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Decision Making Under Uncertainty Russell and Norvig: ch 16, 17 CMSC 671 – Fall 2005 material from Lise Getoor, Jean-Claude Latombe, and Daphne Koller

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Decision Making Under Uncertainty Many environments have multiple possible outcomes Some of these outcomes may be good; others may be bad Some may be very likely; others unlikely What’s a poor agent to do??

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Non-Deterministic vs. Probabilistic Uncertainty ? bac {a,b,c} decision that is best for worst case ? bac {a(p a ),b(p b ),c(p c )} decision that maximizes expected utility value Non-deterministic modelProbabilistic model ~ Adversarial search

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Expected Utility Random variable X with n values x 1,…,x n and distribution (p 1,…,p n ) E.g.: X is the state reached after doing an action A under uncertainty Function U of X E.g., U is the utility of a state The expected utility of A is EU[A] = i=1,…,n p(x i |A)U(x i )

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s0s0 s3s3 s2s2 s1s1 A1 0.20.70.1 1005070 U(S0) = 100 x 0.2 + 50 x 0.7 + 70 x 0.1 = 20 + 35 + 7 = 62 One State/One Action Example

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s0s0 s3s3 s2s2 s1s1 A1 0.20.70.1 1005070 A2 s4s4 0.20.8 80 U1(S0) = 62 U2(S0) = 74 U(S0) = max{U1(S0),U2(S0)} = 74 One State/Two Actions Example

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s0s0 s3s3 s2s2 s1s1 A1 0.20.70.1 1005070 A2 s4s4 0.20.8 80 U1(S0) = 62 – 5 = 57 U2(S0) = 74 – 25 = 49 U(S0) = max{U1(S0),U2(S0)} = 57 -5-25 Introducing Action Costs

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MEU Principle A rational agent should choose the action that maximizes agent’s expected utility This is the basis of the field of decision theory The MEU principle provides a normative criterion for rational choice of action

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Not quite… Must have complete model of: Actions Utilities States Even if you have a complete model, will be computationally intractable In fact, a truly rational agent takes into account the utility of reasoning as well---bounded rationality Nevertheless, great progress has been made in this area recently, and we are able to solve much more complex decision-theoretic problems than ever before

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We’ll look at Decision-Theoretic Planning Simple decision making (ch. 16) Sequential decision making (ch. 17)

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Axioms of Utility Theory Orderability (A>B) (A**B) (B>C) (A>C) Continuity A>B>C p [p,A; 1-p,C] ~ B Substitutability A~B [p,A; 1-p,C]~[p,B; 1-p,C] Monotonicity A>B (p≥q [p,A; 1-p,B] >~ [q,A; 1-q,B]) Decomposability [p,A; 1-p, [q,B; 1-q, C]] ~ [p,A; (1-p)q, B; (1-p)(1-q), C]
**

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Money Versus Utility Money <> Utility More money is better, but not always in a linear relationship to the amount of money Expected Monetary Value Risk-averse – U(L) < U(S EMV(L) ) Risk-seeking – U(L) > U(S EMV(L) ) Risk-neutral – U(L) = U(S EMV(L) )

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Value Function Provides a ranking of alternatives, but not a meaningful metric scale Also known as an “ordinal utility function” Remember the expectiminimax example: Sometimes, only relative judgments (value functions) are necessary At other times, absolute judgments (utility functions) are required

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Multiattribute Utility Theory A given state may have multiple utilities...because of multiple evaluation criteria...because of multiple agents (interested parties) with different utility functions We will talk about this more later in the semester, when we discuss multi-agent systems and game theory

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Decision Networks Extend BNs to handle actions and utilities Also called influence diagrams Use BN inference methods to solve Perform Value of Information calculations

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Decision Networks cont. Chance nodes: random variables, as in BNs Decision nodes: actions that decision maker can take Utility/value nodes: the utility of the outcome state.

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R&N example

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Umbrella Network weather forecast umbrella happiness take/don’t take f w p(f|w) sunny rain 0.3 rainy rain 0.7 sunny no rain 0.8 rainy no rain 0.2 P(rain) = 0.4 U(have,rain) = -25 U(have,~rain) = 0 U(~have, rain) = -100 U(~have, ~rain) = 100 have umbrella P(have|take) = 1.0 P(~have|~take)=1.0

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Evaluating Decision Networks Set the evidence variables for current state For each possible value of the decision node: Set decision node to that value Calculate the posterior probability of the parent nodes of the utility node, using BN inference Calculate the resulting utility for action Return the action with the highest utility

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Decision Making: Umbrella Network weather forecast umbrella happiness take/don’t take f w p(f|w) sunny rain 0.3 rainy rain 0.7 sunny no rain 0.8 rainy no rain 0.2 P(rain) = 0.4 U(have,rain) = -25 U(have,~rain) = 0 U(~have, rain) = -100 U(~have, ~rain) = 100 have umbrella P(have|take) = 1.0 P(~have|~take)=1.0 Should I take my umbrella??

