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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

- 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

- 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

- 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

- 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 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

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

- 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

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?

- 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

- 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

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

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

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

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

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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