Human-awareRobotics

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Informed Search• 2018/01/18

• Chapter 3.5 in R&N 3rd

Ø Announcement:q Slides for this lecture are here:

http://www.public.asu.edu/~yzhan442/teaching/CSE471/Lectures/informed.

pdf

q Project 1 released. Due in two weeks, on Jan 31 by midnight.

Human-awareRobotics

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• Planning agent (goal-based agent) and environment

• Planning problem

• Search

• Search strategies

• Required reading (red means it will be on your exams):

o R&N: Chapter 3.1-3.4

Last time

Human-awareRobotics

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Outline for today

• Heuristics

• Best-first search

• Admissible heuristics

• Graph search and consistency

• Required reading (red means it will be on your exams):

o R&N: Chapter 3.5-3.6

Human-awareRobotics

• Search problem:– States (configurations of the world)– Actions and costs– Successor function (world dynamics)– Start state and goal test

• Search tree:– Nodes: represent plans for reaching states– Plans have costs (sum of action costs)

• Search algorithm:– Systematically builds a search tree– Chooses an ordering of the fringe (unexplored nodes)– Optimal: finds least-cost plans

Planning agent

fromhttp://ai.berkeley.edu

Human-awareRoboticsTree search

Human-awareRobotics

• Remember: UCS explores increasing cost contours

• The good: UCS is complete and optimal!

• The bad:– Explores options in every “direction”– No information about goal location

• We’ll fix it today!

Start Goal

…

c£ 3c£ 2

c£ 1

Issues with UCS

Human-awareRoboticsInformed search

Human-awareRobotics§ A heuristic is:

§ A function that estimates how close a state is to a goal

§ Designed for a particular search problem§ Examples: Manhattan distance, Euclidean

distance for path finding

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5

11.2

Search heuristics

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Outline for today

• Heuristics

• Best-first search

• Admissible heuristics

• Graph search and consistency

• Required reading (red means it will be on your exams):

o R&N: Chapter 3.5-3.6

Human-awareRoboticsBest-first search

Idea:

• Use heuristic for each node to estimate its “desirability”• Expand the most desirable unexpanded node

Special cases:

Greedy searchA* search

Human-awareRoboticsGreedy search

Human-awareRobotics

• Strategy: expand a node that you think is closest to a goal state– Heuristic: estimate of distance to

nearest goal for each state

• A common case:– Best-first takes you straight to the

(wrong) goal

• Worst-case: like a badly-guided DFS

…b

…b

Greedy search

Human-awareRoboticsGreedy search

Human-awareRoboticsGreedy search

Human-awareRoboticsGreedy search

Human-awareRobotics• What nodes does greedy search expand?

– Takes time O(bm) (exponential in effective depth)– A good heuristic can give dramatic improvement!

• How much space does the fringe take?– May keeps all nodes at the bottom tier, so O(bm)

• Is it complete?– No, i.e., stuck in loops (when state space graph has

loops)

• Is it optimal?– No

Properties of greedy search

…b

…b

mtiers

Human-awareRoboticsA* search

Human-awareRobotics• Uniform-cost orders by path cost, or backward cost g(n)• Greedy orders by goal proximity, or forward cost h(n)• A* Search orders by the sum: f(n) = g(n) + h(n)

S a d

b

Gh=5

h=6

h=2

1

8

11

2

h=6 h=0

c

h=7

3

e h=11

S

a

b

c

ed

dG

G

g =0h=6

g =1h=5

g =2h=6

g =3h=7

g =4h=2

g =6h=0

g =9h=1

g =10h=2

g =12h=0

Combining UCS and Greedy

Human-awareRobotics• Should we stop when we enqueue a goal?• No: only stop when we dequeue a goal

S

B

A

G

2

3

2

2h=1

h=2

h=0h=3

When should A* terminate?

Human-awareRobotics

• What went wrong?• Actual bad goal cost < estimated good goal cost• We need estimates to be less than actual costs!

A

GS

1 3h=6

h=0

5

h =7

Is A* optimal

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Outline for today

• Heuristics

• Best-first search

• Admissible heuristics

• Graph search and consistency

• Required reading (red means it will be on your exams):

o R&N: Chapter 3.5-3.6

Human-awareRoboticsAdmissible Heuristic

Human-awareRobotics

Inadmissible (pessimistic) heuristics break optimality by trapping good plans on the fringe

Admissible (optimistic) heuristics slow down bad plans but never outweigh true costs

Admissibility

Human-awareRobotics• A heuristic h is admissible (optimistic) if:

where is the true cost to a nearest goal

• Examples:

• Coming up with admissible heuristics is most of what’s involved in using A* in practice.

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Admissible Heuristic

Human-awareRobotics

• Most of the work in solving hard search problems optimally is in coming up with admissible heuristics

• Often, admissible heuristics are solutions to relaxed problems, where new actions are available

• Inadmissible heuristics are often useful too

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Admissible heuristics

Human-awareRobotics• Heuristic: Number of tiles

misplaced• Why is it admissible?• h(start) = • This is a relaxed-problem heuristic

Averagenodesexpandedwhentheoptimalpathhas……4steps

…8steps

…12steps

UCS 112 6,300 3.6x106

TILES 13 39 227

StartState GoalState

StatisticsfromAndrewMoore

Example

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Human-awareRobotics

• What if we had an easier 8-puzzle where any tile could slide any direction at any time, ignoring other tiles?

