Transcript ppt

Informed Search
ECE457 Applied Artificial Intelligence
Spring 2007
Lecture #3
Outline
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Heuristics
Informed search techniques
More on heuristics
Iterative improvement
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Russell & Norvig, chapter 4
Skip “Genetic algorithms” pages 116-120
(will be covered in Lecture 12)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Recall: Uninformed Search
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Travel blindly until they reach Bucharest
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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An Idea…
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It would be better if the agent knew
whether or not the city it is travelling to
gets it closer to Bucharest
Of course, the agent doesn’t know the
exact distance or path to Bucharest (it
wouldn’t need to search otherwise!)
The agent must estimate the distance
to Bucharest
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Heuristic Function
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More generally:
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We want the search algorithm to be able to
estimate the path cost from the current
node to the goal
This estimate is called a heuristic
function
Cannot be done based on problem
formulation
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Need to add additional information
Informed search
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R. Khoury (2007)
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Heuristic Function
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Heuristic function h(n)
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h(n): estimated cost from node n to goal
h(n1) < h(n2) means it’s probably cheaper
to get to the goal from n1
h(ngoal) = 0
Path cost g(n)
Evaluation function f(n)
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f(n) = g(n)
f(n) = h(n)
f(n) = g(n) + h(n)
ECE457 Applied Artificial Intelligence
Uniform Cost
Greedy Best-First
A*
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Greedy Best-First Search
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f(n) = h(n)
Always expand the node closest to the
goal and ignore path cost
Complete only if m is finite
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Not optimal
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Rarely true in practice
Can go down a long path of cheap actions
Time complexity = O(bm)
Space complexity = O(bm)
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R. Khoury (2007)
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Greedy Best-First Search
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Worst case: goal is last node of the tree
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Number of nodes generated:
b nodes for each node of m levels (entire tree)
Time and space complexity: all generated nodes O(bm)
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A* Search
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f(n) = g(n) + h(n)
Best-first search
Complete
Optimal, given admissible heuristic
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Optimally efficient
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Never overestimates the cost to the goal
No other optimal algorithm will expand less nodes
Time complexity = O(bC*/є)
Space complexity = O(bC*/є)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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A* Search
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Worst case: heuristic is the trivial h(n) = 0
A* becomes Uniform Cost Search
Goal has path cost C*, all other actions have minimum cost of є
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Depth explored before taking action C*: C*/є
Number of generated nodes: O(bC*/є)
Space & time complexity: all generated nodes
C*
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A* Search
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Using a good heuristic can reduce time
complexity
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Can go down to O(bm)
However, space complexity will always
be exponential
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A* runs out of memory before running out
of time
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R. Khoury (2007)
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Iterative Deepening A* Search
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Like Iterative Deepening Search, but
cut-off limit is f-value instead of depth
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Next iteration limit is the smallest f-value of
any node that exceeded the cut-off of
current iteration
Properties
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Complete and optimal like A*
Space complexity of depth-first search
Performs poorly if small action cost (small
step in each iteration)
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Simplified Memory-Bounded A*
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Uses all available memory
When memory limit reached, delete
worst leaf node (highest f-value)
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If equality, delete oldest leaf node
SMA memory problem
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If the entire optimal path fills the memory
and there is only one non-goal leaf node
SMA cannot continue expanding
Goal is not reachable
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Simplified Memory-Bounded A*
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Space complexity known and controlled
by system designer
Complete if shallowest goal depth less
than memory size
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Shallowest goal is reachable
Optimal if optimal goal is reachable
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Example: Greedy Search
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h(n) = straight-line distance
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Example: A* Search
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h(n) = straight-line distance
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Heuristic Function Properties
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Admissible
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Consistency / Monotonicity
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Never overestimate the cost
h(np) ≤ h(nc) + cost(np,nc)
h(np) + g(np) ≤ h(nc) + cost(np,nc) + g(np)
h(np) + g(np) ≤ h(nc) + g(nc)
f(np) ≤ f(nc)
f(n) never decreases as we get closer to the
goal
Domination
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h1(n) ≥ h2(n) for all n
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Creating Heuristic Functions
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Found by relaxing the problem
Straight-line distance to Bucharest
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Eliminate constraint of traveling on roads
8-puzzle
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Move each square that’s out of
place (7)
Move by the number of squares
to get to place (12)
Move some tiles in place
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Creating Heuristic Functions
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Puzzle: move the red block through the exit
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Action: move a block, if the path is clear
A block can be moved any distance along a clear
path in one action
Design a heuristic for this game
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Relax by assuming that the red block can get to
the exit following the path that has the fewest
blocks in the way
Further relax by assuming that each block in the
way requires only one action to be moved out of
the way
But blocks must be moved out of the way! If
there are no blank spots out of the way then
