1. 4/21  Make-up class on this Friday  No class on next Tuesday Progression corresponds to finding a single path in the transition graph What about regression?

2. Interpreting progression and regression in the transition graph In the transition graph (corresponding to the atomic model) –progression search corresponds to finding a single path –Regression search corresponds to simultaneously starting from multiple states (all of which satisfy the goal conditions), and effectively searching in parallel until one of the paths reaches the initial state Alternately, you can see regression as searching in the space of sets of states, with the termination condition being that any of the states is an initial state...In contrast, planning with an incomplete state is also a search in the space of belief states (remember the vaccum world), except the termination condition requires that every state in the belief state is a goal state.

3. CSE 574: Planning & Learning Subbarao Kambhampati Handling Conditional Effects G Conditional effects don’t change the progression much at all –Why? (because the state in which the operator is being applied is known. So you know whether or not the conditional effect actually happens) G Handling conditional effects in regression planning introduces “secondary” preconditions –Consider regressing goals {P,Q} over an action A with two conditional effects: R=>P; J=>~Q –What happens if A has two more effects: U=> P; N=>~Q

4. CSE 574: Planning & Learning Subbarao Kambhampati

5. CSE 574: Planning & Learning Subbarao Kambhampati

6. CSE 574: Planning & Learning Subbarao Kambhampati Handling “lifted” actions (action schemas) G Progression doesn’t change much! –You can generate all the applicable groundings of the operator G Regression changes—can be less committed! –Consider regressing a goal state {P(a),Q(b)} over an action schema A with effects P(x) and ~Q(y) –What happens if the effects were U(x)=>P(x) and M(y)=>~Q(y)

7. Handling “lifted” actions (action schemas) Progression doesn’t change much! –You can generate all the applicable groundings of the operator Regression changes—can be less committed! –Consider regressing a goal state {P(a),Q(b)} over an action schema A with effects P(x) and ~Q(y) –What happens if the effects were U(x)=>P(x) and M(y)=>~Q(y)

8. Don’t look at curved lines for now… Have(cake) ~eaten(cake) ~Have(cake) eaten(cake) Eat No-op Have(cake) eaten(cake) bake ~Have(cake) eaten(cake) Have(cake) ~eaten(cake) Eat No-op Have(cake) ~eaten(cake) Graph has leveled off, when the prop list has not changed from the previous iteration The note that the graph has leveled off now since the last two Prop lists are the same (we could actually have stopped at the Previous level since we already have all possible literals by step 2)

9. Blocks world State variables: Ontable(x) On(x,y) Clear(x) hand-empty holding(x) Stack(x,y) Prec: holding(x), clear(y) eff: on(x,y), ~cl(y), ~holding(x), hand-empty Unstack(x,y) Prec: on(x,y),hand-empty,cl(x) eff: holding(x),~clear(x),clear(y),~hand-empty Pickup(x) Prec: hand-empty,clear(x),ontable(x) eff: holding(x),~ontable(x),~hand-empty,~Clear(x) Putdown(x) Prec: holding(x) eff: Ontable(x), hand-empty,clear(x),~holding(x) Initial state: Complete specification of T/F values to state variables --By convention, variables with F values are omitted Goal state: A partial specification of the desired state variable/value combinations --desired values can be both positive and negative Init: Ontable(A),Ontable(B), Clear(A), Clear(B), hand-empty Goal: ~clear(B), hand-empty All the actions here have only positive preconditions; but this is not necessary

10. onT-A onT-B cl-A cl-B he Pick-A Pick-B onT-A onT-B cl-A cl-B he h-A h-B ~cl-A ~cl-B ~he

11. onT-A onT-B cl-A cl-B he Pick-A Pick-B onT-A onT-B cl-A cl-B he h-A h-B ~cl-A ~cl-B ~he St-A-B St-B-A Ptdn-A Ptdn-B Pick-A onT-A onT-B cl-A cl-B he h-A h-B ~cl-A ~cl-B ~he on-A-B on-B-A Pick-B

12. Progression Regression How do we use reachability heuristics for regression?

13. Neither h lev nor h sum work well always p1p1 p2p2 p3p3 p 99 p 100 B1 q B2 B3 B99 B100 q P1P1 A0A0 P0P0 p1p1 p2p2 p3p3 p 99 p 100 q B* q P1P1 A0A0 P0P0 True cost of {p 1 …p 100 } is 100 (needs 100 actions to reach) H lev says the cost is 1 H sum says the cost is 100 H sum better than H lev True cost of {p 1 …p 100 } is 1 (needs just one action reach) H lev says the cost is 1 H sum says the cost is 100 H lev better than H sum H relax will get it correct both times..

