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- Distributed Constraint Handling and OptimizationAlessandro Farinelli1 Alex Rogers2 Meritxell Vinyals1 1 Computer Science Department University of Verona, Italy 2 Agents, Interaction and Complexity Group School of Electronics and Computer Science University of Southampton,UK Tutorial EASSS 2012 Valencia https://sites.google.com/site/ easss2012optimization/
- OutlineIntroductionDistributed Constraint ReasoningApplications and Exemplar ProblemsComplete algorithms for DCOPsApproximated Algorithms for DCOPsConclusions
- OutlineIntroductionDistributed Constraint ReasoningApplications and Exemplar ProblemsComplete algorithms for DCOPsApproximated Algorithms for DCOPsConclusions View slide
- Constraints• Pervade our everyday lives• Are usually perceived as elements that limit solutions to the problems we face View slide
- ConstraintsFrom a computational point of view, they: • Reduce the space of possible solutions • Encode knowledge about the problem at hand • Are key components for efﬁciently solving hard problems
- Constraint ProcessingMany different disciplines deal with hard computational problems that can be made tractable by carefully considering the constraints that deﬁne the structure of the problem. Planning Operational Automated Reasoning Computer Scheduling Research Decision Theory Vision
- Constraint Processing in Multi-Agent Systems Focus on how constraint processing can be used to address optimization problems in Multi-Agent Systems (MAS) where:A set of agents must come to some agreement, typically via someform of negotiation, about which action each agent should take in order to jointly obtain the best solution for the whole system. M2 M1 A1 A2 A2 A3
- Distributed Constraint Optimization Problems (DCOPs)We will consider Distributed Constraint Optimization Problems (DCOP)where: Each agent negotiates locally with just a subset of other agents (usually called neighbors) that are those that can directly inﬂuence his/her behavior. M2 M1 A2 A1 A3
- Distributed Constraint Optimization Problems (DCOPs)After reading this chapter, you will understand: • The mathematical formulation of a DCOP • The main exact solution techniques for DCOPs • Key differences, beneﬁts and limitations • The main approximate solution techniques for DCOPs • Key differences, beneﬁts and limitations • The quality guarantees these approach provide: • Types of quality guarantees • Frameworks and techniques
- OutlineIntroductionDistributed Constraint ReasoningApplications and Exemplar ProblemsComplete algorithms for DCOPsApproximated Algorithms for DCOPsConclusions
- Constraint NetworksA constraint network N is formally deﬁned as a tuple X, D,C where: • X = {x1 , . . . , xn } is a set of discrete variables; • D = {D1 , . . . , Dn } is a set of variable domains, which enumerate all possible values of the corresponding variables; and • C = {C1 , . . . ,Cm } is a set of constraints; where a constraint Ci is deﬁned on a subset of variables Si ⊆ X which comprise the scope of the constraint • r = |Si | is the arity of the constraint • Two types: hard or soft
- Hard constraints• A hard constraint Cih is a relation Ri that enumerates all the valid joint assignments of all variables in the scope of the constraint. Ri ⊆ Di1 × . . . × Dir Ri xj xk 0 1 1 0
- Soft constraints• A soft constraint Cis is a function Fi that maps every possible joint assignment of all variables in the scope to a real value. Fi : Di1 × . . . × Dir → ℜ Fi xj xk 2 0 0 0 0 1 0 1 0 1 1 1
- Binary Constraint Networks x1• Binary constraint networks are those where: F1,3 ,2 • Each constraint (soft or hard) is deﬁned F1 over two variables.• Every constraint network can be mapped to a binary constraint network x2 x3 4 F1, • requires the addition of variables and constraints F2,4 • may add complexity to the model• They can be represented by a constraint graph x4
- Different objectives, different problems• Constraint Satisfaction Problem (CSP) • Objective: ﬁnd an assignment for all the variables in the network that satisﬁes all constraints.• Constraint Optimization Problem (COP) • Objective: ﬁnd an assignment for all the variables in the network that satisﬁes all constraints and optimizes a global function. • Global function = aggregation (typically sum) of local functions. F(x) = ∑i Fi (xi )
- Distributed Constraint Reasoning A1When operating in adecentralized context: • a set of agents control variables A2 • agents interact to ﬁnd a A4 solution to the constraint network A3
- Distributed Constraint ReasoningTwo types of decentralized problems: • distributed CSP (DCSP) • distributed COP (DCOP) Here, we focus on DCOPs.
