U NCERTAINTY IN S ENSING ( AND ACTION )

A GENDA Planning with belief states Nondeterministic sensing uncertainty Probabilistic sensing uncertainty

B ELIEF S TATE A belief state is the set of all states that an agent think are possible at any given time or at any stage of planning a course of actions, e.g.: To plan a course of actions, the agent searches a space of belief states, instead of a space of states

S ENSOR M ODEL ( DEFINITION #1) State space S The sensor model is a function SENSE: S  2 S that maps each state s  S to a belief state (the set of all states that the agent would think possible if it were actually observing state s) Example: Assume our vacuum robot can perfectly sense the room it is in and if there is dust in it. But it can’t sense if there is dust in the other room SENSE( ) =

S ENSOR M ODEL ( DEFINITION #2) State space S, percept space P The sensor model is a function SENSE: S  P that maps each state s  S to a percept (the percept that the agent would obtain if actually observing state s) We can then define the set of states consistent with the observation P CONSISTENT(P) = { s if SENSE(s)=P } SENSE( ) = CONSISTENT( ) = ??

V ACUUM R OBOT A CTION AND S ENSOR M ODEL Right Appl if s  In(R 1 ) {s 1 = s - In(R 1 ) + In(R 2 ), s 2 = s} [Right does either the right thing, or nothing] Left Appl if s  In(R 2 ) {s 1 = s - In(R 2 ) + In(R 1 ), s 2 = s - In(R 2 ) + In(R 1 ) - Clean(R 2 )} [Left always move the robot to R 1, but it may occasionally deposit dust in R 2 ] Suck(r) Appl s  In(r) {s 1 = s+Clean(r)} [ Suck always does the right thing] The robot perfectly senses the room it is in and whether there is dust in it But it can’t sense if there is dust in the other room State s : any logical conjunction of In(R 1 ), In(R 2 ), Clean(R 1 ), Clean (R 2 ) (notation: + adds an attribute, - removes an attribute)

T RANSITION B ETWEEN B ELIEF S TATES Suppose the robot is initially in state: After sensing this state, its belief state is: Just after executing Left, its belief state will be: After sensing the new state, its belief state will be: or if there is no dust in R 1 if there is dust in R 1

T RANSITION B ETWEEN B ELIEF S TATES Suppose the robot is initially in state: After sensing this state, its belief state is: Just after executing Left, its belief state will be: After sensing the new state, its belief state will be: or if there is no dust in R 1 if there is dust in R 1 Left Clean(R 1 )  Clean(R 1 )

T RANSITION B ETWEEN B ELIEF S TATES How do you propagate the action/sensing operation to obtain the successors of a belief state? Left Clean(R 1 )  Clean(R 1 )

C OMPUTING THE T RANSITION BETWEEN BELIEF STATES Given an action A, and a belief state S = {s 1,…,s n } Result of applying action, without sensing: Take the union of all SUCC(s i,A) for i=1,…,n This gives us a pre-sensing belief state S’ Possible percepts resulting from sensing: {SENSE(s i ’) for s i ’ in S’} (using SENSE definition #2) This gives us a percept set P Possible states both in S’ AND consistent with each possible percept p j in P: S j = {s i | SENSE(s i ’)=p j for s i ’ in S’} i.e., S j = CONSISTENT(p j ) ∩ S’

AND/OR T REE OF B ELIEF S TATES Left Suck goal A goal belief state is one in which all states are goal states An action is applicable to a belief state B if its precondition is achieved in all states in B Right loop goal

P LANNING W ITH P ROBABILISTIC U NCERTAINTY IN S ENSING No motion Perpendicular motion

P ARTIALLY O BSERVABLE MDP S Consider the MDP model with states s  S, actions a  A Reward R(s) Transition model P(s’|s,a) Discount factor  With sensing uncertainty, initial belief state is a probability distributions over state: b(s) b(s i )  0 for all s i  S,  i b(s i ) = 1 Observations are generated according to a sensor model Observation space o  O Sensor model P(o|s) Resulting problem is a Partially Observable Markov Decision Process (POMDP)

POMDP U TILITY F UNCTION A policy  (b)  is defined as a map from belief states to actions Expected discounted reward with policy  : U  (b) = E[  t  t R(S t )] where S t is the random variable indicating the state at time t P(S 0 =s) = b 0 (s) P(S 1 =s) = ?