14. onT-A onT-B cl-A cl-B he Pick-A Pick-B onT-A onT-B cl-A cl-B he h-A h-B ~cl-A ~cl-B ~he St-A-B St-B-A Ptdn-A Ptdn-B Pick-A onT-A onT-B cl-A cl-B he h-A h-B ~cl-A ~cl-B ~he on-A-B on-B-A Pick-B Relaxed plan for our blocks example

15. “Relaxed plan” Suppose you want to find a relaxed plan for supporting literals g1…gm on a k-length PG. You do it this way: –Start at kth level. Pick an action for supporting each gi (the actions don’t have to be distinct—one can support more than one goal). Let the actions chosen be {a1…aj} –Take the union of preconditions of a1…aj. Let these be the set p1…pv. –Repeat the steps 1 and 2 for p1…pv—continue until you reach init prop list. The plan is called “relaxed” because you are assuming that sets of actions can be done together without negative interactions. No backtracking needed! Optimal relaxed plan is still NP-hard

16. h-sum; h-lev; h-relax Given a set of literals {l1…lk} –H-lev is the earliest level in which all of them are present –H-sum is the sum of the earliest level in which each of them are present –H-relax is the length of the plan to support the literals H-lev is lower than or equal to h-relax H-sum is larger than or equal to H-lev H-lev is admissible H-relax is not admissible unless you find optimal relaxed plan –Which is NP-Hard..

17. Planning Graphs for heuristics  Construct planning graph(s) at each search node  Extract relaxed plan to achieve goal for heuristic p5 q5 r5 p6 o pq o pr o 56 p5p5 pqr56pqr56 o pq o pr o 56 pqrst567pqrst567 o ps o qt o 67 q5q5 qtr56qtr56 o qt o qr o 56 qtrsp567qtrsp567 o qs o tp o 67 r5r5 rqp56rqp56 o rq o rp o 56 rqpst567rqpst567 o rs o qt o 67 p6p6 pqr67pqr67 o pq o pr o 67 pqrst678pqrst678 o ps o qt o 78 1 3 4 1 3 o 12 o 34 2 1 3 4 5 o 12 o 34 o 23 o 45 2 3 4 5 3 5 o 34 o 56 3 4 5 o 34 o 45 o 56 66 7 o 67 1 5 1 5 o 12 o 56 2 1 3 5 o 12 o 23 o 56 2 66 7 o 67 G oGoG G oGoG G oGoG G oGoG G oGoG 1 3 3 5 1 5 h( )=5

18. What if actions have non-uniform costs?

19. Challenges in Cost Propagation

21. What if you didn’t have any hard goals..? And got rewards continually? And have stochastic actions? MDPs as Utility-based problem solving agents

22. Planning  PSP  MDPs In addition to actions having costs, we might have goals with rewards, with the understanding that if you achieve a goal, you get the corresponding reward So now, the objective of planning is to find a plan that has the highest net benefit measured as the difference between the cumulative reward for the goals achieved and the cumulative cost of the actions used This problem, called partial satisfaction planning, is both easy (since an “empty” plan is a solution, just not a very good one) and hard (since now the “quality of the plan” in terms of its net benefit is more important) –It is possible to extend the planning graph heuristics to this problem On top of this, we might also want to say that rewards are not limited to just goals achieved in the final state, but can also be gathered for visiting certain good states on the way –Such goals are called “trajectory constraints” Even further, we can consider a scenario where the actions are stochastic –By this time it is not even clear that a sequence of actions is an adequate form for the solution. We need to understand it first at the atomic level—and we shall do so. By the way, this problem is called Markov Decision Process. [MDPs can be done at propositional and relational level, but we won’t discuss that in this class]. If our masochism continues unabated, we can also now say that in addition to actions being stochastic, we have partial observability –This will lead to a generalization of MDP called POMDP (Partially Observable MDP); we won’t cover this in this course...but as long as we are naming things, if we consider actions with durations, we get Semi-MDPs; if we consider other agents, we get decentralized-MDPs (and in each case we can have PO versions..)

