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Computer Science & Engineering, University of Nevada, Reno. CS483/683 Multi-Agent Systems. Lectures 5-6: From Satisfaction to Optimization ADOPT: Asynchronous Distributed Optimization. 2-4 February 2010 Instructor: Kostas Bekris. Distributed Constrained Optimization. DCOP:

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cs483 683 multi agent systems
Computer Science & Engineering, University of Nevada, RenoCS483/683 Multi-Agent Systems
  • Lectures 5-6:
  • From Satisfaction to Optimization
  • ADOPT: Asynchronous Distributed Optimization

2-4 February 2010

Instructor: Kostas Bekris

distributed constrained optimization
Distributed Constrained Optimization
  • DCOP:
    • A set of n variables V={X1, ..., Xn}
    • Each variable has a discrete domain: D1, ..., Dn
    • Each variable is assigned to an agent
      • Only the agent who is assigned a variable has control over its value
      • And knowledge of its domain
    • Assign values to variable so as to optimize a global objective function
      • The optimization of the function satisfies a set of constraints
  • Requirements:
    • Distributed computation using only local communication
    • Fast, asynchronous computation - agents should work in parallel
    • Quality guarantees are needed
      • Provably optimal solutions whenever possible
      • Trade-off between computation and solution quality
        • Bounded-error approximation: Guarantee solution within a distance from optimal, less time than the optimal
adopt asynchronous distributed optimization
ADOPT: Asynchronous Distributed Optimization
  • The objective function is a summation over a set of cost functions
  • F(A)=∑Xi,Xj∈V fij(di,dj)
  • Xi←di, Xj←dj in A
  • We want to find A* that minimizes F(A)
  • e.g. F( { (X1,0), (X2,0), (X3,0), (X4,0) } ) = 4
  • F( { (X1,1), (X2,1), (X3,1), (X4,1) } ) = 0

X1

X2

X3

X4

Constraint Graph

assumptions
Assumptions
  • 1. Summation operation over cost function
    • Associative
    • Commutative
    • Monotonic
      • Cost of a solution can only increase as more costs are aggregated (i.e., we cannot have negative cost)
  • 2. Constraints are at most binary
    • There are ways to extend to constraints that involve a larger number of variables
  • 3. Each agent is assigned a single variable
    • There is a way to extend to the case that an agent must handle multiple variables
key ideas in adopt
Key Ideas in ADOPT
  • 1. Opportunistic best-first search
    • Agents are prioritized in a tree structure
      • an agent has a single parent and multiple children
    • Each agent keeps on choosing the best value based on the current available information
      • i.e., chooses the variable which implies the smallest lower bound
      • lower bounds do not need global information to be estimated
    • Each agent maintains a lower and an upper bound for the cost of its subtrees
      • and informs its parent about its own bounds
    • Strategy allows agents to abandon partial solutions which have not been proven to be suboptimal
      • they may have to reconsider the same assignments into the future

X1

X2

X3

X4

Communication Graph

key ideas in adopt1
Key Ideas in ADOPT
  • 2. Backtrack Threshold
    • When an agent knows from previous search experiences that lb is a lower bound for its subtree
      • inform the subtree agents not to bother searching for a solution whose cost is lower than lb
    • In the general case, remembering these lower bounds for past assignments requires exponential space
      • Approach remembers only one value and then cost is subdivided to children arbitrarily and adapted on the fly as new computations are executed
  • 3. Built-in Termination Detection
    • Keeping track of bounds (lower and upper bound for the cost function) on each agent
      • allows to keep track of the progress towards the optimum solution
      • and automatically terminates when necessary
messages and data structures
Messages and Data Structures

X1

  • VALUE message (like ok?)
  • Send selected value to children along the
  • constraint graph
  • COST message (like NoGood)
  • Send to parents along the communication graph
  • THRESHOLD message
  • Send to children along the communication graph
  • Each agent maintains the “context”
  • (like the “agent_view”)
    • A recorf of higher priority neighbors’ current variable assignment
  • Two contexts are compatible if they do not disagree on any variable assignment

X2

X3

X4

Constraint Graph

X1

X2

X3

X4

Communication Graph

cost message
COST message
  • Xk transmits COST message to Xi
  • Message contains
    • context of Xk
    • lb of Xk
    • up of Xk
  • When Xi receives the message it stores
    • lb(d,Xk)
    • ub(d,Xk)
  • where d is the assignment of Xi in Xk’s context
  • If context of Xi is incompatible with the context of Xk:
    • lb(d,Xk) = 0
    • ub(d,Xk) = ∞

Xi

Xk

costs and bounds
Costs and Bounds

X1

  • Local cost: δ(di) = ∑(Xj,dj) fij(di,dj)
  • di : assignment of agent Xi
  • Xj : higher priority neighbors than Xi
  • Lower bound for value d:
  • ∀ d ∈ Di:
  • LB(d) = δ(d) + ∑ Xk ∈ children lb(d,Xk)
  • similarly for the upper bound for value d
  • Lower bound:
  • LB = min d ∈ Di LB(d)
  • similarly for the upper bound
  • For leaves: LB(d) = UB(d) = δ(d)
  • If not a leaf but has not get received a COST message: LB = δ(d) and UB = ∞