- Distributed Constraint Optimization Problem (DCOP)A DCOP consists of a constraint network N = X, D,C and a set ofagents A = {A1 , . . . , Ak } where each agent: • controls a subset of the variables Xi ⊆ X • is only aware of constraints that involve variable it controls • communicates only with its neighbours
- Distributed Constraint Optimization Problem (DCOP) • Agents are assumed to be fully cooperative • Goal: ﬁnd the assignment that optimizes the global function, not their local local utilities. • Solving a COP is NP-Hard and DCOP is as hard as COP.
- MotivationWhy distribute? • Privacy • Robustness • Scalability
- OutlineIntroductionDistributed Constraint ReasoningApplications and Exemplar ProblemsComplete algorithms for DCOPsApproximated Algorithms for DCOPsConclusions
- Real World ApplicationsMany standard benchmark problems in computer science can bemodeled using the DCOP framework: • graph coloringAs can many real world applications: • human-agent organizations (e.g. meeting scheduling) • sensor networks and robotics (e.g. target tracking)
- OutlineIntroductionDistributed Constraint ReasoningApplications and Exemplar Problems Graph coloring Meeting Scheduling Target TrackingComplete algorithms for DCOPsApproximated Algorithms for DCOPsConclusions
- Graph coloring• Popular benchmark• Simple formulation• Complexity controlled with few parameters: • Number of available colors • Number of nodes • Density (#nodes/#constraints)• Many versions of the problem: • CSP, MaxCSP, COP
- Graph coloring - CSP• Nodes can take k colors• Any two adjacent nodes should have different colors • If it happens this is a conﬂict Yes! No!
- Graph coloring - Max-CSP• Minimize the number of conﬂicts 0 -1
- Graph coloring - COP• Different weights to violated constraints• Preferences for different colors 0 -2 -1-3 -2 -1
- Graph coloring - DCOP • Each node: • controlled by one agent • Each agent: • Preferences for different colors • Communicates with its direct neighbours in the graph -1A1 A3 • A1 and A2 exchange -3 -2 preferences and conﬂicts • A3 and A4 do notA2 A4 communicate -1
- OutlineIntroductionDistributed Constraint ReasoningApplications and Exemplar Problems Graph coloring Meeting Scheduling Target TrackingComplete algorithms for DCOPsApproximated Algorithms for DCOPsConclusions
- Meeting SchedulingMotivation: • Privacy • Robustness • Scalability
- Meeting SchedulingIn large organizations many people, possibly working in different departments, are involved in a number of work meetings.
- Meeting SchedulingPeople might have various private preferences on meeting start times Better after 12:00am
- Meeting SchedulingTwo meetings that share a participant cannot overlap Window: 15:00-18:00 Duration: 2h Window: 15:00-17:00 Duration: 1h
- DCOP formalization for the meeting scheduling problem • A set of agents representing participants • A set of variables representing meeting starting times according to a participant. • Hard Constraints: • Starting meeting times across different agents are equal • Meetings for the same agent are non-overlapping. • Soft Constraints: • Represent agent preferences on meeting starting times. Objective: ﬁnd a valid schedule for the meeting while maximizing the sum of individuals’ preferences.
- OutlineIntroductionDistributed Constraint ReasoningApplications and Exemplar Problems Graph coloring Meeting Scheduling Target TrackingComplete algorithms for DCOPsApproximated Algorithms for DCOPsConclusions
- Target TrackingMotivation: • Privacy • Robustness • Scalability
- Target TrackingA set of sensors tracking a set of targets in order to provide an accurate estimate of their positions. T4 T1 T3 T2 Crucial for surveillance and monitoring applications.
- Target TrackingSensors can have different sensing modalities that impact on the accuracy of the estimation of the targets’ positions. MODES T4 MODES MODES T1 T3 MODES T2
- Target TrackingCollaboration among sensors is crucial to improve system performance T4 T1 T3 T2
- DCOP formalization for the target tracking problem• Agents represent sensors• Variables encode the different sensing modalities of each sensor• Constraints • relate to a speciﬁc target • represent how sensor modalities impacts on the tracking performance• Objective: • Maximize coverage of the environment • Provide accurate estimations of potentially dangerous targets
- Complete AlgorithmsUD Always ﬁnd an optimal solutionD Exhibit an exponentially increasing coordination overhead Very limited scalability on general problems.
