CS 371: Introduction to Artificial Intelligence

Informed Search

Outline

Q/A Time

Overview of My Code, Java Concepts

Informed Search Methods

Memory Bounded Search

Iterative Improvement Search

Heuristic Functions

Game-Playing

Worked Examples

Overview of Code

Abstract classes form “templates” helpful for:

Generalizing problem solving methods (factoring out common code, enabling reuse)

Making the problem formalization clear

Providing a language construct to guide fill-in-the-blank implementation

Abstract Classes for Search

Operator – operator encodings may take on many different forms

State – “search state” or node really. The encoding of the “state” of the agent is internal and unspecified

Searcher – could have added on thread support for anytime algorithm, but kept it simple for now

Informed Search Methods

Best-First Search

Greedy Search

A*

Worked examples

Admissibility, monotonicity, pathmax, optimality, completeness, complexity

Best-First Search

Describe General Search

Evaluation Function f(n) - desirability of node n for expansion

Best-First Search: General Search with queueing ordered by f (A heap rather than a queue, really.)

Best-First Search (cont.)

Let g(n) = pathCost(n). What search do you have if f(n) = g(n)?

Greedy Search

Heuristic Function h(n) - estimated cost of the cheapest path from n to a goal node

Greedy Search: Best-First Search with

f(n) = h(n)

“Always choose the node that looks closest to the goal next.”

Can think of h as the height of a “search terrain”. In greedy search, you would look for the next biggest step down from where you’ve been.

Example: Route Finding

Example: Route Finding (cont.)

h(n) = straight-line distance to goal city

A* Search

Path Cost g(n) - cost from initial state to n

Uniform Cost Search expands minimum g(n)

Greedy Search expands minimum h(n)

Search algorithm “A” expands minimum estimated total cost f(n)=g(n)+h(n)

Worked example

Behavior of A*

Admissible h never overestimates the cost to reach the goal (e.g. route finding, sliding tile puzzle)

h admissible Þ A is called A*

Monotonicity of f – f doesn’t decrease along a path

h admissible Þ f monotonic?

h admissible Þ pathmax equation forces monotonicity:

f(n’) = max(f(n), g(n’)+h(n’))

Monotonicity à search contours

Search Contours

Properties of A*

A* is optimal – finds optimal path to goal

A* is optimally efficient - no other algorithm is guaranteed to expand fewer nodes

A* is complete on locally finite graphs (finite branching factor, minimum cost constant)

Complexity exponential unless h is very accurate (error grows no faster than log of actual path cost)

Iterative Deepening A*

Basic intuition: iterative deepening DFS with increasing f-limit rather than increasing depth-limit.

Heuristic Functions

Fundamental tradeoff

Inventing heuristics: 8-puzzle example

Relaxed problem – problem with constraints removed

Relax: A tile can move from A to B if A adjacent to B and B is empty.

A tile can move from A to B if A adjacent to B.

A tile can move from A to B if A is empty.

A tile can move from A to B.

The cost of an exact solution to a relaxed problem can provide a good heuristic h.

Developing a Heuristic for TripleCross

Pretend tiles can be shifted independently with 4 operators.

Pretend subgoals (solve circle, line of dots, “Binary Arts”) can be solved independently with parallel operations.

Other ideas?

Developing a Heuristic

Tradeoff between quality of h and efficiency of search.

Extreme cases:

h(n) = 0 (admissible, but equivalent to uniform cost search)

h(n) = exact optimal cost remaining (search goes directly to goal, but full problem solved for each step!)

Memory-Bounded Search

Space-complexity the main drawback with A*

IDA* - Iterative deepening search along f contours

Good for simple problems with few contours. What to do if there are many unique path-cost values?

SMA*

Argument: IDA* uses too little memory

SMA* - Simplified Memory-bounded A*

Korf: SMA* 20x slower on 15 tile puzzle

Tradeoff: wasted work searching vs. wasted work checking repeated states.

Can there be a good balance?

Iterative Improvement Search

Hill-Climbing – greedy wandering, no queue, no backtracking

Should be called “hill-descending” or “local optimization”

Gets caught in local minima

Local optimization with random restarts

Local optimization with % random steps

Simulated Annealing

Simulated Annealing – local optimization with % random steps, with % decreasing slowly over time

Analogy to statistical mechanics, process of annealing metals, forming crystalline structures.