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Value of Information (VOI) Suppose an agent’s current knowledge is E. The value of the current best action is The value of the new best action (after new evidence E’ is obtained): The value of information for E’ is therefore:

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Value of Information: Umbrella Network weather forecast umbrella happiness take/don’t take f w p(f|w) sunny rain 0.3 rainy rain 0.7 sunny no rain 0.8 rainy no rain 0.2 P(rain) = 0.4 U(have,rain) = -25 U(have,~rain) = 0 U(~have, rain) = -100 U(~have, ~rain) = 100 have umbrella P(have|take) = 1.0 P(~have|~take)=1.0 What is the value of knowing the weather forecast?

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Sequential Decision Making Finite Horizon Infinite Horizon

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Simple Robot Navigation Problem In each state, the possible actions are U, D, R, and L

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Probabilistic Transition Model In each state, the possible actions are U, D, R, and L The effect of U is as follows (transition model): With probability 0.8 the robot moves up one square (if the robot is already in the top row, then it does not move)

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Probabilistic Transition Model In each state, the possible actions are U, D, R, and L The effect of U is as follows (transition model): With probability 0.8 the robot moves up one square (if the robot is already in the top row, then it does not move) With probability 0.1 the robot moves right one square (if the robot is already in the rightmost row, then it does not move)

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Probabilistic Transition Model In each state, the possible actions are U, D, R, and L The effect of U is as follows (transition model): With probability 0.8 the robot moves up one square (if the robot is already in the top row, then it does not move) With probability 0.1 the robot moves right one square (if the robot is already in the rightmost row, then it does not move) With probability 0.1 the robot moves left one square (if the robot is already in the leftmost row, then it does not move)

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Markov Property The transition properties depend only on the current state, not on previous history (how that state was reached)

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Sequence of Actions Planned sequence of actions: (U, R) 2 3 1 4321 [3,2]

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Sequence of Actions Planned sequence of actions: (U, R) U is executed 2 3 1 4321 [3,2] [4,2][3,3][3,2]

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Histories Planned sequence of actions: (U, R) U has been executed R is executed There are 9 possible sequences of states – called histories – and 6 possible final states for the robot! 4321 2 3 1 [3,2] [4,2][3,3][3,2] [3,3][3,2][4,1][4,2][4,3][3,1]

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Probability of Reaching the Goal P([4,3] | (U,R).[3,2]) = P([4,3] | R.[3,3]) x P([3,3] | U.[3,2]) + P([4,3] | R.[4,2]) x P([4,2] | U.[3,2]) 2 3 1 4321 Note importance of Markov property in this derivation P([3,3] | U.[3,2]) = 0.8 P([4,2] | U.[3,2]) = 0.1 P([4,3] | R.[3,3]) = 0.8 P([4,3] | R.[4,2]) = 0.1 P([4,3] | (U,R).[3,2]) = 0.65

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Utility Function [4,3] provides power supply [4,2] is a sand area from which the robot cannot escape +1 2 3 1 4321

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Utility Function [4,3] provides power supply [4,2] is a sand area from which the robot cannot escape The robot needs to recharge its batteries +1 2 3 1 4321

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Utility Function [4,3] provides power supply [4,2] is a sand area from which the robot cannot escape The robot needs to recharge its batteries [4,3] or [4,2] are terminal states +1 2 3 1 4321

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Utility of a History [4,3] provides power supply [4,2] is a sand area from which the robot cannot escape The robot needs to recharge its batteries [4,3] or [4,2] are terminal states The utility of a history is defined by the utility of the last state (+1 or –1) minus n/25, where n is the number of moves +1 2 3 1 4321

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Utility of an Action Sequence +1 Consider the action sequence (U,R) from [3,2] 2 3 1 4321

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Utility of an Action Sequence +1 Consider the action sequence (U,R) from [3,2] A run produces one among 7 possible histories, each with some probability 2 3 1 4321 [3,2] [4,2][3,3][3,2] [3,3][3,2][4,1][4,2][4,3][3,1]

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Utility of an Action Sequence +1 Consider the action sequence (U,R) from [3,2] A run produces one among 7 possible histories, each with some probability The utility of the sequence is the expected utility of the histories: U = h U h P(h) 2 3 1 4321 [3,2] [4,2][3,3][3,2] [3,3][3,2][4,1][4,2][4,3][3,1]

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Optimal Action Sequence +1 Consider the action sequence (U,R) from [3,2] A run produces one among 7 possible histories, each with some probability The utility of the sequence is the expected utility of the histories The optimal sequence is the one with maximal utility 2 3 1 4321 [3,2] [4,2][3,3][3,2] [3,3][3,2][4,1][4,2][4,3][3,1]

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Optimal Action Sequence +1 Consider the action sequence (U,R) from [3,2] A run produces one among 7 possible histories, each with some probability The utility of the sequence is the expected utility of the histories The optimal sequence is the one with maximal utility But is the optimal action sequence what we want to compute? 2 3 1 4321 [3,2] [4,2][3,3][3,2] [3,3][3,2][4,1][4,2][4,3][3,1] only if the sequence is executed blindly!

