CS 6300 Artificial Intelligence Spring 2018

Transcription

1 Expectimax Search CS 6300 Artificial Intelligence Spring 2018 Tucker Hermans Many slides courtesy of Pieter Abbeel and Dan Klein

2 Expectimax Search Trees What if we don t know what the result of an action will be? E.g., In solitaire, next card is unknown In minesweeper, mine locations In pacman, the ghosts act randomly Can do expectimax search to maximize average score Max nodes as in minimax search Chance nodes, like min nodes, except the outcome is uncertain Calculate expected utilities: weighted average (expectation) of values of children Later, we ll learn how to formalize these underlying problems as a Markov Decision Processes max average 2

3 Expectimax Pseudocode def value(s) if s is a max node return maxvalue(s) if s is an exp node return expvalue(s) if s is a terminal node return evaluation(s) def maxvalue(s) values = [value(s ) for s in successors(s)] return max(values) def expvalue(s) values = [value(s ) for s in successors(s)] weights = [probability(s, s ) for s in successors(s)] return expectation(values, weights)

4 Reminder: Probabilities 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

5 Reminder: Expectations Real valued functions of random variables: Expectation of a function of a random variable Example: Expected value of a fair die roll X P f 1 1/ / / / / /6 6 5

6 Expectimax Quantities 6

7 Expectimax Pruning? 7

8 Depth-Limited Expectimax 8

9 What Utilities to Use? For minimax, 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 expectimax, we need magnitudes to be meaningful 9

10 What Probabilities to Use? In expectimax search, we have a probabilistic model of how the opponent (or environment) will behave in any state Model could be a simple uniform distribution (roll a die) Model could be sophisticated and require a great deal of computation We have a node for every outcome out of our control: opponent or environment The model might say that adversarial actions are likely! For now, assume for any state we magically have a distribution to assign probabilities to opponent actions / environment outcomes Having a probabilistic belief about an agent s action does not mean that agent is flipping any coins! 10

11 Expectimax for Pacman Notice that we ve gotten away from thinking that the ghosts are trying to minimize pacman s score Instead, they are now a part of the environment Pacman has a belief (distribution) over how they will act Quiz: Can we see minimax as a special case of expectimax? Quiz: what would pacman s computation look like if we assumed that the ghosts were doing 1-ply minimax and taking the result 80% of the time, otherwise moving randomly? If you take this further, you end up calculating belief distributions over your opponents belief distributions over your belief distributions, etc Can get unmanageable very quickly! 11

12 Expectimax for Pacman Results from playing 5 games Pacman used depth 4 search with an eval function that avoids trouble Ghost used depth 2 search with an eval function that seeks Pacman 12

13 Mixed Layer Types E.g. backgammon Expectiminimax (!) Environment is an extra player that moves after each agent Chance nodes take expectations, otherwise like minimax 13

14 Stochastic Two-Player Dice rolls increase b: 21 possible rolls with 2 dice Backgammon» 20 legal moves Depth 2 = 20 x (21 x 20) x 10 9 As depth increases, probability of reaching a given node shrinks So value of lookahead is diminished So limiting depth is less damaging But pruning is less possible TDGammon uses depth-2 search + very good eval function + reinforcement learning: worldchampion level play First AI world champion in any game! 14

15 Multi-Agent Utilities Similar to minimax: Terminals have utility tuples Each player maximizes its own utility and propagates (or backs up) nodes from children Can give rise to cooperation and competition dynamically 1,6,6 7,1,2 6,1,2 7,2,1 5,1,7 1,5,2 7,7,1 5,2,5 15

16 Expectimax: Formal Definition Chance node s i : expected value of outcomes s ij V * (s i )= E [V * (s i )]= j p ij V * (s ij ) The * refers to optimal values SB use V instead of U for utility Max node s: choose best action a i based on values of chance nodes s 1, s 2, s 3 V * (s)= max a i V * (s i ) = max a i j p ij V * (s ij ) 16

17 Action Uncertainty: Grid World The agent lives in a grid Terminal states: +/- 1 Walls block the agent s path The agent s actions do not always go as planned: 80% of the time, the action North takes the agent North (if there is no wall there) 10% of the time, North takes the agent West; 10% East If there is a wall in the direction the agent would have been taken, the agent stays put Tucker Hermans Transition probability T (s,a,s ') Probability that action a from state s leads to state s' 17

18 Grid Word: Action Uncertainty 18

19 Expectimax with Action Uncertainty q-state (s,a i ): state-action pair replaces chance node V * (s)= max a i j T (s,a i,s ij )V * (s ij ) 19

20 Grid World: Reward Function The agent lives in a grid Terminal states: +/- 1 Walls block the agent s path The agent s actions do not always go as planned: 80% of the time, the action North takes the agent North (if there is no wall there) 10% of the time, North takes the agent West; 10% East If there is a wall in the direction the agent would have been taken, the agent stays put Transition probability T (s,a,s ') Probability that action a from state s leads to state s' Reward function R(s,a, s' ) Small reward for each transition Big reward entering terminal state Tucker Hermans 20

21 Expectimax with Reward Function A (small) reward is accrued for each transition. This small reward is added to the expected value of the node V * (s)= max a i j T (s,a i,s ij )( R(s,a i, s ij )+V * (s ij )) 21

22 Reward Function: Special Cases Rewards depend only on the successor state: (It doesn't matter how you get to the goal) Rewards depend only on the start state: (It doesn't matter where you're going) R(s, a, s') = R(s') R(s, a, s') = R(s) Rewards depend only on the action: R(s, a, s') = R(a) (Suitable for continuous cases, e.g., mpg while driving) 22

23 The Bellman Equations Richard Bellman ( ) Work done at RAND Optimal rewards = maximize over first action and then follow optimal policy Bellman equations: a s, a s s,a,s s 23