Optimally Solving Dec-POMDPs as Continuous-State MDPs

Transcription

1 Optimally Solving Dec-POMDPs as Continuous-State MDPs Jilles Dibangoye (1), Chris Amato (2), Olivier Buffet (1) and François Charpillet (1) (1) Inria, Université de Lorraine France (2) MIT, CSAIL USA IJCAI August 8, 2013 Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

2 Outline Outline 1 Background Overview Decentralized POMDPs Existing methods 2 Dec-POMDPs as continuous-state MDPs Overview Solving the occupancy MDP Exploiting multiagent structure 3 Experiments 4 Conclusion Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

3 Background Overview General overview Agents situated in a world, receiving information and choosing actions Uncertainty about outcomes and sensors Sequential domains Cooperative multi-agent Decision-theoretic approach Developing approaches that scale to real-world domains Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

4 Background Overview Cooperative multiagent problems Each agent s choice affects all others, but must be made using only local information Communication may be costly, slow or noisy Domains of interest robotics, disaster response, networks,... Surveillance Area Communication Area Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, 2013 Base Area 4 / 20

5 Background Decentralized POMDPs Multi-Agent Decision Making Under Uncertainty Decentralized partially observable Markov decision process (Dec-POMDP) Sequential decision-making At each stage, each agent takes an action and receives: A local observation A joint immediate reward a 1 r o 1 a n Environment o n Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

6 Background Decentralized POMDPs Multi-Agent Decision Making Under Uncertainty Dec-POMDP definition Dec-POMDP I, S, {A i }, {Z i }, p, r, o, b 0, T I, a finite set of agents S, a finite set of states A i, each agent s finite set of actions Z i, each agent s finite set of observations p, the state transition model: Pr(s s, a) o, the observation model: Pr( o s, a) r, the reward model: R(s, a) b 0, initial state distribution T, planning horizon Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

7 Dec-POMDP solutions Background Decentralized POMDPs History θi t = ai 0, oi 1,..., a t 1 i, oi t Local policy: each agent maps histories to actions, π i : Θ i A i State is unknown, so beneficial to remember history π i, a sequence of decision rules π i = πi 0,..., π T 1 i mapping histories to actions, πi t (θi t ) = a i Joint policy π = π 1,..., π n with individual (local) agent policies π i Goal is to maximize expected cumulative reward over a finite horizon a 1 o 1 o 2 a 1 o 1 o 2 a 2 a 3 a 2 a 3 o 1 o 2 o 1 o 2 o 1 o 2 o 1 o 2 a 3 a 2 a 1 a 1 a 3 a 2 a 1 a 1 Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

8 POMDPs Background a o,r Decentralized POMDPs Environment Subclass of Dec-POMDPs with only one agent Agent maintain s belief state (distributions over states) Policy = mapping from histories or belief states π : B A Can solve a POMDP as a continuous-state belief MDP V π (b) = R(b, a) + o Pr(b b, a, o) Pr(o b, a)v π (b ) Structure: piecewise linear convex (PWLC) value function s 1 s 2 S Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

9 Background Decentralized POMDPs Example: 2-Agent Navigation Meeting in a grid States: grid cell pairs Actions: move,,,, stay Transitions: noisy Observations: red lines Rewards: negative unless sharing the same square Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

10 Background Decentralized POMDPs Challenges in solving Dec-POMDPs Partial observability makes the problem difficult to solve No common state estimate (centralized belief state) or concise sufficient statistic Each agent depends on the others Can t directly transform Dec-POMDPs into a continuous-state MDP from a single agent s perspective Therefore, Dec-POMDPs are fundamentally different and more complex (NEXP instead of PSPACE) Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

11 Current methods Background Existing methods Assume an offline planning phase that is centralized Generate explicit policy representations (trees) for each agent Search bottom up (DP) or top down (heuristic search) Often use game-theoretic ideas from the perspective of a single agent Search in the space of policies for the optimal set Top down Bottom up t=0 a1 a1 o1 o2 o1 o2 t=1 a1 a2 a1 a2 o1 o2 o1 o2 o1 o2 o1 o2 t=2 a1 a2 a2 a2 a1 a2 a2 a2 old new Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

12 Dec-POMDPs as continuous-state MDPs Overview Overview of our approach Current methods don t take full advantage of centralized planning phase Overview Push common information into an occupancy state Move local information into action selection as decision rules Formalize Dec-POMDPs as continuous-state MDPs with a PWLC value function Exploit multiagent structure in representation, making it scalable This doesn t use explicit policy representations or construct policies from a single agent s perspective Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

