Informed search algorithms
CISC 6525 Artificial Intelligence
Informed search algorithms:
Local, A* and Adversarial
Informed Search
Best-first & greedy best-first search,
A* search & Heuristics – Chapter 3 (3.5-6)
Local search algorithms – Chapter 4 (4.1)
Hill-climbing search
Simulated annealing search
Local beam search
Genetic algorithms
Adversarial search – Chapter 5 (5.1-4)
Games trees & Optimality
α-β pruning
Imperfect, real-time decisions
Review: Tree search
A search strategy is defined by picking the order of node expansion
Best-first search
Idea: use an evaluation function f(n) for each node
estimate of “desirability”
Expand most desirable unexpanded node
Implementation:
Order the nodes in fringe in decreasing order of desirability
Special cases:
greedy best-first search
A* search
Romania with step costs in km
100
Greedy best-first search
Evaluation function f(n) = h(n) (heuristic)
= estimate of cost: from n to goal
e.g., hSLD(n) = straight-line distance from n to Bucharest
Greedy best-first search expands the node that appears to be closest to goal
Straight line distance
Greedy best-first search example
Greedy best-first search example
Greedy best-first search example
Greedy best-first search example
Properties of greedy best-first search
Complete? No – can get stuck in loops,
E.g., Consider Iasi to Fagaras:
Iasi Neamt Iasi Neamt ……..
Time? O(bm), but a good heuristic can give dramatic improvement
Space? O(bm) — keeps all nodes in memory
Optimal? No
A* search
Idea: avoid expanding paths that are already expensive
Evaluation function f(n) = g(n) + h(n)
g(n) = cost so far to reach n
h(n) = estimated cost from n to goal
f(n) = estimated total cost of path through n to goal
A*: Romania with step costs in km
A* Evaluation: f(n) = g(n) + h(n)
100
A* search example
A* search example
A* search example
A* search example
A* search example
A* search example
A*: In Class Exercise
A* Evaluation: f(n) = g(n) + h(n)
100
Admissible heuristics
A heuristic h(n) is admissible if for every node n,
h(n) ≤ h*(n), where h*(n) is the true cost to reach the goal state from n.
An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimistic
Example: hSLD(n) (never overestimates the actual road distance)
Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal
Optimality of A* (proof)
Suppose some suboptimal goal G2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G.
f(G2) = g(G2) since h(G2) = 0
g(G2) > g(G) since G2 is suboptimal
f(G) = g(G) since h(G) = 0
f(G2) > f(G) from above
Optimality of A* (proof)
Suppose some suboptimal goal G2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G.
f(G2) > f(G) from above
h(n) ≤ h*(n) since h is admissible
g(n) + h(n) ≤ g(n) + h* (n)
f(n) ≤ f(G)
Hence f(G2) > f(n), and A* will never select G2 for expansion
Consistent heuristics
A heuristic is consistent if for every node n, every successor n’ of n generated by any action a,
h(n) ≤ c(n,a,n’) + h(n’)
If h is consistent, we have
f(n’) = g(n’) + h(n’)
= g(n) + c(n,a,n’) + h(n’)
≥ g(n) + h(n)
= f(n)
i.e., f(n) is non-decreasing along any path.
Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal
Optimality of A*
A* expands nodes in order of increasing f value
Gradually adds “f-contours” of nodes
Contour i has all nodes with f=fi, where fi < fi+1
Properties of A*
Complete? Yes (unless there are infinitely many nodes with f ≤ f(G) )
Time? Exponential
Space? Keeps all nodes in memory
Optimal? Yes
Admissible heuristics
E.g., for the 8-puzzle:
h1(n) = number of misplaced tiles
h2(n) = total Manhattan distance
(i.e., no. of squares from desired location of each tile)
h1(S) = ?
h2(S) = ?
Admissible heuristics
E.g., for the 8-puzzle:
h1(n) = number of misplaced tiles
h2(n) = total Manhattan distance
(i.e., no. of squares from desired location of each tile)
h1(S) = ? 8
h2(S) = ? 3+1+2+2+2+3+3+2 = 18
Performance measure for a heuristic
N+1 = 1 + b* + b*2 … + b*d
(b*)d+1 = N, so b* = (N)1/(d+1)
If A* generates solution at depth d=5 and expands N=52 nodes, then
b*6 = 52,
b* = (52)1/6 = 1.92
Effective Branching Factor, b*
Full binary tree height h N = 2^h+1 -1 => N+1 = 2^h+1
30
Dominance
If h2(n) ≥ h1(n) for all n (both admissible)
then h2 dominates h1
h2 is better for search
Typical search costs (average number of nodes expanded):
d=12 IDS = 3,644,035 nodes
A*(h1) = 227 nodes
A*(h2) = 73 nodes
d=24 IDS = too many nodes
A*(h1) = 39,135 nodes
A*(h2) = 1,641 nodes
Relaxed problems
A problem with fewer restrictions on the actions is called a relaxed problem
The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem
If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h1(n) gives the shortest solution
If the rules are relaxed so that a tile can move to any adjacent square, then h2(n) gives the shortest solution
Local search algorithms
In many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution
State space = set of “complete” configurations
Find configuration satisfying constraints, e.g., n-queens
In such cases, we can use local search algorithms
keep a single “current” state, try to improve it
Example: n-queens
Put n queens on an n × n board with no two queens on the same row, column, or diagonal
Example: n-queens
Put n queens on an n × n board with no two queens on the same row, column, or diagonal
Example: n-queens
Put n queens on an n × n board with no two queens on the same row, column, or diagonal
Hill-climbing search
“Like climbing Everest ..