POMDP U TILITY F UNCTION A policy  (b)  is defined as a map from belief states to actions Expected discounted reward with policy  : U  (b) = E[  t  t R(S t )] where S t is the random variable indicating the state at time t P(S 0 =s) = b 0 (s) P(S 1 =s) = P(s|  (b  ),b 0 ) =  s’ P(s|s’,  (b 0 )) P(S 0 =s’) =  s’ P(s|s’,  (b 0 )) b 0 (s’)

POMDP U TILITY F UNCTION A policy  (b)  is defined as a map from belief states to actions Expected discounted reward with policy  : U  (b) = E[  t  t R(S t )] where S t is the random variable indicating the state at time t P(S 0 =s) = b 0 (s) P(S 1 =s) =  s’ P(s|s’,  (b)) b 0 (s’) P(S 2 =s) = ?

POMDP U TILITY F UNCTION A policy  (b)  is defined as a map from belief states to actions Expected discounted reward with policy  : U  (b) = E[  t  t R(S t )] where S t is the random variable indicating the state at time t P(S 0 =s) = b 0 (s) P(S 1 =s) =  s’ P(s|s’,  (b)) b 0 (s’) What belief states could the robot take on after 1 step?

b0b0 Predict b 1 (s)=  s’ P(s|s’,  (b 0 )) b 0 (s’) Choose action  (b 0 ) b1b1

b0b0 oAoA oBoB oCoC oDoD Predict b 1 (s)=  s’ P(s|s’,  (b 0 )) b 0 (s’) Choose action  (b 0 ) b1b1 Receive observation

b0b0 P(o A |b 1 ) Predict b 1 (s)=  s’ P(s|s’,  (b 0 )) b 0 (s’) Choose action  (b 0 ) b1b1 Receive observation b 1,A P(o B |b 1 )P(o C |b 1 )P(o D |b 1 ) b 1,B b 1,C b 1,D

b0b0 Predict b 1 (s)=  s’ P(s|s’,  (b 0 )) b 0 (s’) Choose action  (b 0 ) b1b1 Update belief b 1,A (s) = P(s|b 1,o A ) P(o A |b 1 )P(o B |b 1 )P(o C |b 1 )P(o D |b 1 ) Receive observation b 1,A b 1,B b 1,C b 1,D b 1,B (s) = P(s|b 1,o B ) b 1,C (s) = P(s|b 1,o C ) b 1,D (s) = P(s|b 1,o D )

b0b0 Predict b 1 (s)=  s’ P(s|s’,  (b 0 )) b 0 (s’) Choose action  (b 0 ) b1b1 Update belief P(o A |b 1 )P(o B |b 1 )P(o C |b 1 )P(o D |b 1 ) Receive observation P(o|b) =  s P(o|s)b(s) P(s|b,o) = P(o|s)P(s|b)/P(o|b) = 1/Z P(o|s) b(s) b 1,A (s) = P(s|b 1,o A ) b 1,B (s) = P(s|b 1,o B ) b 1,C (s) = P(s|b 1,o C ) b 1,D (s) = P(s|b 1,o D ) b 1,A b 1,B b 1,C b 1,D

B ELIEF - SPACE SEARCH TREE Each belief node has |A| action node successors Each action node has |O| belief successors Each (action,observation) pair (a,o) requires predict/update step similar to HMMs Matrix/vector formulation: b(s): a vector b of length |S| P(s’|s,a): a set of |S|x|S| matrices T a P(o k |s): a vector o k of length |S| b a = T a b (predict) P(o k |b a ) = o k T b a (probability of observation) b a,k = diag( o k ) b a / ( o k T b a ) (update) Denote this operation as b a,o

R ECEDING HORIZON SEARCH Expand belief-space search tree to some depth h Use an evaluation function on leaf beliefs to estimate utilities For internal nodes, back up estimated utilities: U(b) = E[R(s)|b] +  max a  A  o  O P(o|b a )U(b a,o )

QMDP E VALUATION F UNCTION One possible evaluation function is to compute the expectation of the underlying MDP value function over the leaf belief states f(b) =  s U MDP (s) b(s) “Averaging over clairvoyance” Assumes the problem becomes instantly fully observable after 1 action Is optimistic: U(b)  f(b) Approaches POMDP value function as state and sensing uncertainty decreases In extreme h=1 case, this is called the QMDP policy

QMDP P OLICY (L ITTMAN, C ASSANDRA, K AELBLING 1995 )

W ORST - CASE C OMPLEXITY Infinite-horizon undiscounted POMDPs are undecideable (reduction to halting problem) Exact solution to infinite-horizon discounted POMDPs are intractable even for low |S| Finite horizon: O(|S| 2 |A| h |O| h ) Receding horizon approximation: one-step regret is O(  h ) Approximate solution: becoming tractable for |S| in millions  -vector point-based techniques Monte Carlo tree search …Beyond scope of course…

S CHEDULE 11/29: Robotics 12/1: Guest lecture: Mike Gasser, Natural Language Processing 12/6: Review 12/8: Final project presentations, review