Adversarial Search CE417: Introduction to Artificial Intelligence - - PowerPoint PPT Presentation
Adversarial Search CE417: Introduction to Artificial Intelligence Sharif University of Technology Spring 2018 Soleymani Artificial Intelligence: A Modern Approach , 3 rd Edition, Chapter 5 Most slides have been adopted from Klein and
CE417: Introduction to Artificial Intelligence Sharif University of Technology Spring 2018
“Artificial Intelligence: A Modern Approach”, 3rd Edition, Chapter 5 Most slides have been adopted from Klein and Abdeel, CS188, UC Berkeley.
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Competitive multi-agent environments
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Many different kinds of games! Axes:
Deterministic or stochastic? One, two, or more players? Zero sum? Perfect information (can you see the state)?
Want algorithms for calculating a strategy (policy) which recommends a
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Zero-Sum Games
Agents have opposite utilities (values on outcomes)
Lets us think of a single value that one maximizes and the other minimizes
Adversarial, pure competition
General Games
Agents have independent utilities (values on outcomes)
Cooperation, indifference, competition, and more are all possible
More later on non-zero-sum games
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agents act alternately
agents’ goals are in conflict: sum of utility values at the end of the
fully observable
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Many possible formalizations, one is:
States: S (start at s0) Players: P={1...N} (usually take turns) Actions:A (may depend on player / state) Transition Function: SxA S TerminalTest: S {t,f} Terminal Utilities: SxP R
Solution for a player is a policy: S A
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Zero-sum games: 𝑄
1 gets 𝑉(𝑢), 𝑄2 gets 𝐷 − 𝑉(𝑢) for terminal node 𝑢
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1: 𝑌
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A state-space search tree Players alternate turns Compute each node’s minimax value:
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Best achievable payoff against best play
Maximizes the worst-case outcome for MAX
It works for zero-sum games
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𝑏∈𝐵𝐷𝑈𝐽𝑃𝑂𝑇(𝑡𝑢𝑏𝑢𝑓) 𝑁𝐽𝑂_𝑊𝐵𝑀𝑉𝐹(𝑆𝐹𝑇𝑉𝑀𝑈(𝑡𝑢𝑏𝑢𝑓, 𝑏))
Finding exact solution is completely infeasible
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Branch & bound algorithm Prunes away branches that cannot influence the final decision
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General configuration (MIN version)
We’re computing the MIN-VALUE at some node n
We’re looping over n’s children
Who cares about n’s value? MAX
Let a be the best value that MAX can get at any choice point along the current path from the root
If n becomes worse than a, MAX will avoid it, so we can stop considering n’s other children (it’s already bad enough that it won’t be played)
MAX version is symmetric
a n
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𝑛 2) 39
?
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(4-ply look-ahead is a hopeless chess player!)
It is still hopeless
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Instead, search only to a limited depth in the tree Replace terminal utilities with an evaluation
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Cut off test instead of terminal test (e.g., depth limit)
Heuristic function evaluation instead of utility function
Evaluation functions are always imperfect The deeper in the tree the evaluation
An important example of the tradeoff
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Evaluation functions score non-terminals in depth-limited search Ideal function: returns the actual minimax value of the position In practice: typically weighted linear sum of features: e.g. f1(s) = (num white queens – num black queens), etc.
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A danger of replanning agents!
He knows his score will go up by eating the dot now (west, east)
He knows his score will go up just as much by eating the dot later (east, west)
There are no point-scoring opportunities after eating the dot (within the horizon, two here)
Therefore, waiting seems just as good as eating: he may go east, then back west in the next round of replanning!
A better evaluation function can solve this problem
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Why wouldn’t we know what the result of an action will be?
Explicit randomness: rolling dice Unpredictable opponents: the ghosts respond randomly Actions can fail: when moving a robot, wheels might slip
Values
Expectimax search: compute the average score under
Max nodes as in minimax search Chance nodes are like min nodes but the outcome is uncertain Calculate their expected utilities
i.e. take weighted average (expectation) of children
Later, we’ll learn how to formalize the underlying uncertain-
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A random variable represents an event whose outcome is unknown
A probability distribution is an assignment of weights to outcomes
Example:Traffic on freeway
Random variable:T = whether there’s traffic
Outcomes:T in {none, light, heavy}
Distribution: P(T=none) = 0.25, P(T=light) = 0.50, P(T=heavy) = 0.25
Some laws of probability (more later):
Probabilities are always non-negative
Probabilities over all possible outcomes sum to one
As we get more evidence, probabilities may change:
P(T=heavy) = 0.25, P(T=heavy | Hour=8am) = 0.60
We’ll talk about methods for reasoning and updating probabilities later
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The expected value of a function of a random variable is the
Example: How long to get to the airport?
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In expectimax search, we have a probabilistic model
Model could be a simple uniform distribution (roll a die)
Model could be sophisticated and require a great deal of computation
We have a chance node for any outcome out of our control:
The model might say that adversarial actions are likely!
For now, assume each chance node magically comes
In expectimax search, we have a probabilistic model
Model could be a simple uniform distribution (roll a die)
Model could be sophisticated and require a great deal of computation
We have a chance node for any outcome out of our control:
The model might say that adversarial actions are likely!
For now, assume each chance node magically comes
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1/2 1/3 1/6
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Adversarial Ghost Random Ghost Minimax Pacman Won 5/5
Won 5/5
Expectimax Pacman Won 1/5
Won 5/5
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E.g. Backgammon Expectiminimax
Environment is an extra “random agent”
Each
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It is consistent with the definition of rational agents trying to
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𝑠
For worst-case minimax reasoning, terminal function scale doesn’t matter We just want better states to have higher evaluations (get the ordering right) We call this insensitivity to monotonic transformations For average-case expectimax reasoning, we need magnitudes to be meaningful
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Backgammon: 𝑐 ≈ 20 (can be up to 4000 for double dice rolls), 𝑜 = 21
3-plies is manageable (≈ 108 nodes)
Forming detailed plans of actions may be pointless
Limiting depth is not such damaging particularly when the probability values (for
But pruning is not straightforward.
Dice rolls increase b: 21 possible rolls with 2 dice
Backgammon 20 legal moves Depth 2 = 20 x (21 x 20)3 = 1.2 x 109
As depth increases, probability of reaching a given
So usefulness of search is diminished So limiting depth is less damaging But pruning is trickier…
Historic AI: TDGammon uses depth-2 search + very
Image: Wikipedia
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What if the game is not zero-sum, or has multiple players? Generalization of minimax:
Terminals have utility tuples
Node values are also utility tuples
Each player maximizes its own component
Can give rise to cooperation and competition dynamically…
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In 1997, Deep Blue defeated Kasparov.
ran on a parallel computer doing alpha-beta search. reaches depth 14 plies routinely.
techniques to extend the effective search depth
Hydra: Reaches depth 18 plies using more heuristics.
Chinook (ran on a regular PC and uses alpha-beta search) ended 40-
Since 2007, Chinook has been able to play perfectly by using alpha-beta
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Logistello defeated the human world champion by
Human champions are no match for computers at
Human
MOGO avoided alpha-beta search and used Monte
AlphaGo (2016) has beaten professionals without
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TD-Gammon (1992) was competitive with top
Depth 2 or 3 search along with a good evaluation
In 1998, GIB was 12th in a filed of 35 in the par
In 2005, Jack defeated three out of seven top
In 2006, Quackle defeated the former world