Applet demo by Dudu Noy

http://www.math.tau.ac.il/~dudunoy/ex2/tsp.html

Another applet demo by Wesley Perkins:

http://www.starbath.com/Optimization/Gacut.htm

Genetic Algorithms

Analogy to genetic evolution:

Encoding (of characteristics, behaviors)

Population

Fitness (evaluation function, survival of fittest)

Crossover, mutation from one generation to the next

Koza video

Biomorph demo by Jean-Philippe Rennard

http://www.rennard.org/alife/english/jpmain.html?border.html&entete.html&gavgb.html


	Q/A Time
	Overview of My Code, Java Concepts
	Informed Search Methods
	Memory Bounded Search
	Iterative Improvement Search
	Heuristic Functions
	Game-Playing
	Worked Examples


	Abstract classes form “templates” helpful for:
		Generalizing problem solving methods (factoring out common code, enabling reuse)
		Making the problem formalization clear
		Providing a language construct to guide fill-in-the-blank implementation


	Operator – operator encodings may take on many different forms
	State – “search state” or node really. The encoding of the “state” of the agent is internal and unspecified
	Searcher – could have added on thread support for anytime algorithm, but kept it simple for now


	Best-First Search
	Greedy Search
	A*
	Worked examples
	Admissibility, monotonicity, pathmax, optimality, completeness, complexity


	Describe General Search
	Evaluation Function f(n) - desirability of node n for expansion
	Best-First Search: General Search with queueing ordered by f (A heap rather than a queue, really.)


	Let g(n) = pathCost(n). What search do you have if f(n) = g(n)?


	Heuristic Function h(n) - estimated cost of the cheapest path from n to a goal node
	Greedy Search: Best-First Search with
	f(n) = h(n)
	“Always choose the node that looks closest to the goal next.”
	Can think of h as the height of a “search terrain”. In greedy search, you would look for the next biggest step down from where you’ve been.


	Path Cost g(n) - cost from initial state to n
	Uniform Cost Search expands minimum g(n)
	Greedy Search expands minimum h(n)
	Search algorithm “A” expands minimum estimated total cost f(n)=g(n)+h(n)
	Worked example


	Admissible h never overestimates the cost to reach the goal (e.g. route finding, sliding tile puzzle)
	h admissible Þ A is called A*
	Monotonicity of f – f doesn’t decrease along a path
	h admissible Þ f monotonic?
	h admissible Þ pathmax equation forces monotonicity:
		f(n’) = max(f(n), g(n’)+h(n’))
	Monotonicity à search contours


	A* is optimal – finds optimal path to goal
	A* is optimally efficient - no other algorithm is guaranteed to expand fewer nodes
	A* is complete on locally finite graphs (finite branching factor, minimum cost constant)
	Complexity exponential unless h is very accurate (error grows no faster than log of actual path cost)


	Basic intuition: iterative deepening DFS with increasing f-limit rather than increasing depth-limit.


	Fundamental tradeoff
	Inventing heuristics: 8-puzzle example
	Relaxed problem – problem with constraints removed
	Relax: A tile can move from A to B if A adjacent to B and B is empty.
		A tile can move from A to B if A adjacent to B.
		A tile can move from A to B if A is empty.
		A tile can move from A to B.
	The cost of an exact solution to a relaxed problem can provide a good heuristic h.


	Pretend tiles can be shifted independently with 4 operators.
	Pretend subgoals (solve circle, line of dots, “Binary Arts”) can be solved independently with parallel operations.
	Other ideas?


	Tradeoff between quality of h and efficiency of search.
	Extreme cases:
		h(n) = 0 (admissible, but equivalent to uniform cost search)
		h(n) = exact optimal cost remaining (search goes directly to goal, but full problem solved for each step!)


	Space-complexity the main drawback with A*
	IDA* - Iterative deepening search along f contours
	Good for simple problems with few contours. What to do if there are many unique path-cost values?


	Argument: IDA* uses too little memory
	SMA* - Simplified Memory-bounded A*
	Korf: SMA* 20x slower on 15 tile puzzle
	Tradeoff: wasted work searching vs. wasted work checking repeated states.
	Can there be a good balance?


	Hill-Climbing – greedy wandering, no queue, no backtracking
	Should be called “hill-descending” or “local optimization”
	Gets caught in local minima
	Local optimization with random restarts
	Local optimization with % random steps


	Simulated Annealing – local optimization with % random steps, with % decreasing slowly over time
	Analogy to statistical mechanics, process of annealing metals, forming crystalline structures.
	Applet demo by Dudu Noy
		http://www.math.tau.ac.il/~dudunoy/ex2/tsp.html
	Another applet demo by Wesley Perkins:
		http://www.starbath.com/Optimization/Gacut.htm


	Analogy to genetic evolution:
		Encoding (of characteristics, behaviors)
		Population
		Fitness (evaluation function, survival of fittest)
		Crossover, mutation from one generation to the next
	Koza video
	Biomorph demo by Jean-Philippe Rennard
		http://www.rennard.org/alife/english/jpmain.html?border.html&entete.html&gavgb.html