach the goal
        • estimates the cost of solving the problem is less than it actually is
        • e.g., never overestimates the actual road distances
      • A* using Tree-Search is optimal if h ( n ) is admissible
      • could get suboptimal solutions using Graph-Search
        • might discard the optimal path to a repeated state if it is not the first one generated
        • a simple solution is to discard the more expensive of any two paths found to the same node (extra memory)
  • : Straight line distance heuristic
  • A* search example
  • Optimality of A*
    • Consistency (monotonicity)
      • n’ is any successor of n, general triangle inequality ( n , n’ , and the goal)
      • consistent heuristic is also admissible
    • A* using Graph-Search is optimal if h ( n ) is consistent
      • the values of f ( n ) along any path are nondecreasing
  • Properties of A*
    • Suppose C * is the cost of the optimal solution path
      • A* expands all nodes with f ( n ) < C *
      • A* might expand some of nodes with f ( n ) = C * on the “goal contour”
      • A* will expand no nodes with f ( n ) > C *, which are pruned !
      • Pruning : eliminating possibilities from consideration without examination
    • A* is optimally efficient for any given heuristic function
      • no other optimal algorithm is guaranteed to expand fewer nodes than A*
      • an algorithm might miss the optimal solution if it does not expand all nodes with f ( n ) < C *
    • A* is complete
    • Time complexity
      • exponential number of nodes within the goal contour
    • Space complexity
      • keeps all generated nodes in memory
      • runs out of space long before runs out of time
  • Memory-bounded heuristic search
    • Iterative-deepening A* (IDA*)
      • uses f -value ( g + h ) as the cutoff
    • Recursive best-first search (RBFS)
      • replaces the f -value of each node along the path with the best f -value of its children
      • remembers the f -value of the best leaf in the “forgotten” subtree so that it can reexpand it later if necessary
      • is efficient than IDA* but generates excessive nodes
      • changes mind : go back to pick up the second-best path due to the extension ( f -value increased) of current best path
      • optimal if h ( n ) is admissible
      • space complexity is O ( bd )
      • time complexity depends on the accuracy of h ( n ) and how often the current best path is changed
    • Exponential time complexity of Both IDA* and RBFS
      • cannot check repeated states other than those on the current path when search on Graphs – Should have used more memory (to store the nodes visited)!
  • : Straight line distance heuristic
  • RBFS example
  • Memory-bounded heuristic search (cont’d)
    • SMA* – Simplified MA* (Memory-bounded A*)
      • expands the best leaf node until memory is full
      • then drops the worst leaf node – the one has the highest f -value
      • regenerates the subtree only when all other paths have been shown to look worse than the path it has forgotten
      • complete and optimal if there is a solution reachable
      • might be the best general-purpose algorithm for finding optimal solutions
    • If there is no way to balance the trade off between time an memory, drop the optimality requirement !
  • (Admissible) Heuristic Functions h 1 ? h 2 ? = the number of misplaced tiles = total Manhattan (city block) distance = 7 tiles are out of position = 4+0+3+3+1+0+2+1 = 14
  • Effect of heuristic accuracy
      • Effective branching factor b*
        • total # of nodes generated by A* is N , the solution depth is d
        • b* is b that a uniform tree of depth d containing N +1 nodes would have
        • well-designed heuristic would have a value close to 1
        • h 2 is better than h 1 based on the b*
      • Domination
        • h 2 dominates h 1 if for any node n
        • A* using h 2 will never expand more nodes than A* using h 1
        • every node n with will be expanded
        • the larger the better, as long as it does not overestimate!
  • Inventing admissible heuristic functions
      • h 1 and h 2 are solutions to relaxed (simplified) version of the puzzle.
        • If the rules of the 8-puzze are relaxed so that a tie can move anywhere , then h 1 gives the shortest solution
        • If the rules are relaxed so that a tile can move to any adjacent square , then h 2 gives the shortest solution
      • Relaxed problem : A problem with fewer restrictions on the actions
        • Admissible heuristics for the original problem can be derived from the optimal (exact) solution to a relaxed problem
        • Key point : the optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the original problem
        • Which should we choose if none of the h 1 … h m dominates any of the others?
        • We can have the best of all worlds, i.e., use whichever function is most accurate on the current node
      • Subproblem *
        • Admissible heuristics for the original problem can also be derived from the solution cost of the subproblem.