- Complete Algorithms • Completely decentralised • Search-based. • Synchronous: SyncBB, AND/OR search • Asynchronous: ADOPT, NCBB and AFB. • Dynamic programming. • Partially decentralised • OptAPONext, we focus on completely decentralised algorithms
- Decentralised Complete AlgorithmsSearch-based Dynamic programming • Uses distributed search • Uses distributed inference • Exchange individual values • Exchange constraints • Small messages but • Few messages but . . . exponentially many . . . exponentially largeRepresentative: ADOPT [Modi Representative: DPOP [Petcuet al., 2005] and Faltings, 2005]
- OutlineIntroductionDistributed Constraint ReasoningApplications and Exemplar ProblemsComplete algorithms for DCOPs Search Based: ADOPT Dynamic Programming DPOPApproximated Algorithms for DCOPsConclusions
- ADOPTADOPT (Asynchronous Distributed OPTimization) [Modi et al., 2005]: • Distributed backtrack search using a best-ﬁrst strategy • Best value based on local information: • Lower/upper bound estimates of each possible value of its variable • Backtrack thresholds used to speed up the search of previously explored solutions. • Termination conditions that check if the bound interval is less than a given valid error bound (0 if optimal)
- ADOPT by example 4 variables (4 agents): x1 , x2 , x3 , x4 with D = {0, 1} 4 identical cost functions x1 Fi, j xi xj 2 0 0 F1,3 ,2 F1 0 0 1 0 1 0 x2 x3 1 1 1 4 F1,F2,4 Goal: ﬁnd a variable assignment with minimal costx4 Solution: x1 = 1, x2 = 0, x3 = 0 and x4 = 1 giving total cost 1.
- DFS arrangement• Before executing ADOPT, agents must be arranged in a depth ﬁrst search (DFS) tree.• DFS trees have been frequently used in optimization because they have two interesting properties: • Agents in different branches of the tree do not share any constraints; • Every constraint network admits a DFS tree.
- ADOPT by example A1 (root) x1 → ← Fi, j xi xj ent par par ent , 2 0 0 F1,3 ,2 , chil hil d F1 0 0 1 d→ ←c 0 1 0 x2 x3 1 1 1 A2 A3 4 F1, → arentF2,4 → il d , p ← chx4 DFS arrangement A4
- Cost functionsThe local cost function for an agent Ai (δ (xi )) is the sum of the values of constraints involving only higher neighbors in the DFS.
- ADOPT by example A1 δ (x1 ) = 0δ (x1 , x2 ) = F1,2 (x1 , x2 ) A2 A3 δ (x1 , x3 ) = F1,3 (x1 , x3 ) A4 δ (x1 , x2 , x4 ) = F1,4 (x1 , x4 ) + F2,4 (x2 , x4 )
- Initialization Each agent initially chooses a random value for their variables andinitialize the lower and upper bounds to zero and inﬁnity respectively. x1 = 0, LB = 0,UB = ∞ A1 x2 = 0, LB = 0,UB = ∞ A2 A3 x3 = 0, LB = 0,UB = ∞x4 = 0, LB = 0,UB = ∞ A4
- ADOPT by example Value messages are sent by an agent to all its neighbors that are lower in the DFS tree x1 = 0 A1 ← =0 x 1−− − − −− =→ x1 0 A1 sends three value message to A2 , A3 and x2 = 0 A2 A3 x3 = 0 ←−0 A4 informing them that its −− x1 = ←−0 −− current value is 0. x2 =x4 = 0 A4
- ADOPT by example Current Context: a partial variable assignment maintained by each agent that records the assignment of all higher neighbours in the DFS. A1 • Updated by each VALUE messagec2 : {x1 = 0} A2 A3 • If current context is not c3 : {x1 = 0} compatible with some child context, the latter is re-initialized A4 (also the child bound) c4 : {x1 = 0, x2 = 0}
- ADOPT by example Each agent Ai sends a cost message to its parent A p A1 , c2 ] [0, 0 Each cost message reports: [0, ∞ , c 3] • The minimum lower bound (LB) • The maximum upper bound (UB) A2 A3 • The context (ci ) 4] c[0, 0, [LB,UP, ci ] A4
- Lower bound computationEach agent evaluates for each possible value of its variable: • its local cost function with respect to the current context • adding all the compatible lower bound messages received from children. Analogous computation for upper bounds
- ADOPT by example Consider the lower bound in the cost message sent by A4 : A1 • Recall that A4 local cost function is: δ (x1 , x2 , x4 ) = F1,4 (x1 , x4 ) + F2,4 (x2 , x4 ) • Restricted to the current context c4 = {(x1 = 0, x2 = 0)}: λ (0, 0, x4 ) = F1,4 (0, x4 ) + F2,4 (0, x4 ). A2 A3 • For x4 = 0: λ (0, 0, 0) = F1,4 (0, 0) + F2,4 (0, 0) = 2 + 2 = 4. 4] c • For x4 = 1:[0, 0, λ (0, 0, 1) = F1,4 (0, 1) + F2,4 (0, 1) = 0 + 0 = 0. A4 Then the minimum lower bound across variable values is LB = 0.