• Total Manhattan distance

• Why is it admissible?

• h(start) = 3+1+2+…=18

Averagenodesexpandedwhentheoptimalpathhas…

…4steps …8steps …12stepsTILES 13 39 227MANHATTAN 12 25 73

StartState GoalState

Example

Human-awareRobotics

• How about using the actual cost as a heuristic?– Would it be admissible?– What’s wrong with it?

• With A*: a trade-off between quality of estimate and work per node– As heuristics get closer to the true cost, you will expand fewer

nodes but usually do more work per node to compute the heuristic itself

Example

Human-awareRobotics• Dominance: ha ≥ hc if

• Heuristics form a semi-lattice:– Max of admissible heuristics is

admissible

• Trivial heuristics– Bottom of lattice is the zero

heuristic (what does this give us?)– Top of lattice is the exact

heuristic

Dominance

Human-awareRoboticsOptimality of A*

Human-awareRobotics

Assume:• A is an optimal goal node• B is a suboptimal goal

node• h is admissible

Claim:

• A will exit the fringe before B

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Optimality of A*

Human-awareRobotics

Proof:• Imagine B is on the fringe• Some ancestor n of A is on the

fringe, too (maybe A!)• Claim: n will be expanded

before B1. f(n) is less or equal to f(A)

Admissibilityofh

…

h=0atagoal

Optimality of A*

Human-awareRobotics

Proof:• Imagine B is on the fringe• Some ancestor n of A is on the

fringe, too (maybe A!)• Claim: n will be expanded

before B1. f(n) is less or equal to f(A)2. f(A) is less than f(B)

B issuboptimalh=0atagoal

…

Optimality of A*

Human-awareRobotics

Proof:• Imagine B is on the fringe• Some ancestor n of A is on the

fringe, too (maybe A!)• Claim: n will be expanded

before B1. f(n) is less or equal to f(A)2. f(A) is less than f(B)3. n expands before B

• All ancestors of A expand before B

• A expands before B• A* search is optimal

…

Optimality of A*

Human-awareRobotics

…b

…b

Uniform-Cost A*

UCS and A* contour

Human-awareRobotics

• Uniform-cost expands equally in all “directions”

• A* expands mainly toward the goal, but does hedge its bets to ensure optimality

Start Goal

Start Goal

UCS and A* contour

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Greedy UniformCost A*

Comparison

Human-awareRobotics

Greedy UniformCost A*

Comparison

S

G

UsingManhattandistanceastheheuristic

• Whenwillgreedyperformbetter?• WhenwilluniformcostbeatA*?• Howyoubreaktiesmayalsobeimportant

Canonlymovetoadjacentcells

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Outline for today

• Heuristics

• Best-first search

• Admissible heuristics

• Graph search and consistency

• Required reading (red means it will be on your exams):

o R&N: Chapter 3.5-3.6

Human-awareRoboticsGraph search

Human-awareRoboticsGraph search vs tree search

Human-awareRobotics

• In BFS, for example, we shouldn’t bother expanding the circled nodes (why?)

S

a

b

d p

a

c

e

p

h

f

r

q

q c G

a

qe

p

h

f

r

q

q c G

a

Graph search

Human-awareRobotics

S

A

B

C

G

1

1

1

23

h=2

h=1

h=4

h=1

h=0

S(0+2)

A(1+4) B(1+1)

C(2+1)

G(5+0)

C(3+1)

G(6+0)

Statespacegraph Searchtree

A* graph search gone wrong

Human-awareRobotics• Main idea: estimated heuristic costs ≤ actual costs

– Admissibility: heuristic cost ≤ actual cost to

goal

h(A) ≤ actual cost from A to G

– Consistency: heuristic “arc” cost ≤ actual cost

for each arc

h(A) – h(C) ≤ cost(A to C)

• Consequences of consistency:

– The f value along a path never decreases

h(A) ≤ cost(A to C) + h(C)

– A* graph search is optimal

3

A

C

G

h=4 h=11

h=2

Consistency

Human-awareRobotics

• Sketch: consider what A* does with a consistent heuristic:

– Fact 1: In tree search, A* expands nodes in increasing total f value (f-contours)

– Fact 2: For every state s, nodes that reach s optimally are expanded before s is expanded

– Result: A* graph search is optimal

…

f£ 3

f£ 2

f£ 1

Optimality of A* graph search

Human-awareRobotics

• Tree search:– A* is optimal if heuristic is admissible– UCS is a special case (h = 0)

• Graph search:– A* optimal if heuristic is consistent– UCS optimal (h = 0 is consistent)

• Consistency implies admissibility

• In general, most natural admissible heuristics tend to be consistent, especially if from relaxed problems

Optimality

Human-awareRobotics

• Video games• Path finding / routing problems• Robot motion planning• Resource planning problems• Language analysis• Machine translation• Speech recognition• …

A* applications

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Summary

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Outline for today

• Heuristics

• Best-first search

• Admissible heuristics

• Graph search and consistency

• Required reading (red means it will be on your exams):

o R&N: Chapter 3.5-3.6