      • Learning from experience *
  • Local search algorithms and optimization
    • Systematic search algorithms
      • to find (or given) the goal and to find the path to that goal
    • Local search algorithms
      • the path to the goal is irrelevant , e.g., n -queens problem
      • state space = set of “complete” configurations
      • keep a single “current” state and try to improve it, e.g., move to its neighbors
      • Key advantages:
        • use very little (constant) memory
        • find reasonable solutions in large or infinite (continuous) state spaces
      • (pure) Optimization problem:
        • to find the best state (optimal configuration ) based on an objective function , e.g. reproductive fitness – Darwinian, no goal test and path cost
  • Local search – example
  • Local search – state space landscape
      • elevation = the value of the objective function or heuristic cost function
    global minimum heuristic cost function
      • A complete local search algorithm finds a solution if one exists
      • A optimal algorithm finds a global minimum or maximum
    • moves in the direction of increasing value until a “ peak ”
      • current node data structure only records the state and its objective function
      • neither remember the history nor look beyond the immediate neighbors
      • like climbing Mount Everest in thick fog with amnesia
    Hill-climbing search
    • complete-state formulation for 8-queens
      • successor function returns all possible states generated by moving a single queen to another square in the same column (8 x 7 = 56 successors for each state)
      • the heuristic cost function h is the number of pairs of queens that are attacking each other
    Hill-climbing search - example best moves reduce h = 17 to h = 12 local minimum with h = 1
  • Hill-climbing search – greedy local search
    • Hill climbing, the greedy local search, often gets stuck
      • Local maxima : a peak that is higher than each of its neighboring states, but lower than the global maximum
      • Ridges : a sequence of local maxima that is difficult to navigate
      • Plateau : a flat area of the state space landscape
        • a flat local maximum : no uphill exit exists
        • a shoulder : possible to make progress
      • can only solve 14% of 8-queen instance but fast (4 steps to S and 3 to F )
  • Hill-climbing search – improvement
    • Allows sideways move : with hope that the plateau is a shoulder
      • could stuck in an infinite loop when it reaches a flat local maximum
      • limits the number of consecutive sideways moves
      • can solve 94% of 8-queen instances but slow (21 steps to S and 64 to F )
    • Variations
      • stochastic hill climbing
        • chooses at random ; probability of selection depends on the steepness
      • first choice hill climbing
        • randomly generates successors to find a better one
      • All the hill climbing algorithms discussed so far are incomplete
        • fail to find a goal when one exists because they get stuck on local maxima
      • Random-restart hill climbing
        • conducts a series of hill-climbing searches; randomly generated initial states
      • Have to give up the global optimality
        • landscape consists of a large amount of porcupines on a flat floor
        • NP-hard problems
  • Simulated annealing search
      • combine hill climbing ( efficiency ) with random walk ( completeness )
      • annealing : harden metals by heating metals to a high temperature and gradually cooling them
      • getting a ping-pong ball into the deepest crevice in a humpy surface
        • shake the surface to get the ball out of the local minima
        • not too hard to dislodge it from the global minimum
      • simulated annealing :
        • start by shaking hard (at a high temperature) and then gradually reduce the intensity of the shaking (lower the temperature)
        • escape the local minima by allowing some “bad” moves
        • but gradually reduce their size and frequency
  • Simulated annealing search - Implementation
      • Always accept the good moves
      • The probability to accept a bad move
        • decreases exponentially with the “ badness ” of the move
        • decreases exponentially with the “ temperature ” T (decreasing)
      • finds a global optimum with probability approaching 1 if the schedule lowers T slowly enough
  • Local beam search
      • Local beam search : keeps track of k states rather than just one
        • generates all the successors of all k states
        • selects the k best successors from the complete list and repeats
        • quickly abandons unfruitful searches and moves to the space where the most progress is being made
        • – “ Come over here, the grass is greener! ”
        • lack of diversity among the k states
      • stochastic beam search : chooses k successors at random , with the probability of choosing a given successor having an increasing value
      • natural selection: the successors (offspring) if a state (organism) populate the next generation according to is value (fitness).
  • Genetic algorithms
      • Genetic Algorithms (GA): successor states are generated by combining two parents states.
        • population : s set of k randomly generated states
        • each state, called individual , is represented as a string over a finite alphabet, e.g. a string of 0s and 1s; 8-queens : 24 bits or 8 digits for their positions
        • fitness (evaluation) function : return higher values for better states,
        • e.g., the number of nonattacking pairs of queens
        • randomly choosing two pairs for reproducing based on the probability ; proportional to fitness score; not choosing the similar ones too early
  • Genetic algorithms (cont’d)
      • schema : a substring in which some of the positions can be left unspecified
        • instances : strings that match the schema
        • GA works best when schemas correspond to meaningful components of a solution.
      • a crossover point is randomly chosen from the positions in the string
        • larger steps in the state space early and smaller steps later
      • each location is subject to random mutation with a small independent probability