- ADOPT by example Each agent asynchronously chooses the value of its variable that minimizes its lower bound. A2 computes for each possible value of its A1 variable its local function restricted to the , c2 ] current context c2 = {(x1 = 0)} [0, 2 (λ (0, x2 ) = F1,2 (0, x2 )) and adding lower bound message from A4 (lb).x2 = 0 → 1 A2 A3 • For x2 = 0: LB(x2 = 0) = λ (0, x2 = 0) + lb(x2 = 0) = 2 + 0 = 2. 1 x2 = • For x2 = 1: LB(x2 = 1) = λ (0, x2 = 1) + 0 = 0 + 0 = 0. A4 A2 changes its value to x2 = 1 with LB = 0.
- Backtrack thresholds The search strategy is based on lower boundsProblem • Values abandoned before proven to be suboptimal • Lower/upper bounds only stored for the current contextSolution • Backtrack thresholds: used to speed up the search of previously explored solutions.
- ADOPT by example x1 = 0 → 1 → 0 A1 A1 changes its value and the context with x1 = 0 is visited again. • Reconstructing from scratch is inefﬁcient A2 A3 • Remembering solutions is expensiveA4
- Backtrack thresholdsSolution: Backtrack thresholds • Lower bound previously determined by children • Polynomial space • Control backtracking to efﬁciently search • Key point: do not change value until LB(currentvalue)> threshold
- A child agent will not change its variable value so long as cost is less than the backtrack threshold given to it by its parent. LB(x1 = 0) = 1 A1 t (x 1 = 1 2 0) = 0) = t (x1 = 1 2 1 1LB(x2 = 0) > 2 ? A2 A3 LB(x3 = 0) > 2 ? A4
- Rebalance incorrect thresholdHow to correctly subdivide threshold among children? • Parent distributes the accumulated bound among children • Arbitrarily/Using some heuristics • Correct subdivision as feedback is received from children • LB < t(CONT EXT ) • t(CONT EXT ) = ∑Ci t(CONT EXT ) + δ
- Backtrack Threshold Computation A1 (2) 2 t 1 =1 = 0) = 1 (x 1 =(1) LB 1 • When A1 receives a new lower bound x 0) = 0 (2) t ( from A2 rebalances thresholds • A1 resends threshold messages to A2 A2 A3 and A3 A4
- ADOPT extensions• BnB-ADOPT [Yeoh et al., 2008] reduces computation time by using depth-ﬁrst search with branch and bound strategy• [Ali et al., 2005] suggest the use of preprocessing techniques for guiding ADOPT search and show that this can result in a consistent increase in performance.
- OutlineIntroductionDistributed Constraint ReasoningApplications and Exemplar ProblemsComplete algorithms for DCOPs Search Based: ADOPT Dynamic Programming DPOPApproximated Algorithms for DCOPsConclusions
- DPOPDPOP (Dynamic Programming Optimization Protocol) [Petcu andFaltings, 2005]: • Based on the dynamic programming paradigm. • Special case of Bucket Tree Elimination Algorithm (BTE) [Dechter, 2003].
- DPOP by example x1 x1 Fi, j xi xj ← P2 , child → F1,3 ,2 F1 → ← PP3 2 0 0 4 il d , PP ,4 0 0 1 F2,3F1 , pseud x2 x3 0 1 0 ud och 1 1 1 ochil d x2 ← pse ,4F2 → ← → => P3, 4 ,P chi ld chi ldx4 DFS arrangement ← → Objective: ﬁnd assignment x4 x3 with maximal value
- DPOP phasesGiven a DFS tree structure, DPOP runs in two phases: • Util propagation: agents exchange util messages up the tree. • Aim: aggregate all info so that root agent can choose optimal value • Value propagation: agents exchange value messages down the tree. • Aim: propagate info so that all agents can make their choice given choices of ancestors
- Sepi : set of agents preceding Ai in the pseudo-tree order that are connected with Ai or with a descendant of Ai . x1 Sep1 = 0 / ← P2 , child → → ← PP3 4 il d , PP , pseud ud och ochil d x2 Sep2 = {x1 } ← pse → ← → P3, 4 ,P chi ld chi ld ← → Sep4 = {x1 , x2 } x4 x3 Sep3 = {x1 , x2 }
- Util messageThe Util message Ui→ j that agent Ai sends to its parent A j can becomputed as: Ui→ j (Sepi ) = max Uk→i ⊗ Fi,p xi Ak ∈Ci A p ∈Pi ∪PPiSize exponential All incoming messages Shared constraints with in Sepi from children parents/pseudoparentsThe ⊗ operator is a join operator that sums up functions with differentbut overlapping scores consistently.
