CAP Plan, Activity, and Intent Recognition

Transcription

1 CAP Plan, Activity, and Intent Recognition Lecture 10: Sequential Decision-Making Under Uncertainty (part 1) MDPs and POMDPs Instructor: Dr. Gita Sukthankar SP2-1

2 Reminder Turn-in questionnaire Homework (due Thurs): 1 page descriing the improvements that you plan to make to your project in the next half of the semester SP2-2

3 Model: POMDP Applications? Strengths? Weaknesses? How does a POMDP differ from an HMM? SP2-3

4 Model: POMDP Applications? Human-root interaction Dialog management Assistive technology Agent interaction Strengths? Integrates action selection directly with state estimation Weaknesses? Intractale for real-world domains How does a POMDP differ from an HMM? MDP and POMDP are for calculating optimal decisions from sequences of oservations; HMMs are for recognizing hidden state.from sequences of oservations. MDP and POMDP: actions and rewards SP2-4

5 Markov Decision Processes Classical planning models: logical representation of transition systems goal-ased ojectives plans as sequences Markov decision processes generalize this view controllale, stochastic transition system general ojective functions (rewards) that allow tradeoffs with transition proailities to e made more general solution concepts (policies) SP2-5

6 Markov Decision Processes An MDP has four components, S, A, R, Pr: (finite) state set S ( S = n) (finite) action set A ( A = m) transition function Pr(s,a,t) each Pr(s,a,-) is a distriution over S represented y set of n x n stochastic matrices ounded, real-valued reward function R(s) represented y an n-vector can e generalized to include action costs: R(s,a) can e stochastic (ut replacale y expectation) Model easily generalizale to countale or continuous state and action spaces SP2-6

7 System Dynamics Finite State Space S State s 1013 : Loc = 236 Joe needs printout Craig needs coffee... SP2-7

8 System Dynamics Finite Action Space A Pick up Printouts? Go to Coffee Room? Go to charger? SP2-8

9 System Dynamics Transition Proailities: Pr(s i, a, s j ) Pro. = 0.95 SP2-9

10 System Dynamics Transition Proailities: Pr(s i, a, s k ) Pro. = 0.05 s 1 s 2... s n s s s n SP2-10

11 Reward Process Reward Function: R(s i ) - action costs possile Reward = -10 R s 1 12 s s n 10 SP2-11

12 Assumptions Markovian dynamics (history independence) Pr(S t+1 A t,s t,a t-1,s t-1,..., S 0 ) = Pr(S t+1 A t,s t ) Markovian reward process Pr(R t A t,s t,a t-1,s t-1,..., S 0 ) = Pr(R t A t,s t ) Stationary dynamics and reward Pr(S t+1 A t,s t ) = Pr(S t +1 A t,s t ) for all t, t Full oservaility though we can t predict what state we will reach when we execute an action, once it is realized, we know what it is SP2-12

13 Policies Nonstationary policy π:s x T A π(s,t) is action to do at state s with t-stages-to-go Stationary policy π:s A π(s) is action to do at state s (regardless of time) analogous to reactive or universal plan These assume or have these properties: full oservaility history-independence deterministic action choices MDP and POMDPs are methods for calculating the optimal lookup tales (policies). SP2-13

14 Value of a Policy How good is a policy π? How do we measure accumulated reward? Value function V: S R associates value with each state (sometimes S x T) V π (s) denotes value of policy at state s how good is it to e at state s? depends on immediate reward, ut also what you achieve susequently expected accumulated reward over horizon of interest note V π (s) R(s); it measures utility SP2-14

15 Value of a Policy (con t) Common formulations of value: Finite horizon n: total expected reward given π Infinite horizon discounted: discounting keeps total ounded SP2-15

16 Value Iteration (Bellman 1957) Markov property allows exploitation of DP principle for optimal policy construction no need to enumerate A Tn possile policies Value Iteration V V 0 k ( s) ( s) = R( s), = π *( s, k) R( s) = s max k + Pr( s, a, s') V ( ') ' 1 s s a arg max a Pr( s, a, s' s') V Vk is optimal k-stage-to-go value function Bellman ackup k 1 ( s') SP2-16

17 Value Iteration V t-2 V t-1 V t V t+1 s s s s4 V t (s4) = R(s4)+max { 0.7 V t+1 (s1) V t+1 (s4) 0.4 V t+1 (s2) V t+1 (s3) } SP2-17

18 Value Iteration V t-2 V t-1 V t V t+1 s s s s4 Π t (s4) = max { } SP2-18

19 Value Iteration SP2-19

20 Complexity T iterations At each iteration A computations of n x n matrix times n-vector: O( A n 3 ) Total O(T A n3) Can exploit sparsity of matrix: O(T A n2) SP2-20

21 MDP Application: Electric Elves Calculating optimal transfer of control policy in an adjustale autonomy application Dynamically adjusts users meetings State of world is known; future actions of users are unknown SP2-21