13 Dec-POMDPs as continuous-state MDPs Overview Centralized Sufficient Statistic Policy π, sequence of decentralized decision rules, π = π 0,..., π T 1 Joint history θ t = θ t 1,..., θ t n, with π t (θ t ) = a 1,..., a n a3 a2 o1 a1 o2 a3 o1 o2 o1 o2 a2 a1 a1 a3 a2 o1 a1 o2 a3 o1 o2 o1 o2 a2 a1 a1 An occupancy state is a distribution η(s, θ t ) = Pr(s, θ t π 0:t 1, b 0 ) The occupancy state is a sufficient statistic: Can optimize future policy π t:t over η rather than initial belief and past joint policies Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

14 Dec-POMDPs as continuous-state MDPs Overview Dec-POMDPs as continuous-state MDPs Occupancy state η t (s, θ t ) = Pr(s, θ t π 0:t 1, η 0 ) with η 0 = b 0 Transform Dec-POMDP into a continuous-state MDP s MDP : η a MDP : π t (decentralized decision rules) T MDP : Pr(η t π t 1, η t 1 ) Deterministic with P(η t, π t ) = η t+1 R MDP : s,θ t η t (s, θ t )R(s, π t (θ t )) Centralized sufficient statistic (the occupancy state) Decision rules ensure decentralization Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

15 Dec-POMDPs as continuous-state MDPs Overview Piecewise linear convexity Bellman optimality operator: V t (η t ) = max π t D R MDP(η t, π t ) + V t+1(p(η t, π t )) 1- Operator preserves PWLC property (piecewise linearity and convexity) 2- R MDP (η t, π t ) is linear PWLC value function SxΘ t POMDP algorithms can be used! Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

16 Dec-POMDPs as continuous-state MDPs Solving the occupancy MDP Solving the occupancy MDP Feature-based heuristic search value iteration (FB-HSVI) Based on heuristic search value iteration (Smith and Simmons, UAI 04) Sample occupancy distributions starting from the initial occupancy Update upper bounds based on decision rules (on the way down) Update lower bounds (on the way back up) Stop when bounds converge for initial occupancy υ(η) η, β lower bound Λ = {β k } k β 0 β1 β 2 η ῡ(η) 0.2 v v2 upper bound Γ = {(η k, v k )} k (η 1, v 1 ) (η 2, v 2 ) η = 0.2 η η 2 Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

17 Scaling up Dec-POMDPs as continuous-state MDPs Exploiting multiagent structure The occupancy MDP has very large action and state spaces Two key ideas to deal with these combinatorial explosions: 1 State reduction through history compression Compress histories of the same length (Oliehoek et al., JAIR 13) Reduce history length without loss 2 More efficient action selection Generating a greedy decision rule for an occupancy state as a weighted constraint satisfaction problem Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

18 Experiments Experiments Tested 3 versions of our algorithm Algorithm 0: HSVI with occupancy MDP Algorithm 1: HSVI with efficient action selection Algorithm 2: HSVI with efficient action selection + feature-based state space Comparison algorithms Forward search: GMAA*-ICE (Spaan et al., IJCAI 2011) Dynamic programming: IPG (Amato et al., ICAPS 2009), LPC (Boularias and Chaib-draa, ICAPS 2008) Optimization: MILP (Aras and Dutech, JAIR 2010) Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20

19 Experiments Experiments Optimal υ within ε = 0.01 The multi-agent tiger problem ( S =2, Z =4, A =9, K =3) T MILP LPC IPG ICE FB-HSVI( ) ( 0 ) The recycling-robot problem ( S =4, Z =4, A =9, K =1) The mars-rovers problem ( S =256, Z =81, A =36, K =3) Table 2: Experiments comparing the computation times (in Time and value on benchmarks Blank space = algorithm over time (200s) Red for fastest and previously unsolvable horizons K is the largest history window Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20 used

20 Conclusion Conclusion Summary Dec-POMDPs are powerful multiagent models Formulated Dec-POMDPs as continuous-state MDPs with PWLC value function POMDP (and continuous MDP) methods can now be applied Can also take advantage of multiagent structure in the problem Our approach shows significantly improved scalability Future work Approximate solutions (bounds on the solution quality) More concise statistics Subclasses like TI Dec-MDPs in our AAMAS-13 paper Just observation histories as in Oliehoek, IJCAI 13 Dibangoye, Amato, Buffet and Charpillet Optimally Solving Dec-POMDPs as MDPs August 8, / 20