Informed Search

Informed Search

ECE457 Applied Artificial IntelligenceFall 2007

Lecture #3

ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 2

Outline Heuristics Informed search techniques More on heuristics Iterative improvement

Russell & Norvig, chapter 4 Skip “Genetic algorithms” pages 116-

120 (will be covered in Lecture 12)


Recall: Uninformed Search Travel blindly until they reach

Bucharest


An Idea… 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


Heuristic Function More generally:

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 Need to add additional information Informed search


Heuristic Function Heuristic function h(n)

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)

f(n) = g(n) Uniform Cost f(n) = h(n) Greedy Best-First f(n) = g(n) + h(n) A*


Greedy Best-First Search f(n) = h(n) Always expand the node closest to

the goal and ignore path cost Complete only if m is finite

Rarely true in practice Not optimal

Can go down a long path of cheap actions

Time complexity = O(bm) Space complexity = O(bm)


Greedy Best-First Search Worst case: goal is last node of the tree

Number of nodes generated:b nodes for each node of m levels (entire tree)

Time and space complexity: all generated nodes O(bm)


A* Search f(n) = g(n) + h(n) Best-first search Complete Optimal, given admissible heuristic

Never overestimates the cost to the goal Optimally efficient

No other optimal algorithm will expand less nodes

Time complexity = O(bC*/є) Space complexity = O(bC*/є)


A* Search 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

Depth explored before taking action C*: C*/є Number of generated nodes: O(bC*/є) Space & time complexity: all generated nodes

C* є

є є

є є

є є є є

є є

є є є є


A* Search Using a good heuristic can reduce

time complexity Can go down to O(bm)

However, space complexity will always be exponential A* runs out of memory before running

out of time


Iterative Deepening A* Search Like Iterative Deepening Search, but

cut-off limit is f-value instead of depth Next iteration limit is the smallest f-value of

any node that exceeded the cut-off of current iteration

Properties Complete and optimal like A* Space complexity of depth-first search

(because it’s possible to delete nodes and paths from memory when we explore down to the cut-off limit)

Performs poorly if small action cost (small step in each iteration)


Simplified Memory-Bounded A* Uses all available memory When memory limit reached, delete

worst leaf node (highest f-value) If equality, delete oldest leaf node

SMA memory problem 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


Simplified Memory-Bounded A* Space complexity known and

controlled by system designer Complete if shallowest goal depth

less than memory size Shallowest goal is reachable

Optimal if optimal goal is reachable


Example: Greedy Search h(n) = straight-line distance


Example: A* Search h(n) = straight-line distance


Heuristic Function Properties Admissible

Never overestimate the cost Consistency / Monotonicity

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 h1(n) ≥ h2(n) for all n


Creating Heuristic Functions Found by relaxing the problem Straight-line distance to Bucharest

Eliminate constraint of traveling on roads

8-puzzle 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


Creating Heuristic Functions

Puzzle: move the red block through the exit

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

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 out-of-the-way blank spots needed



State

g(n) 0 70 82

h(n) 6 3 3

Cost 87 17 5



State

g(n) 4 20 24

h(n) 8 7 8

Cost 83 67 63


Path to the Goal Sometimes the path to the goal is

irrelevant Only the solution matters

n-queen puzzle


Different Search Problem No longer minimizing path cost Improve quality of state

Minimize state cost Maximize state payoff

Iterative improvement


Example: Iterative Improvement

Minimize cost: number of attacks


Example: Travelling Salesman

Tree search method Start with home city Visit next city until

optimal round trip Iterative

improvement method Start with random

round trip Swap cities until

optimal round trip


Graphic Visualisation State value / state plot: state space

“State” axis can be states or specific properties Neighbouring states on the axis are states linked

by actions or with similar property values State values are computed using a heuristic and

do not include path cost

Valu

e

State


Graphic Visualisation State value / state plot: state

spaceValue

State

Global maximum

Global minimum

Local maxima

Local minima

Plateau


Graphic Visualisation If state payoff is a complex

mathematical function depending on one state property-1x * x2 + sin2(x)/x + (1000-x)*cos(5x)/5x –

x/10 State space:

x [10, 80] Max: x = 74

payoff = 66.3193


Graphic Visualisation More complex state spaces can

have several dimensions Example: States are X-Y coordinates,

state value is Z coordinate


Graphic Visualisation Each state is a point on

the map Each state’s value is

the distance to the CN Tower Locations in water

always have the worst value because we can’t swim

2D state space X-Y coordinates of the

agent Z coordinate for state

value Red = minimum distance Blue = maximum

distance


Hill Climbing (Gradient Descent)

Simple but efficient local optimization strategy

Always take the action that most improves the state


Hill Climbing (Gradient Descent)

Generate random initial state Each iteration

Generate and evaluate neighbours at step size

Move to neighbour with greatest increase/decrease (i.e. take one step)

End when there are no better neighbours


Example: Travelling to Toronto

Trying to get to downtown Toronto Take steps toward the CN Tower


Hill Climbing (Gradient Descent) Advantages

Fast No search tree

Disadvantages Gets stuck in local optimum Does not allow worse moves Solution dependant on initial state Selecting step size

Common improvements Random restarts Intelligently-chosen initial state Decreasing step size


Simulated Annealing 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 Stochastic search method Simulates annealing process in

metallurgy


Annealing 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)


Simulated AnnealingAnnealing Simulated Annealing

Atoms moving Agent modifying state

towards minimum-energy location in

crystal

towards state with global optimal value

while avoiding bad position.

while avoiding local optimum.

Atoms are more likely to move out of a bad

position

Agents are more likely to accept bad moves

if the metal’s temperature is high.

if the “temperature” control parameter has

a high value.


Simulated AnnealingAnnealing Simulated Annealing

The metal’s temperature starts hot,

The “temperature” control parameter starts with a high

value,

then it cools off then it decreases

continuously incrementally

over time with each iteration of the search

until the metal is room temperature

until it reaches a pre-set threshold.


Simulated Annealing Allow some bad moves

Bad enough to get out of local optimum Not so bad as to get out of global

optimum Probability of accepting bad moves

given Badness of the move (i.e. variation in

state value V) Temperature T P = e-V/T


Simulated Annealing Generate random initial state and high

temperature Each iteration

Generate and evaluate a random neighbour If neighbour better than current state

Accept Else (if neighbour worse than current state)

Accept with probability e-V/T

Reduce temperature End when temperature less than

threshold


Simulated Annealing Advantages

Avoids local optima Very good at finding high-quality solutions Very good for hard problems with complex

state value functions Disadvantage

Can be very slow in practice


Simulated Annealing Application Traveling-wave tube (TWT)

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 Power efficiency becomes a key issue TWT research group at NASA working for over

30 years on improving power efficiency


Simulated Annealing Application Optimizing TWT efficiency

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 Doubled TWT efficiency More flexible then past tappers

Maximize overall power efficiency Maximize efficiency over various bandwidth Maximize efficiency while minimize signal distortion


Assumptions Goal-based agent Environment

Fully observable Deterministic Sequential Static Discrete Single agent


Assumptions Updated Utility-based agent Environment

Fully observable Deterministic Sequential Static Discrete / Continuous Single agent

Informed Search

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