22 Recognizing User Intent MDP POMDP SP2-22

23 POMDPs Partially oservale Markov Decision Process (POMDP): a stochastic system Σ = (S, A, P) as efore A finite set O of oservations P a (o s) = proaility of oservation o in state s after executing action a Require that for each a and s, o in O P a (o s) = 1 O models partial oservaility The controller can t oserve s directly; it can only oserve o The same oservation o can occur in more than one state Why do the oservations depend on the action a? Why do we have P a (o s) rather than P(o s)? This is a way to model sensing actions, which do not change the state ut return information make some oservation availale (e.g., from a sensor) SP2-23

24 Example of a Sensing Action Suppose there are a state s 1 action a 1, and oservation o 1 with the following properties: For every state s, Pa 1 (s s) = 1 a 1 does not change the state Pa 1 (o 1 s 1 ) = 1, and Pa 1 (o 1 s) = 0 for every state s s 1 After performing a 1, o 1 occurs if and only if we re in state s 1 Then to tell if you re in state s 1, just perform action a 1 and see whether you oserve o 1 Two states s and s are indistinguishale if for every o and a, P a (o s) = P a (o s ) SP2-24

25 Belief States At each point we will have a proaility distriution (s) over the states in S (s) is called a elief state (our elief aout what state we re in) Basic properties: 0 (s) 1 for every s in S s in S (s) = 1 Definitions: a = the elief state after doing action a in elief state Thus a (s) = P(in s after doing a in ) = s' in S P a (s s') (s') a (o) = P(oserve o after doing a in ) Marginalize over states = s in S P a (o s) (s) ao (s) = P(in s after doing a in and oserving o) Belief states are n-dimensional vectors representing the proaility of eing in every state.. SP2-25

26 Belief States (Continued) Recall that in general, P(x y,z) P(y z) = P(x,y z) Thus P a (o s) a (s) = P(oserve o after doing a in s) P(in s after doing a in ) = P(in s and oserve o after doing a in ) Similarly, ao (s) a (o) = P(in s after doing a in and oserving o) * P(oserve o after doing a in ) = P(in s and oserve o after doing a in ) Thus ao (s) = P a (o s) a (s) / a (o) Formula for updating elief state Can use this to distinguish states that would otherwise e indistinguishale SP2-26

27 state Example Root r1 can move etween l1 and l2 move(r1,l1,l2) move(r1,l2,l1) There may e a container c1 in location l2 in(c1,l2) a = move(r1,l1,l2) O = {full, empty} full: c1 is present empty: c1 is asent areviate full as f, and empty as e a a state a a a SP2-27

28 Example (Continued) state Neither move action returns useful oservations For every state s and for a = either move action, P a (f s) = P a (e s) = P a (f s) = P a (e s) = 0.5 a = move(r1,l1,l2) Thus if there are no other actions, then state a s1 and s2 are indistinguishale s3 and s4 are indistinguishale a a a a SP2-28

29 Example (Continued) state Suppose there s a sensing action see that works perfectly in location l2 P see (f s4) = P see (e s3) = 1 P see (f s3) = P see (e s4) = 0 see does not work elsewhere P see (f s1) = P see (e s1) = P see (f s2) = P see (e s2) = 0.5 Then s1 and s2 are still indistinguishale s3 and s4 are now distinguishale a a SP2-29 a = move(r1,l1,l2) state a a a

30 Example (Continued) By itself, see doesn t tell us the state with certainty seee (s3) = P see (e s3) * see (s3) / see (e) = 1 * 0.25 / 0.5 = 0.5 If we first do a=move(l1,l2) then do see, this will tell the state with certainty state a = move(r1,l1,l2) Let ' = a ' seee (s3) = P see (e s3) * ' see (s3) / ' see (e) = 1 * 0.5 / 0.5 = 1 a state ' = a a a a SP2-30

31 Modified Example state Suppose we know the initial elief state is Policy to tell if there s a container in l2: π = {(, move(r1,l1,l2)), (', see)} a = move(r1,l1,l2) state ' = a a a a a SP2-31

32 Solving POMDPs Information-state MDPs Belief states of POMDP are states in new MDP Continuous state space Discretise Policy-tree algorithms SP2-32

33 Policy Trees Policy tree: an agent s non-stationary t-step policy Tree(a,T) create a new policy tree with action a at root and oservation z=t(z) Vp vector for value function for policy tree p with one component per state Act(p) action at root of tree p Sutree(p,z) sutree of p after os z Stval(a,z,p) vector for proaility-weighted value of tree p after a,z SP2-33

34 Application: Nurseot Root assists elderly patients Model uncertainty aout the user s dialog and position Exploit hierarchical structure to handle large state space SP2-34

35 Value Functions 2-state 3-state SP2-35

36 References Most slides were taken from Eyal Amir s course, CS 598, Decision Making under Uncertainty (lectures 12 and 13) L. Kaeling, M. Littman, and A. Cassandra, Planning and Acting in Partially Oservale Stochastic Domains, Artificial Intelligence, Volume 101, pp , 1998 SP2-36