23. [can generalize to have action costs C(a,s)] If M ij matrix is not known a priori, then we have a reinforcement learning scenario.. Repeat

24. What does a solution to an MDP look like? The solution should tell the optimal action to do in each state (called a “Policy”) –Policy is a function from states to actions (* see finite horizon case below*) –Not a sequence of actions anymore Needed because of the non-deterministic actions –If there are |S| states and |A| actions that we can do at each state, then there are |A| |S| policies How do we get the best policy? –Pick the policy that gives the maximal expected reward –For each policy  Simulate the policy (take actions suggested by the policy) to get behavior traces Evaluate the behavior traces Take the average value of the behavior traces. We will concentrate on infinite horizon problems (infinite horizon doesn’t necessarily mean that that all behavior traces are infinite. They could be finite and end in a sink state)

25. Optimal Policies depend on rewards.. -- Repeat - -

26. 4/23 If you are twenty and not a liberal, you are heartless. If you are sixty and not a conservative, you are mindless. --Winston Churchill But why is Rao putting this here? He better not be hinting that the campus republicans are heartless or geriatric.. Make-up class: Tomorrow (Friday) 10:30—11:45 in DCDC Conference Room 175 [pass Bisonwitches, turn right]

27. Horizon & Policy We said policy is a function from states to actions.. but we sort of lied. Best policy is non-stationary, i.e., depends on how long the agent has to “live” – which is called “horizon” More generally, a policy is a mapping from  –So, if we have a horizon of k, then we will have k policies If the horizon is infinite, then policies must all be the same.. (So infinite horizon case is easy!) If you are twenty and not a liberal, you are heartless If you are sixty and not a conservative, you are mindless --Churchill

28. What does a solution to an MDP look like? The solution should tell the optimal action to do in each state (called a “Policy”) –Policy is a function from states to actions (* see finite horizon case below*) –Not a sequence of actions anymore Needed because of the non-deterministic actions –If there are |S| states and |A| actions that we can do at each state, then there are |A| |S| policies How do we get the best policy? –Pick the policy that gives the maximal expected reward –For each policy  Simulate the policy (take actions suggested by the policy) to get behavior traces Evaluate the behavior traces Take the average value of the behavior traces. We will concentrate on infinite horizon problems (infinite horizon doesn’t necessarily mean that that all behavior traces are infinite. They could be finite and end in a sink state)

29. Horizon & Policy How long should behavior traces be? –Each trace is no longer than k (Finite Horizon case) Policy will be horizon-dependent (optimal action depends not just on what state you are in, but how far is your horizon) –Eg: Financial portfolio advice for yuppies vs. retirees. –No limit on the size of the trace (Infinite horizon case) Policy is not horizon dependent We will concentrate on infinite horizon problems (infinite horizon doesn’t necessarily mean that that all behavior traces are infinite. They could be finite and end in a sink state) If you are twenty and not a liberal, you are heartless If you are sixty and not a conservative, you are mindless --Churchill

30. How to evaluate a policy? Step 1: Define utility of a sequence of states in terms of their rewards –Assume “stationarity” of preferences If you prefer future f1 to f2 starting tomorrow, you should prefer them the same way even if they start today –Then, only two reasonable ways to define Utility of a sequence of states –U(s 1, s 2  s n ) =  n R(s i ) –U(s 1, s 2  s n ) =  n ° i R(s i ) (0 · ° · 1) Maximum utility bounded from above by R max /(1 - °) Step 2: Utility of a policy ¼ is the expected utility of the behaviors exhibited by an agent following it. E [  1 t=0 ° t R(s t ) | ¼ ] Step 3: Optimal policy ¼ * is the one that maximizes the expectation: argmax ¼ E [  1 t=0 ° t R(s t ) | ¼ ] –Since there are only A |s| different policies, you can evaluate them all in finite time (Haa haa..)