Hill-climbing search
“Like climbing Everest in thick fog …
Hill-climbing search
“Like climbing Everest in thick fog with amnesia.”
Example: n-queens
Put n queens on an n × n board with no two queens on the same row, column, or diagonal
Hill-climbing search: 8-queens problem
h = number of pairs of queens that are attacking each other, either directly or indirectly
Hill-climbing search: 8-queens problem
A local minimum with h = 1
Hill-climbing search
Problem: depending on initial state, can get stuck in local maxima
Hill-climbing search
Problem: depending on initial state, can get stuck in local maxima
Hill-climbing search
Problem: depending on initial state, can get stuck in local maxima
Hill-climbing search
Problem: depending on initial state, can get stuck in local maxima
Local beam search
Keep track of k states rather than just one
Start with k randomly generated states
At each iteration, all the successors of all k states are generated
If any one is a goal state, stop; else select the k best successors from the complete list and repeat.
Simulated annealing search
Idea: escape local maxima by allowing some “bad” moves but gradually decrease their frequency
Properties of simulated annealing search
One can prove: If T decreases slowly enough, then simulated annealing search will find a global optimum with probability approaching 1
Widely used in VLSI layout, airline scheduling, etc
Genetic algorithms
A successor state is generated by combining two parent states
Start with k randomly generated states (population)
A state is represented as a string over a finite alphabet (often a string of 0s and 1s)
Evaluation function (fitness function). Higher values for better states.
Produce the next generation of states by selection, crossover, and mutation
Genetic algorithms
Fitness function: number of non-attacking pairs of queens (min = 0, max = 8 × 7/2 = 28)
24/(24+23+20+11) = 31%
23/(24+23+20+11) = 29% etc
Genetic algorithms
Adversarial Search
Games!
“Unpredictable” opponent specify a move for every possible opponent reply
Time limits unlikely to find goal, must approximate
Deterministic games in practice
Checkers: Chinook ended 40-year-reign of human world champion Marion Tinsley in 1994. Used a precomputed endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 444 billion positions.
Chess: Deep Blue defeated human world champion Garry Kasparov in a six-game match in 1997. Deep Blue searches 200 million positions per second, uses very sophisticated evaluation, and undisclosed methods for extending some lines of search up to 40 ply. Current programs even better.
Othello: human champions refuse to compete against computers, who are too good.
Go: In the 2017 Future of Go Summit, AlphaGo beat Ke Jie, the world No.1 ranked player at the time, in a three-game match. alphaGo uses Monte Carlo and neural network techniques to learn how to refine its search to winning games.
Game tree (2-player, deterministic, turns)
Minimax
Perfect play for deterministic games
Idea: choose move to position with highest minimax value
= best achievable payoff against best play
E.g., 2-ply game:
Minimax algorithm
Properties of minimax
Complete? Yes (if tree is finite)
Optimal? Yes (against an optimal opponent)
Time complexity? O(bm)
Space complexity? O(bm) (depth-first exploration)
For chess, b ≈ 35, m ≈100 for “reasonable” games
exact solution completely infeasible
α-β pruning example
α-β pruning example
α-β pruning example
α-β pruning example
α-β pruning example
Properties of α-β
Pruning does not affect final result
Good move ordering improves effectiveness of pruning
With “perfect ordering,” time complexity = O(bm/2)
doubles depth of search
A simple example of the value of reasoning about which computations are relevant (a form of metareasoning)
Why is it called α-β?
α is the value of the best (i.e., highest-value) choice found so far at any choice point along the path for max
If v is worse than α, max will avoid it
prune that branch
Define β similarly for min
Why is it called α-β?
β is the value of the best (i.e., lowest-value) choice found so far at any choice point along the path for min
If v is worse than α, min will avoid it
prune that branch
MIN
MIN
MAX
MAX
The α-β algorithm
The α-β algorithm
Resource limits
Suppose we have 100 secs, explore 104 nodes/sec
106 nodes per move
Standard approach:
cutoff test:
e.g., depth limit (perhaps add quiescence search)
evaluation function
= estimated desirability of position
Evaluation functions
For chess, typically linear weighted sum of features
Eval(s) = w1 f1(s) + w2 f2(s) + … + wn fn(s)
e.g., w1 = 9 with
f1(s) = (number of white queens) – (number of black queens), etc.
Cutting off search
MinimaxCutoff is identical to MinimaxValue except
Terminal? is replaced by Cutoff?
Utility is replaced by Eval
Does it work in practice?
bm = 106, b=35 m=4
4-ply lookahead is a hopeless chess player!
4-ply ≈ human novice
8-ply ≈ typical PC, human master
12-ply ≈ Deep Blue, Kasparov
/docProps/thumbnail.jpeg