another block will need to be moved
h(n) = 1 (cost of moving the red block to the exit)
+ 1 for each block in the way + 1 for each 2 outof-the-way blank spots needed
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Creating Heuristic Functions
State
g(n)
h(n)
Cost
0
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Creating Heuristic Functions
State
g(n)
h(n)
Cost
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Path to the Goal
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Sometimes the path to the goal is
irrelevant
Only the solution matters
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n-queen puzzle
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Different Search Problem
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No longer minimizing path cost
Improve quality of state
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Minimize state cost
Maximize state payoff
Iterative improvement
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Example: Iterative Improvement
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Minimize cost: number of attacks
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Example: Travelling Salesman
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Tree search method
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Iterative improvement
method
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ECE457 Applied Artificial Intelligence
Start with home city
Visit next city until
optimal round trip
Start with random
round trip
Swap cities until optimal
round trip
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Graphic Visualisation
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State value / state plot: state space
Value
Global maximum
Local maxima
Plateau
Global
minimum
State
Local minima
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Graphic Visualisation
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More complex state spaces can have
several dimensions
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Example: States are X-Y coordinates, state
value is Z coordinate
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Hill Climbing (Gradient Descent)
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Simple but efficient local optimization
strategy
Always take the action that most
improves the state
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R. Khoury (2007)
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Hill Climbing (Gradient Descent)
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Generate random initial state
Each iteration
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Generate and evaluate neighbours at step
size
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Move to neighbour with greatest
increase/decrease (i.e. take one step)
End when there are no better
neighbours
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Example: Travelling to Toronto
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Trying to get to downtown Toronto
Take steps toward the CN Tower
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Hill Climbing (Gradient Descent)
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Advantages
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Disadvantages
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Fast
No search tree
Gets stuck in local optimum
Does not allow worse moves
Solution dependant on initial state
Selecting step size
Common improvements
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Random restarts
Intelligently-chosen initial state
Decreasing step size
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Simulated Annealing
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Problem with hill climbing: local best
move doesn’t lead to optimal goal
Solution: allow bad moves
Simulated annealing is a popular way of
doing that
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Stochastic search method
Simulates annealing process in metallurgy
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Annealing
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Tempering technique in metallurgy
Weakness and defects come from
atoms of crystals freezing in the wrong
place (local optimum)
Heating to unstuck the atoms (escape
local optimum)
Slow cooling to allow atoms to get to
better place (global optimum)
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Simulated Annealing
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Allow some bad moves
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Bad enough to get out of local optimum
Not so bad as to get out of global optimum
Probability of accepting bad moves
given
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Badness of the move (i.e. variation in state
value V)
Temperature T
P = e-V/T
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Simulated Annealing
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Generate random initial state and high
temperature
Each iteration
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Generate and evaluate a random neighbour
If neighbour better than current state
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Else (if neighbour worse than current state)
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Accept
Accept with probability e-V/T
Reduce temperature
End when temperature less than threshold
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R. Khoury (2007)
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Simulated Annealing
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Advantages
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Avoids local optima
Very good at finding high-quality solutions
Very good for hard problems with complex state
value functions
Disadvantage
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Can be very slow in practice
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
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Simulated Annealing Application
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Traveling-wave tube (TWT)
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Uses focused electron beam to amplify
electromagnetic communication waves
Produces high-power radio frequency (RF)
signals
Critical components in deep-space
probes and communication satellites
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Power efficiency becomes a key issue
TWT research group at NASA working for
over 30 years on improving power efficiency
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Simulated Annealing Application
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Optimizing TWT efficiency
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Synchronize electron velocity and phase velocity of
RF wave
Using “phase velocity tapper” to control and
decrease RF wave phase velocity
Improving tapper design improves synchronization,
improves efficiency of TWT
Tapper with simulated annealing algorithm to
optimize synchronization
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Doubled TWT efficiency
More flexible then past tappers
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Maximize overall power efficiency
Maximize efficiency over various bandwidth
Maximize efficiency while minimize signal distortion
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Assumptions
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Goal-based agent
Environment
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Fully observable
Deterministic
Sequential
Static
Discrete
Single agent
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Assumptions Updated
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Utility-based agent
Environment
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Fully observable
Deterministic
Sequential
Static
Discrete / Continuous
Single agent
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