31. How to handle unbounded state sequences? If we don’t have a horizon, then we can have potentially infinitely long state sequences. Three ways to handle them 1.Use discounted reward model ( i th state in the sequence contributes only ° i R(s i ) 2.Assume that the policy is proper (i.e., each sequence terminates into an absorbing state with non-zero probability). 3.Consider “average reward per-step”

32. Why are they called Markov decision processes? Markov property means that state contains all the information (to decide the reward or the transition) –Reward of a state S n is independent of the path used to get to S n –Effect of doing an action A in state S n doesn’t depend on the way we reached state S n –(As a consequence of the above) Maximal expected utility of a state S doesn’t depend on the path used to get to S Markov properties are assumed (to make life simple) –It is possible to have non-markovian rewards (e.g. you will get a reward in state S i only if you came to S i through S J E.g. If you picked up a coupon before going to the theater, then you will get a reward –It is possible to convert non-markovian rewards into markovian ones, but it leads to a blow-up in the state space. In the theater example above, add “coupon” as part of the state (it becomes an additional state variable—increasing the state space two-fold). –It is also possible to have non-markovian effects—especially if you have partial observability E.g. Suppose there are two states of the world where the agent can get banana smell

33. (Value) How about deterministic case? U(s i ) is the shortest path to the goal Repeat (“sequence of states” = “behavior”)

35. Utility of a State The (long term) utility of a state s with respect to a policy \pi is the expected value of all state sequences starting with s –U ¼ (s) = E [  1 t=0 ° t R(s t ) | ¼, s 0 =s ] The true utility of a state s is just its utility w.r.t optimal policy U(s) =U ¼ *(s) Thus, U and ¼ * are closely related – ¼ * (s) = argmax a  s’ M a ss’ U(s’) As are utilities of neighboring states –U(s) = R(s) + ° argmax a  s’ M a ss’ U(s’) Bellman Eqn

36. Optimal Utility depends on Optimal Policy If you go to Tiger Hill near Darjeeling, and only look towards the direction the Sun is rising, you may not understand what the brouhaha is all about; but if you look the other side, you see this enchanting view of Kanchanjunga

37. Think of these as h*() values… Called value function U* Think of these as related to h* values Repeat U* is the maximal expected utility (value) assuming optimal policy

38. (Value) How about deterministic case? U(s i ) is the shortest path to the goal Repeat (“sequence of states” = “behavior”)

39. Bellman Equations as a basis for computing optimal policy Qn: Is there a simpler way than having to evaluate |A| |S| policies? –Yes… The Optimal Value and Optimal Policy are related by the Bellman Equations –U(s) = R(s) + ° argmax a  s’ M a ss’ U(s’) – ¼ * (s) = argmax a  s’ M a ss’ U(s’) The equations can be solved exactly through –“value iteration” (iteratively compute U and then compute ¼ * ) – “policy iteration” ( iterate over policies) –Or solve approximately through “real-time dynamic programming”

40. .8.1 U(i) = R(i) + ° max j M a ij U(j) + °

41. Why are values coming down first? Why are some states reaching optimal value faster? Updates can be done synchronously OR asynchronously --convergence guaranteed as long as each state updated infinitely often.8.1

42. Value Iteration Demo http://www.cs.ubc.ca/spider/poole/demos/ mdp/vi.htmlhttp://www.cs.ubc.ca/spider/poole/demos/ mdp/vi.html Things to note –The way the values change (states far from absorbing states may first reduce and then increase their values) –The convergence speed difference between Policy and value

43. Terminating Value Iteration The basic idea is to terminate the value iteration when the values have “converged” (i.e., not changing much from iteration to iteration) –Set a threshold  and stop when the change across two consecutive iterations is less than  –There is a minor problem since value is a vector We can bound the maximum change that is allowed in any of the dimensions between two successive iterations by  Max norm ||.|| of a vector is the maximal value among all its dimensions. We are basically terminating when ||U i – U i+1 || < 

44. 4/28 (held on 4/24)  Policy Iteration  Real-time Dynamic Programming  Min-max Search  Alpha-beta pruning

45. Policies converge earlier than values There are finite number of policies but infinite number of value functions. So entire regions of value vector are mapped to a specific policy So policies may be converging faster than values. Search in the space of policies Given a utility vector U i we can compute the greedy policy  ui The policy loss of  ui is ||U  ui  U*|| (max norm difference of two vectors is the maximum amount by which they differ on any dimension) V(S 1 ) V(S 2 ) Consider an MDP with 2 states and 2 actions P1P1 P2P2 P3P3 P4P4 U*U*

46. We can either solve the linear eqns exactly, or solve them approximately by running the value iteration a few times (the update wont have the “max” operation) n linear equations with n unknowns.