- Join operatorF2,4 x2 x4 max{x4 } (F1,4 ⊗ F2,4 ) F1,4 ⊗ F2,42 0 0 x1 x2 x4 x1 x2 x40 0 1 4 0 0 0 0 0 00 1 0 max(4,0) 0 0 0 1 0 0 11 1 1 Project 2 0 1 0 0 1 0 Add max(2,1) 1 0 1 1 out x4 0 1 1F1,4 x1 x4 2 1 0 0 1 0 02 0 0 max(2,2) 2 1 0 1 1 0 10 0 1 0 1 1 0 1 1 00 1 0 max(0,2) 2 1 1 1 1 1 11 1 1
- Complexity exponential to the largest Sepi . Largest Sepi = induced width of the DFS tree ordering used. A1 (root) U2→1 x1 Sep2 U1→2 ←− −− 0 10 Sep4 1 5 A2 max(U3→2 ⊗U4→2 ⊗ F1,2 ) x2U4→2 x1 x2 U3→2 x1 x2 Sep3 U← − −− 2 4 0 0 4 0 0 → − 4→ 3 − − → U 2 2 0 1 2 0 1 2 1 0 2 1 0 2 1 1 2 1 1 A4 A3 1 max(F1,3 ⊗ F2,3 ) 2 max(F1,4 ⊗ F2,4 ) x3 x4
- Value messageKeeping ﬁxed the value of parent/pseudoparents, ﬁnds the value thatmaximizes the computed cost function in the util phase: ∗ xi = arg max ∑ U j→i (xi , x∗ ) + p ∑ Fi, j (xi , x∗ ) j xi A j ∈Ci A j ∈Pi ∪PPiwhere x∗ = A j ∈Pi ∪PPi {x∗ } is the set of optimal values for Ai ’s parent p jand pseudoparents received from Ai ’s parent.Propagates this value through children down the tree: ∗ ∗ Vi→ j = {xi = xi } ∪ {xs = xs } xs ∈Sepi ∩Sep j
- ∗ x1 = max U1→2 (x1 ) A1 x1 V1→2 −→ − ∗ ∗ ∗ ∗ x2 = max(U3→2 (x1 , x2 ) ⊗U4→2 (x1 , x2 ) ⊗ F1,2 (x1 , x2 )) A2 x2 V 2→→ −4 − − ←2→ − V 3 A4 A3 ∗ ∗ ∗x4 = max(F1,4 (x1 , x4 ) ⊗ F2,4 (x2 , x4 )) ∗ ∗ ∗ x3 = max(F1,3 (x1 , x3 ) ⊗ F2,3 (x2 , x3 )) x4 x3
- DPOP extensions• MB-DPOP [Petcu and Faltings, 2007] trades-off message size against the number of messages.• A-DPOP trades-off message size against solution quality [Petcu and Faltings, 2005(2)].
- Conclusions• Constraint processing • exploit problem structure to solve hard problems efﬁciently• DCOP framework • applies constraint processing to solve decision making problems in Multi-Agent Systems • increasingly being applied within real world problems.
- References I• [Modi et al., 2005] P. J. Modi, W. Shen, M. Tambe, and M.Yokoo. ADOPT: Asynchronous distributed constraint optimization with quality guarantees. Artiﬁcial Intelligence Jour- nal, (161):149-180, 2005.• [Yeoh et al., 2008] W. Yeoh, A. Felner, and S. Koenig. BnB-ADOPT: An asynchronous branch-and-bound DCOP algorithm. In Proceedings of the Seventh International Joint Conference on Autonomous Agents and Multiagent Systems, pages 591Ð598, 2008.• [Ali et al., 2005] S. M. Ali, S. Koenig, and M. Tambe. Preprocessing techniques for accelerating the DCOP algorithm ADOPT. In Proceedings of the Fourth International Joint Conference on Autonomous Agents and Multiagent Systems, pages 1041Ð1048, 2005.• [Petcu and Faltings, 2005] A. Petcu and B. Faltings. DPOP: A scalable method for multiagent constraint opti- mization. In Proceedings of the Nineteenth International Joint Conference on Arti- ﬁcial Intelligence, pages 266-271, 2005.• [Dechter, 2003] R. Dechter. Constraint Processing. Morgan Kaufmann, 2003.
