Solving Large Markov Decision Processes Yilan Gu Dept. of Computer Science University of Toronto April 12, 2004

2 Outline Introduction: what’s the problem ? Temporal abstraction Logical representation of MDPs Potential future directions

3 Markov Decision Processes (MDPs) Decision-theoretic planning and learning problems are often modeled in MDPs. An MDP is a model M = consisting  a set of environment states S,  a set of actions A,  a transition function T: S  A  S  [0,1] T(s,a,s’) = Pr (s’| s,a),  a reward function R: S  A  R. A policy is a function  : S  A. Expected cumulative reward -- value function V  : S  R. The Bellman Eq.: V  (s) = R(s,  (s)) +   s’ T(s,  (s),s’) V  (s’)

4 MDP Example S = {(1,1), (1,2), …,(8,8)} A = {up, down, left, right} e.g., T((2,2),up,(1,2)) = 0.8, T((2,2),up,(2,1))=0.1, T((2,2),up,(2,3)) = 0.1, T((2,2),up,s’) = 0 for s’  (1,2), (2,1), (2,3) …… R((1,8)) = 1, R(s)= -1 for s  (1,8). Fig. The 8*8 grid world up 0.8 (2,2) Notice: explicit representation of the model

5 Conventional Solution Algorithms for MDPs Goal: looking for optimal policy  * so that V*(s) = V  * (s)  V  (s) for all s  S and  Conventional algorithms  Dynamic programming: value iteration and policy iteration,  Decision tree search algorithm, etc.  Example: Value iteration Beginning with arbitrary V 0 ; In each iteration n>0: for every s  S, Q n (s,a) := R(s, a) +   s’ T(s, a, s’) V n-1 (s’) for any a; V n (s) := max a Q n (s,a) ; When n  , V n (s)  V*(s). Problem: it does not scale up!

6 Solving Large MDPs (Part I) Temporal abstraction approaches (basic idea)  Solving MDPs hierarchically  Using complex actions or subtasks to compress the scales of the state spaces Representing and solving MDPs in a logical way (basic idea)  Logically representing environment features  Aggregating ‘similar’ states  Representing effect of actions compactly by using logical structures, and eliminating unaffected features during reasoning

7 Options (Macro-Actions)  Partition { S 1, S 2, S 3, S 4 }  A macro-action -- a local policy  i : S i  A on region S i  E.g., EPer ( S 1 ) = {(3,5),(5,3)}  Discounted transition model T i : S i  {  i }  Eper( S i )  [0,1]  Discounted reward model R i : S i  {  i }  R Example S1S1            

8 Abstract MDP M’= S ’=  Eper( S i ), e.g., {(4,3),(3,4),(5,3),(3,5),(6,4),(4,6), (5,6),(6,5)}. A ’=  A i, where A i is a set of macro-actions on region S i. Transition model T’: S ’  A ’  S ’  [0,1] T’(s,  i, s’) = T i (s,  i,s’) if s  S i, s’  Eper( S i ); T’(s,  i, s’) = 0 otherwise. Reward model R’: S ’  A ’  R R’(s,  i ) = R i (s,  i ) for any s’  Eper( S i ).

9 Other Temporal Abstraction Approaches Options [Sutton 1995; Singh, Sutton and Precup 1999] Macro-actions [Hauskrecht et al. 1998; Parr 1998] –Fixed policies Hierarchical abstract machines (HAMs) [Parr and Russell 1997; Andre and Russell 2001,2002] –Finite controllers MAXQ methods [Dietterich 1998, 2000] –Goal-oriented subtasks Etc.

10 Solving Large MDPs (Part II) Temporal abstraction approaches (basic idea)  Solving MDPs hierarchically  Using complex actions or subtasks to compress the scale of the state spaces Representing and solving MDPs logically (basic idea)  Logically representing environment features  Aggregating ‘similar’ states  Representing effect of actions compactly by using logical structures, and eliminating unaffected features during reasoning

11 First-Order MDPs Using the stochastic situation calculus to model decision-theoretic planning problems Underlying model : first-order MDPs (FOMDPs) Solving FOMDPs using symbolic dynamic programming

12 Stochastic Situation Calculus (I) Using choice axioms to specify possible outcomes n i (x) of any stochastic action a(x) Example: choice(delCoff(x),a)  a = delCoffS(x)  a = delCoffF(x) Situations: S 0, do(a,s) Fluents F(x,s) – modeling environment features compactly Examples: office(x,s), coffeeReq(x,s), holdingCoffee(s) Basic action theory:  Using successor state axioms to describe the effect of the actions ’ outcomes on each fluent coffeeReq(x,do(a,s))  coffeeReq(x,s)   a = delCoffS(x)

13 Stochastic Situation Calculus (II) Asserting probabilities (may be depended on conditions of current situation Example: prob(delCoffS(x), delCoff(x), s) = case [hot, 0.9;  hot, 0.7] Specifying rewards/costs conditionally Example: R(do(a,s)) = case[  x. a = delCoffS(x), 10;   x. a = delCoffS(x), 0] stGolog programs, policies proc  (x) if  holdingCoffee then getCoffee else ( ?(coffeeReq(x)) ; delCoffee(x)) end proc

14 Symbolic Dynamic Programming Representing value function V n-1 (s) logically case [  1 (s),v 1 ; … ;  m (s),v m ] Input: the system described in stochastic SitCal and V n-1 (s) Output (also in case format): –Q-functions Q n (a(x),s)= R(s)+   i prob(n i (x),a(x),s) V n-1 (do(n i (x),s)) –Value function V n (s) V n (s) = (  a)(  b) Q n (a,s)  Q n (b,s)

15 Other Logical Representations First-order MDPs [e.g., Boutilier et al. 2000; Boutilier, Reiter and Price 2001] Factored MDPs [e.g., Boutilier and Dearden 1994, Boutilier, Dearden and Goldszmit 1995; Hoey et al. 1999] Relational MDPs [e.g., Guestrin et al. 2003] Integrated Bayesian Agent Language (IBAL) [Pfeffer 2001] Etc [e.g., Bacchus 1993, Poole 1995].

16 Our Attempt: Combining temporal abstraction with logical representations of MDPs.

17 Motivation cityB cityA living(X, houseA) houseB inCity(Y,cityA)

18 Prior Work MAXQ approaches [Dietterich 2000] and PHAMs method [Andre and Sutton 2001] –Using variables to represent state features –Propositional representations Extending DTGolog with options [Ferrein, Fritz and Lakemeyer 2003] –Specifying options with the SitCal and Golog programs –Benefit: reusable when entering the exact same region –Shortage: options are based on explicit regions, and therefore not reusable under ‘similar’ regions

19 Our Idea and Potential Directions Given any stGolog program (a macro-action schema) Example : proc getCoffee(X) if  holdingCoffee then getCoffee else ( while coffeeReq(X) do delCoffee(X) ) end proc Basic Idea – inspired by macro-actions [Boutilier et al 1998]:  Analyzing the macro-action to find what has been affected by the macro-action Example: holdingCoffee, coffeeReq(X)  Preprocessing discounted transition and reward models Example: tr( holdingCoffee  coffeeReq(X), getCoffee(X),  holdingCoffee   coffeeReq(X) )

20 (Continue)  using and re-using macro-actions as primitive actions Benefit:  Schematic  Free variables in the macro-actions can represent a class of objects which have same characteristics  Even for infinite objects  Reusable in similar regions, other than the exact region

THE END Thank you!