47. Bellman equations when actions have costs The model discussed in class ignores action costs and only thinks of state rewards –C(s,a) is the cost of doing action a in state s Assume costs are just negative rewards.. –The Bellman equation then becomes U(s) = R(s) + ° max a [ -C(s,a) +  s’ R(s’) M a ss’ ] Notice that the only difference is that -C(s,a) is now inside the maximization With this model, we can talk about “partial satisfaction” planning problems where –Actions have costs; goals have utilities and the optimal plan may not satisfy all goals.

48. Incomplete observability (the dreaded POMDPs) To model partial observability, all we need to do is to look at MDP in the space of belief states (belief states are fully observable even when world states are not) –Policy maps belief states to actions In practice, this causes (humongous) problems –The space of belief states is “continuous” (even if the underlying world is discrete and finite). {GET IT? GET IT??} –Even approximate policies are hard to find (PSPACE- hard). Problems with few dozen world states are hard to solve currently –“Depth-limited” exploration (such as that done in adversarial games) are the only option… Belief state = { s 1 :0.3, s 2 :0.4; s 4 :0.3} This figure basically shows that belief states change as we take actions 5 LEFTs5 UPs5 rights

49. MDPs and Deterministic Search Problem solving agent search corresponds to what special case of MDP? –Actions are deterministic; Goal states are all equally valued, and are all sink states. Is it worth solving the problem using MDPs? –The construction of optimal policy is an overkill The policy, in effect, gives us the optimal path from every state to the goal state(s)) –The value function, or its approximations, on the other hand are useful. How? As heuristics for the problem solving agent’s search This shows an interesting connection between dynamic programming and “state search” paradigms –DP solves many related problems on the way to solving the one problem we want –State search tries to solve just the problem we want –We can use DP to find heuristics to run state search..

50. Real Time Dynamic Programming Value and Policy iteration are the bed-rock methods for solving MDPs. Both give optimality guarantees –Both of them tend to be very inefficient for large (several thousand state) MDPs (Polynomial in |S|  ) Many ideas are used to improve the efficiency while giving up optimality guarantees –E.g. Consider the part of the policy for more likely states (envelope extension method) –Interleave “search” and “execution” (Real Time Dynamic Programming) Do limited-depth analysis based on reachability to find the value of a state (and there by the best action you should be doing—which is the action that is sending you the best value) The values of the leaf nodes are set to be their immediate rewards –Alternatively some admissible estimate of the value function (h*) If all the leaf nodes are terminal nodes, then the backed up value will be true optimal value. Otherwise, it is an approximation… RTDP For leaf nodes, can use R(s) or some heuristic value h(s)

51. What if you see this as a game? The expected value computation is fine if you are maximizing “expected” return If you are --if you are risk-averse? (and think “nature” is out to get you) V 2 = min(V 3,V 4 ) If you are perpetual optimist then V 2 = max(V 3,V 4 ) If you have deterministic actions then RTDP becomes RTA* (if you use h(.) to evaluate leaves

52. Incomplete observability (the dreaded POMDPs) To model partial observability, all we need to do is to look at MDP in the space of belief states (belief states are fully observable even when world states are not) –Policy maps belief states to actions In practice, this causes (humongous) problems –The space of belief states is “continuous” (even if the underlying world is discrete and finite). {GET IT? GET IT??} –Even approximate policies are hard to find (PSPACE- hard). Problems with few dozen world states are hard to solve currently –“Depth-limited” exploration (such as that done in adversarial games) are the only option… Belief state = { s 1 :0.3, s 2 :0.4; s 4 :0.3} This figure basically shows that belief states change as we take actions 5 LEFTs 5 UPs