- References II• [Petcu and Faltings, 2005(2)] A. Petcu and B. Faltings. A-DPOP: Approximations in distributed optimization. In Principles and Practice of Constraint Programming, pages 802-806, 2005.• [Petcu and Faltings, 2007] A. Petcu and B. Faltings. MB-DPOP: A new memory-bounded algorithm for distributed optimization. In Proceedings of the Twentieth International Joint Confer- ence on Artiﬁcial Intelligence, pages 1452-1457, 2007.• [S. Fitzpatrick and L. Meetrens, 2003] S. Fitzpatrick and L. Meetrens. Distributed Sensor Networks: A multiagent perspective, chapter Distributed coordination through anarchic optimization, pages 257- 293. Kluwer Academic, 2003.• [R. T. Maheswaran et al., 2004] R. T. Maheswaran, J. P. Pearce, and M. Tambe. Distributed algorithms for DCOP: A graphical game-based approach. In Proceedings of the Seventeenth International Conference on Parallel and Distributed Computing Systems, pages 432-439, 2004.
- Approximate Algorithms: outline• No guarantees – DSA-1, MGM-1 (exchange individual assignments) – Max-Sum (exchange functions)• Off-Line guarantees – K-optimality and extensions• On-Line Guarantees – Bounded max-sum
- Why Approximate Algorithms• Motivations – Often optimality in practical applications is not achievable – Fast good enough solutions are all we can have• Example – Graph coloring – Medium size problem (about 20 nodes, three colors per node) – Number of states to visit for optimal solution in the worst case 3^20 = 3 billions of states• Key problem – Provides guarantees on solution quality
- Exemplar Application: Surveillance• Event Detection – Vehicles passing on a road• Energy Constraints – Sense/Sleep modes – Recharge when sleeping• Coordination – Activity can be detected by single sensor duty cycle – Roads have different time traffic loads Good Schedule• Aim [Rogers et al. 10] Bad Schedule – Focus on road with more traffic load Heavy traffic road small road
- Surveillance demo
- Guarantees on solution quality• Key Concept: bound the optimal solution – Assume a maximization problem – optimal solution, a solution – – percentage of optimality • [0,1] • The higher the better – approximation ratio • >= 1 • The lower the better – is the bound
- Types of Guarantees Instance-specific Accuracy: high alpha Generality: less use of Bounded Max- instance specific knowledge Sum DaCSAAccuracy Instance-generic No guarantees K-optimality MGM-1, T-optimality DSA-1, Max-Sum Region Opt. Generality
- Centralized Local Greedy approaches• Greedy local search – Start from random solution – Do local changes if global solution improves – Local: change the value of a subset of variables, usually one -1 -1 -1 -4 -1 00 -2 -1 -1 -2 0
- Centralized Local Greedy approaches• Problems – Local minima – Standard solutions: RandomWalk, Simulated Annealing -1 -1 -2 -1 -1 -1 -1 -1 -1 -1 -1
- Distributed Local Greedy approaches• Local knowledge• Parallel execution: – A greedy local move might be harmful/useless – Need coordination -1 -1 -1 -4 -1 0 0 -2 -2 0 0 -2 -2 -4
- Distributed Stochastic Algorithm• Greedy local search with activation probability to mitigate issues with parallel executions• DSA-1: change value of one variable at time• Initialize agents with a random assignment and communicate values to neighbors• Each agent: – Generates a random number and execute only if rnd less than activation probability – When executing changes value maximizing local gain – Communicate possible variable change to neighbors
- DSA-1: Execution Examplernd > ¼ ? rnd > ¼ ? rnd > ¼ ? rnd > ¼ ? -1 -1 -1 P = 1/4 -10 -2
- DSA-1: discussion• Extremely “cheap” (computation/communication)• Good performance in various domains – e.g. target tracking [Fitzpatrick Meertens 03, Zhang et al. 03], – Shows an anytime property (not guaranteed) – Benchmarking technique for coordination• Problems – Activation probablity must be tuned [Zhang et al. 03] – No general rule, hard to characterise results across domains
- Maximum Gain Message (MGM-1)• Coordinate to decide who is going to move – Compute and exchange possible gains – Agent with maximum (positive) gain executes• Analysis [Maheswaran et al. 04] – Empirically, similar to DSA – More communication (but still linear) – No Threshold to set – Guaranteed to be monotonic (Anytime behavior)
- MGM-1: Example -1 -10 -2 -1 -1 -1 -1 0 -2G = -2 G=0 G=0 G=2
- Local greedy approaches• Exchange local values for variables – Similar to search based methods (e.g. ADOPT)• Consider only local information when maximizing – Values of neighbors• Anytime behaviors• Could result in very bad solutions