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Accessible or observable state Repeat: s sensed state If s is terminal then exit a choose action (given s) Perform a Reactive Agent Algorithm

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Policy (Reactive/Closed-Loop Strategy) A policy is a complete mapping from states to actions +1 2 3 1 4321

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Repeat: s sensed state If s is terminal then exit a (s) Perform a Reactive Agent Algorithm

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Optimal Policy +1 A policy is a complete mapping from states to actions The optimal policy * is the one that always yields a history (ending at a terminal state) with maximal expected utility 2 3 1 4321 Makes sense because of Markov property Note that [3,2] is a “dangerous” state that the optimal policy tries to avoid

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Optimal Policy +1 A policy is a complete mapping from states to actions The optimal policy * is the one that always yields a history with maximal expected utility 2 3 1 4321 This problem is called a Markov Decision Problem (MDP) How to compute *?

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Additive Utility History H = (s 0,s 1,…,s n ) The utility of H is additive iff: U (s 0,s 1,…,s n ) = R (0) + U (s 1,…,s n ) = R (i) Reward

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Additive Utility History H = (s 0,s 1,…,s n ) The utility of H is additive iff: U (s 0,s 1,…,s n ) = R (0) + U (s 1,…,s n ) = R (i) Robot navigation example: R (n) = +1 if s n = [4,3] R (n) = -1 if s n = [4,2] R (i) = -1/25 if i = 0, …, n-1

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Principle of Max Expected Utility History H = (s 0,s 1,…,s n ) Utility of H: U (s 0,s 1,…,s n ) = R (i) First-step analysis U (i) = R (i) + max a k P (k | a.i) U (k) *(i) = arg max a k P (k | a.i) U (k) +1

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Value Iteration Initialize the utility of each non-terminal state s i to U 0 (i) = 0 For t = 0, 1, 2, …, do: U t+1 (i) R (i) + max a k P (k | a.i) U t (k) +1 2 3 1 4321

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Value Iteration Initialize the utility of each non-terminal state s i to U 0 (i) = 0 For t = 0, 1, 2, …, do: U t+1 (i) R (i) + max a k P (k | a.i) U t (k) U t ([3,1]) t 0302010 0.611 0.5 0 +1 2 3 1 4321 0.7050.6550.3880.611 0.762 0.8120.8680.918 0.660 Note the importance of terminal states and connectivity of the state-transition graph

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Policy Iteration Pick a policy at random

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Policy Iteration Pick a policy at random Repeat: Compute the utility of each state for U t+1 (i) R (i) + k P (k | (i).i) U t (k)

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Policy Iteration Pick a policy at random Repeat: Compute the utility of each state for U t+1 (i) R (i) + k P (k | (i).i) U t (k) Compute the policy ’ given these utilities ’(i) = arg max a k P (k | a.i) U (k)

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Policy Iteration Pick a policy at random Repeat: Compute the utility of each state for U t+1 (i) R (i) + k P (k | (i).i) U t (k) Compute the policy ’ given these utilities ’(i) = arg max a k P(k | a.i) U (k) If ’ = then return Or solve the set of linear equations: U (i) = R (i) + k P (k | (i).i) U (k) (often a sparse system)

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n-Step decision process Assume that: Each state reached after n steps is terminal, hence has known utility There is a single initial state Any two states reached after i and j steps are different 0123

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n-Step Decision Process For j = n-1, n-2, …, 0 do: For every state s i attained after step j Compute the utility of s i Label that state with the corresponding action *(i) = arg max a k P(k | a.i) U (k) U (i) = R (i) + max a k P(k | a.i) U (k) 0123

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What is the Difference? *(i) = arg max a k P (k | a.i) U (k) U (i) = R (i) + max a k P (k | a.i) U (k) 0123 +1

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Infinite Horizon +1 2 3 1 4321 In many problems, e.g., the robot navigation example, histories are potentially unbounded and the same state can be reached many times One trick: Use discounting to make infinite Horizon problem mathematically tractable What if the robot lives forever?

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Example: Tracking a Target target robot The robot must keep the target in view The target’s trajectory is not known in advance The environment may or may not be known An optimal policy cannot be computed ahead of time: - The environment might be unknown -The environment may only be partially observable -The target may not wait A policy must be computed “on-the-fly”

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POMDP (Partially Observable Markov Decision Problem) A sensing operation returns multiple states, with a probability distribution Choosing the action that maximizes the expected utility of this state distribution assuming “state utilities” computed as above is not good enough, and actually does not make sense (is not rational)

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Example: Target Tracking There is uncertainty in the robot’s and target’s positions; this uncertainty grows with further motion There is a risk that the target may escape behind the corner, requiring the robot to move appropriately But there is a positioning landmark nearby. Should the robot try to reduce its position uncertainty?

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Summary Decision making under uncertainty Utility function Optimal policy Maximal expected utility Value iteration Policy iteration

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