53. Von Neuman (Min-Max theorem) Claude Shannon (finite look-ahead) Chaturanga, India (~550AD) (Proto-Chess) John McCarthy (  pruning) Donald Knuth (  analysis)

54. What if you see this as a game? The expected value computation is fine if you are maximizing “expected” return If you are --if you are risk-averse? (and think “nature” is out to get you) V 2 = min(V 3,V 4 ) If you are perpetual optimist then V 2 = max(V 3,V 4 ) Review

55. Game Playing (Adversarial Search) Perfect play –Do minmax on the complete game tree Alpha-Beta pruning (a neat idea that is the bane of many a CSE471 student) Resource limits –Do limited depth lookahead –Apply evaluation functions at the leaf nodes –Do minmax Miscellaneous –Games of Chance –Status of computer games..

56. Snakes-and-ladders is perfect information with chance  think of the utter boringness of deterministic snakes and ladders  Not that the normal snakes-and-ladders has any real scope for showing your thinking power (your only action is dictated by the dice—so the dice can play it as a solitaire—at most they need your hand..). Kriegspiel (blind-fold chess)

58. Searching Tic Tac Toe using Minmax A game is considered Solved if it can be shown that the MAX player has a winning (or at least Non-losing) Strategy This means that the backed-up Value in the Full min-max Tree is +ve

59. 2 <= 2 Cut 14 <= 14 5 <= 5 2 <= 2 Whenever a node gets its “true” value, its parent’s bound gets updated When all children of a node have been evaluated (or a cut off occurs below that node), the current bound of that node is its true value Two types of cutoffs: If a min node n has bound =j, then cutoff occurs as long as j >=k If a max node n has bound >=k, and a min ancestor of n, say m, has a bound <=j, then cutoff occurs as long as j<=k

60. Another alpha-beta example Project 2 assigned

61. Click for an animation of Alpha-beta search in action on Tic-Tac-Toen animation of Alpha-beta search in action on Tic-Tac-Toe (order nodes in terms of their static eval values)

63. 4/24 class ended here

64. 

66. How does it feel to be black and poor? –A. Very bad –B. Somewhat bad –C. Neither bad nor good –D. Somewhat good –E. Very good –F. F*** you

70. Evaluation Functions: TicTacToe If win for Max +infty If lose for Max -infty If draw for Max 0 Else # rows/cols/diags open for Max - #rows/cols/diags open for Min

73. What depth should we go to? --Deeper the better (but why?) Should we go to uniform depth? --Go deeper in branches where the game is in a flux (backed up values are changing fast) [Called “Quiescence” ] Can we avoid the horizon effect?

74. Why is “deeper” better? Possible reasons –Taking mins/maxes of the evaluation values of the leaf nodes improves their collective accuracy –Going deeper makes the agent notice “traps” thus significantly improving the evaluation accuracy All evaluation functions first check for termination states before computing the non-terminal evaluation

75. (just as human weight lifters refuse to compete against cranes)

77. MDPs and Deterministic Search Problem solving agent search corresponds to what special case of MDP? –Actions are deterministic; Goal states are all equally valued, and are all sink states. Is it worth solving the problem using MDPs? –The construction of optimal policy is an overkill The policy, in effect, gives us the optimal path from every state to the goal state(s)) –The value function, or its approximations, on the other hand are useful. How? As heuristics for the problem solving agent’s search This shows an interesting connection between dynamic programming and “state search” paradigms –DP solves many related problems on the way to solving the one problem we want –State search tries to solve just the problem we want –We can use DP to find heuristics to run state search..

78. RTA* (RTDP with deterministic actions and leaves evaluated by f(.)) Sn m k G S n m G=1 H=2 F=3 G=1 H=2 F=3 k G=2 H=3 F=5 infty --Grow the tree to depth d --Apply f-evaluation for the leaf nodes --propagate f-values up to the parent nodes f(parent) = min( f(children)) RTA* is a special case of RTDP --It is useful for acting in determinostic, dynamic worlds --While RTDP is useful for actiong in stochastic, dynamic worlds

79. End of Gametrees

80. (so is MDP policy)

83. Multi-player Games Everyone maximizes their utility --How does this compare to 2-player games? (Max’s utility is negative of Min’s)

84. Expecti-Max