- Max-sumAgents iteratively computes local functions that depend only on the variable they control X1 X2 Choose arg max X4 X3 Shared constraint All incoming messages except x2 All incoming messages
- Factor Graph and GDL• Factor Graph – [Kschischang, Frey, Loeliger 01] – Computational framework to represent factored computation – Bipartite graph, Variable - Factor H ( X1, X 2 , X 3 ) = H ( X1) + H ( X 2 | X1) + H ( X 3 | X1 ) H ( X 2 | X1) x1 x2 H ( X1) x1 x2 x3 H ( X 3 | X1) x3
- Max-Sum on acyclic graphs• Max-sum Optimal on acyclic graphs – Different branches are H ( X 2 | X1) independent H ( X1) x1 x2 – Each agent can build a correct estimation of its contribution to the global problem (z functions)• Message equations very similar to Util messages in DPOP x3 – GDL generalizes DPOP [Vinyals et al. 2010a] H ( X 3 | X1) sum up info from other nodes local maximization step
- (Loopy) Max-sum Performance• Good performance on loopy networks [Farinelli et al. 08] – When it converges very good results • Interesting results when only one cycle [Weiss 00] – We could remove cycle but pay an exponential price (see DPOP) – Java Library for max-sum http://code.google.com/p/jmaxsum/
- Max-Sum for low power devices• Low overhead – Msgs number/size• Asynchronous computation – Agents take decisions whenever new messages arrive• Robust to message loss
- Max-sum on hardware
- Max-Sum for UAVsTask Assignment for UAVs [Delle Fave et al 12] Video Streaming Max-Sum Interest points
- Task utility Task completionPriority Urgency First assigned UAVs reaches task Last assigned UAVs leaves task (consider battery life) 24
- Factor Graph Representation PDA2 UAV2 X2 PDA1 T1 U 2 T2 U1UAV1 U3 T3 X1 PDA3
- UAVs Demo
- Quality guarantees for approx. techniques• Key area of research• Address trade-off between guarantees and computational effort• Particularly important for many real world applications – Critical (e.g. Search and rescue) – Constrained resource (e.g. Embedded devices) – Dynamic settings
- Instance-generic guarantees Instance-specific Bounded Max- Characterise solution quality without Sum running the algorithm DaCSAAccuracy Instance-generic No guarantees K-optimality MGM-1, T-optimality DSA-1, Max-Sum Region Opt. Generality
- K-Optimality framework• Given a characterization of solution gives bound on solution quality [Pearce and Tambe 07]• Characterization of solution: k-optimal• K-optimal solution: – Corresponding value of the objective function can not be improved by changing the assignment of k or less variables.
- K-Optimal solutions 1 1 1 1 1 1 1 1 2-optimal ? Yes 3-optimal ? No 2 2 0 0 1 0 0 2
- Bounds for K-Optimality For any DCOP with non-negative rewards [Pearce and Tambe 07] Number of agents Maximum arity of constraints K-optimal solutionBinary Network (m=2):
- K-Optimality Discussion• Need algorithms for computing k-optimal solutions – DSA-1, MGM-1 k=1; DSA-2, MGM-2 k=2 [Maheswaran et al. 04] – DALO for generic k (and t-optimality) [Kiekintveld et al. 10]• The higher k the more complex the computation (exponential)Percentage of Optimal:• The higher k the better• The higher the number ofagents the worst
- Trade-off between generality and solution quality• K-optimality based on worst case analysis• assuming more knowledge gives much better bounds• Knowledge on structure [Pearce and Tambe 07]
- Trade-off between generality and solution quality• Knowledge on reward [Bowring et al. 08]• Beta: ratio of least minimum reward to the maximum
- Off-Line Guarantees: Region Optimality• k-optimality: use size as a criterion for optimality• t-optimality: use distance to a central agent in the constraint graph• Region Optimality: define regions based on general criteria (e.g. S-size bounded distance) [Vinyals et al 11]• Ack: Meritxell Vinyals 3-size regions 1-distance regions C regions x0 x0 x1 x2 x1 x2 x3 x3 x0 x0 x1 x2 x1 x2 x3 x3x0 x1 x2 x3 x0 x1 x2 x3
- Size-Bounded Distance• Region optimality can explore new regions: s-size bounded distance• One region per agent, largest t- 3-size bounded distance distance group whose size is less than s x0 x0 x1 x2 x1 x2• S-Size-bounded distance x3 x3 – C-DALO extension of DALO for general regions t=1 t=0 – Can provides better bounds and x0 x0 keep under control size and x1 x2 x1 x2 number of regions x3 x3 t=0 t=1
- Max-Sum and Region Optimality• Can use region optimality to provide bounds for Max- sum [Vinyals et al 10b]• Upon convergence Max-Sum is optimal on SLT regions of the graph [Weiss 00]• Single Loops and Trees (SLT): all groups of agents whose vertex induced subgraph contains at most one cycle x0 x0 x0 x1 x2 x1 x2 x1 x2 x3 x3 x0 x0 x3 x1 x2 x1 x2 x3 x3