X2

X3

X4

Constraint Graph

X1

X2

X3

X4

Communication Graph

when does xi change value
When does Xi change value?
  • Whenever LB(di) exceeds the backtrack threshold value, Xi changes its variable value to one with smaller lower bound
  • The threshold is updated with the following three ways:
    • Its value can increase whenever Xi determines that LB is greater than the current threshold
      • guarantees that there is always a variable with a lower bound than the threshold
    • Its value can decrease whenever Xi determines that UB is lower than the current threshold
  • Invariant: LB ≤ threshold ≤ UB
    • Its value is also updated whenever a THRESHOLD message is received from a parent
      • a parent subdivides its own threshold value among its children
      • t(d,Xk): the threshold on cost allocated by parent Xi to child Xk
      • then the value of t(d,Xk) respects the following invariants:
        • threshold = δ(di) + ∑ Xk ∈ children t(di,Xk)
        • ∀ d ∈ Di, ∀ Xk ∈ children: lb(d,Xk) ≤ t(d,Xk) ≤ ub(d,Xk)
example
Example
  • All agents begin concurrently choosing 0.
  • Each agents send a VALUE message to lower priority agents along the constraint graph
  • We will follow one specific execution path - there are many possible

X1

X1

X2

X2

X3

X4

X3

X4

Constraint Graph

Communication Graph

example1
Example

X1=0

  • X2 receives X1’s VALUE message
  • and records this value to its context
  • X2’s context: {X1=0}
  • Then it computes bounds:
    • LB(0) = δ(0) + lb(0,X3) + lb(0,X4) = 1
    • LB(1) = δ(1) + lb(1,X3) + lb(1,X4) = 2
    • LB(0) < LB(1) ⇒ LB = LB(0) = 1
  • Similarly: UB = ∞
  • threshold was set to LB(0), equal to 1 so the invariant holds
  • Transmits a COST message to X1:
  • COST( {X1=0}, 1, ∞ )

X2=0

X4=0

X3=0

example2
Example

X1=0

  • X3 receives X1’s and X2’s VALUE messages
  • and records these values to its context
  • X3’s context: {X1=0, X2=0}
  • Then it computes bounds:
    • LB(0) = δ(0) = 1 + 1 = 2
    • LB(1) = δ(1) = 2 + 2 = 4
    • LB(0) < LB(1) ⇒ LB = LB(0) = 2
  • Similarly: UB = 2
  • threshold was LB(0) = 2 so the invariant holds
  • Transmits a COST message to X2:
  • COST( {X1=0, X2=0}, 2, 2 )
  • Similarly with X4... but no reference to X1

X2=0

X4=0

X3=0

example3
Example

X1=1

  • X1 receives X2’s COST message
  • COST( {X1=0}, 1, ∞ )
    • test if compatible with its current context
    • store: lb(0,X2) = 1 and ub(0,X2) = ∞
  • Then it computes bounds:
    • LB(0) = δ(0) + lb(0,X2) = 0 + 1 = 1
    • LB(1) = δ(1) + lb(1,X2) = 0 + 0 = 0
    • LB(1) < LB(0) ⇒ LB = LB(1) = 0
  • Similarly: UB = ∞
  • threshold was 0, but LB(0) = 1:
    • violation of the invariant, change assignment
  • Send VALUE messages to children

X2=0

X4=0

X3=0

example4
Example

X1=1

  • Assume COST messages from X3 and X4 are delayed... instead VALUE message from X1 arrives first at X2
    • Current context at X2: {X1=1}
  • When X2 receives the COST messages from X3 its context will be incompatible with X2’s
    • the bounds in the message will not be stored
  • The message from X4 is not incompatible:
    • store lb(0,x4) = 1 and up(0,x4) = 1
  • Then it computes bounds:
    • LB(0) = δ(0) + lb(0,X3) + lb(0,X4) = 2+ 0 +1 = 3
    • LB(1) = δ(1) + lb(1,X3) + lb(1,X4) = 0 +0 +0 = 0
    • LB(1) < LB(0) ⇒ LB = LB(1) = 1
  • Similarly: UB = ∞

X2=1

X4=0

X3=0

example5
Example

X1=1

  • X2 will inform X3 and X4 about the changes.
  • Similar changes will take place on X3 and X4:
    • 1 will be selected as the value
  • COST messages will be transmitted:
    • from X4 to X2: ( {X2=1}, 0, 0 )
    • from X3 to X1 and X2: ( {X1=1,X2=1}, 0, 0 )
    • from X2 to X1: ( {X1=1}, 0, 0)
      • this is after receiving the COST messages from X3 and X4
  • Upon receipt of the COST message of X2 at X1:
    • LB = UB = threshold = 0
  • X1 sends TERMINATE messages to other agents.

X2=1

X4=1

X3=1

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