- Bounds for Max-Sum• Complete: same as 3-size optimality• bipartite• 2D grids
- Variable Disjoint CyclesVery high quality guarantees if smallest cycle is large
- Instance-specific guarantees Instance-specific Bounded Max- Characterise solution quality after/while Sum running the algorithm DaCSAAccuracy Instance-generic No guarantees K-optimality MGM-1, T-optimality DSA-1, Max-Sum Region Opt. Generality
- Bounded Max-SumAim: Remove cycles from Factor Graph avoidingexponential computation/communication (e.g. no junction tree)Key Idea: solve a relaxed problem instance [Rogers et al.11]X1 F2 X3 X1 F2 X3 Build Spanning treeF1 X2 F3 F1 X2 F3 Compute Bound Run Max-Sum X1 X2 X3 Optimal solution on tree
- Factor Graph Annotation• Compute a weight for each edge – maximum possible impact X1 F2 X3 of the variable on the w21 w23 function w11 w22 w33 w12 w32 F1 X2 F3
- Factor Graph Modification• Build a Maximum Spanning Tree – Keep higher weights X1 F2 X3• Cut remaining w21 w23 dependencies – Compute• Modify functions w11 w22 w33• Compute bound w12 w32 F1 X2 F3 W = w22 + w23
- Results: Random Binary Network OptimalBound is significant Approx. – Approx. ratio is Lower Bound typically 1.23 (81 %) Upper Bound Comparison with k-optimal with knowledge on reward structure Much more accurate less general
- Discussion• Discussion with other data-dependent techniques – BnB-ADOPT [Yeoh et al 09] • Fix an error bound and execute until the error bound is met • Worst case computation remains exponential – ADPOP [Petcu and Faltings 05b] • Can fix message size (and thus computation) or error bound and leave the other parameter free• Divide and coordinate [Vinyals et al 10] – Divide problems among agents and negotiate agreement by exchanging utility – Provides anytime quality guarantees
- Summary• Approximation techniques crucial for practical applications: surveillance, rescue, etc.• DSA, MGM, Max-Sum heuristic approaches – Low coordination overhead, acceptable performance – No guarantees (convergence, solution quality)• Instance generic guarantees: – K-optimality framework – Loose bounds for large scale systems• Instance specific guarantees – Bounded max-sum, ADPOP, BnB-ADOPT – Performance depend on specific instance
- References IDOCPs for MRS• [Delle Fave et al 12] A methodology for deploying the max-sum algorithm and a case study on unmanned aerial vehicles. In, IAAI 2012• [Taylor et al. 11] Distributed On-line Multi-Agent Optimization Under Uncertainty: Balancing Exploration and Exploitation, Advances in Complex SystemsMGM• [Maheswaran et al. 04] Distributed Algorithms for DCOP: A Graphical Game-Based Approach, PDCS-2004DSA• [Fitzpatrick and Meertens 03] Distributed Coordination through Anarchic Optimization, Distributed Sensor Networks: a multiagent perspective.• [Zhang et al. 03] A Comparative Study of Distributed Constraint algorithms, Distributed Sensor Networks: a multiagent perspective.Max-Sum• [Stranders at al 09] Decentralised Coordination of Mobile Sensors Using the Max-Sum Algorithm, AAAI 09• [Rogers et al. 10] Self-organising Sensors for Wide Area Surveillance Using the Max-sum Algorithm, LNCS 6090 Self-Organizing Architectures• [Farinelli et al. 08] Decentralised coordination of low-power embedded devices using the max-sum algorithm, AAMAS 08
- References IIInstance-based Approximation• [Yeoh et al. 09] Trading off solution quality for faster computation in DCOP search algorithms, IJCAI 09• [Petcu and Faltings 05b] A-DPOP: Approximations in Distributed Optimization, CP 2005• [Rogers et al. 11] Bounded approximate decentralised coordination via the max-sum algorithm, Artificial Intelligence 2011.Instance-generic Approximation• [Vinyals et al 10b] Worst-case bounds on the quality of max-product fixed-points, NIPS 10• [Vinyals et al 11] Quality guarantees for region optimal algorithms, AAMAS 11• [Pearce and Tambe 07] Quality Guarantees on k-Optimal Solutions for Distributed Constraint Optimization Problems, IJCAI 07• [Bowring et al. 08] On K-Optimal Distributed Constraint Optimization Algorithms: New Bounds and Algorithms, AAMAS 08• [Weiss 00] Correctness of local probability propagation in graphical models with loops, Neural Computation• [Kiekintveld et al. 10] Asynchronous Algorithms for Approximate Distributed Constraint Optimization with Quality Bounds, AAMAS 10

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