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Distributed Stochastic Optimization via Correlated SchedulingPowerPoint Presentation

Distributed Stochastic Optimization via Correlated Scheduling

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Distributed Stochastic Optimization

via Correlated Scheduling

1

Fusion

Center

Observation ω1(t)

1

Observation ω2(t)

2

Michael J. Neely

University of Southern California

http://www-bcf.usc.edu/~mjneely

Distributed sensor reports

2

ω1(t)

1

Fusion

Center

ω2(t)

2

- ωi(t) = 0/1 if sensor i observes the event on slot t
- Pi(t) = 0/1 if sensor i reports on slot t
- Utility: U(t) = min[P1(t)ω1(t) + (1/2)P2(t)ω2(t),1]

Redundant reports do not increase utility.

Distributed sensor reports

3

ω1(t)

1

Fusion

Center

ω2(t)

2

- ωi(t) = 0/1 if sensor i observes the event on slot t
- Pi(t) = 0/1 if sensor i reports on slot t
- Utility: U(t) = min[P1(t)ω1(t) + (1/2)P2(t)ω2(t),1]

Maximize: U

Subject to: P1 ≤ c

P2 ≤ c

Main ideas for this example

4

- Utility function is non-separable.
- Redundant reports do not bring extra utility.
- A centralized algorithm would never send redundant reports (it wastes power).
- A distributed algorithm faces these challenges:
- Sensor 2 does not know if sensor 1 observed an event.
- Sensor 2 does not know if sensor 1 reported anything.

Assumed structure

5

Agree

on plan

t

0

1

2

4

3

Coordinate on a plan before time 0.

Distributively implement plan after time 0.

Example “plans”

6

Agree

on plan

t

0

1

2

4

3

- Example plan:
- Sensor 1:
- t=even Do not report.
- t=odd Report if ω1(t)=1.
- Sensor 2:
- t=even Report with probp if ω2(t)=1
- t=odd: Do not report.

Common source of randomness

7

Day 1

Day 2

- Example: 1 slot = 1 day
- Each person looks at Boston Globe every day:
- If first letter is a “T” Plan 1
- If first letter is an “S” Plan 2
- Etc.

Specific example

8

- Assume:
- Pr[ω1(t)=1] = ¾, Pr[ω2(t)=1] = ½
- ω1(t), ω2(t)independent
- Power constraint c = 1/3

- Approach 1: Independent reporting
- If ω1(t)=1, sensor 1 reports with probability θ1
- If ω2(t)=1, sensor 2 reports with probabilityθ2
- Optimizing θ1, θ2 gives u = 4/9 ≈ 0.44444

Approach 2: Correlated reporting

9

- Pure strategy 1:
- Sensor 1 reports if and only if ω1(t)=1.
- Sensor 2 does not report.
- Pure strategy 2:
- Sensor 1 does not report.
- Sensor 2 reports if and only if ω2(t)=1.
- Pure strategy 3:
- Sensor 1 reports if and only if ω1(t)=1.
- Sensor 2 reports if and only if ω2(t)=1.

Approach 2: Correlated reporting

10

- X(t) = iid random variable (commonly known):
- Pr[X(t)=1] = θ1
- Pr[X(t)=2] = θ2
- Pr[X(t)=3] = θ3

- On slot t:
- Sensors observe X(t)
- If X(t)=k, sensors use pure strategy k.

Optimizing θ1, θ2, θ3 gives u = 23/48 ≈ 0.47917

Summary of approaches

11

u

Strategy

Independent reporting

Correlated reporting

Centralized reporting

0.44444

0.47917

0.5

Summary of approaches

12

u

Strategy

Independent reporting

Correlated reporting

Centralized reporting

0.44444

0.47917

0.5

It can be shown that this is optimal over all

distributed strategies!

General distributed optimization

13

Maximize: U

Subject to: Pk ≤ c for k in {1, …, K}

ω(t) = (ω1(t), …, ωΝ(t))

π(ω) = Pr[ω(t) = (ω1, …, ωΝ)]

α(t) = (α1(t), …, αΝ(t))

U(t) = u(α(t), ω(t))

Pk(t) = pk(α(t), ω(t))

Pure strategies

14

A pure strategy is a deterministic vector-valued function:

g(ω) = (g1(ω1), g2(ω2), …, gΝ(ωΝ))

Let M = # pure strategies:

M = |A1||Ω1| x |A2||Ω2| x ... x|AN||ΩN|

Optimality Theorem

15

- There exist:
- K+1 pure strategies g(m)(ω)
- Probabilities θ1, θ2, …, θK+1
- such that the following distributed algorithm is optimal:
- X(t) = iid, Pr[X(t)=m] = θm
- Each user observes X(t)
- If X(t)=m use strategy g(m)(ω).

LP and complexity reduction

16

- The probabilities can be found by an LP
- Unfortunately, the LP has M variables
- If (ω1(t), …, ωΝ(t)) are mutually independent and the utility function satisfies a preferred action property, complexity can be reduced
- Example N=2 users, |A1|=|A2|=2
- --Old complexity = 2|Ω1|+|Ω2|
- --New complexity = (|Ω1|+1)(|Ω2|+1)

Discussion of Theorem 1

17

- Theorem 1 solves the problem for distributed scheduling, but:
- Requires an offline LP to be solved before time 0.
- Requires full knowledge of π(ω) probabilities.

Online Dynamic Approach

18

- We want an algorithm that:
- Operates online
- Does not need π(ω) probabilities.
- Can adapt when these probabilities change.

- Such an algorithm must use feedback:
- Assume feedback is a fixed delay D.
- Assume D>1.
- Such feedback cannot improve average utility beyond the distributed optimum.

Lyapunov optimization approach

19

- Define K virtual queues Q1(t), …, QK(t).
- Every slot t, observe queues and choose strategy m in {1, …, M} to maximize a weighted sum of queues.
- Update queues with delayed feedback:
- Qk(t+1) = max[Qk(t) + Pk(t-D) - c, 0]

Lyapunov optimization approach

20

- Define K virtual queues Q1(t), …, QK(t).
- Every slot t, observe queues and choose strategy m in {1, …, M} to maximize a weighted sum of queues.
- Update queues with delayed feedback:
- Qk(t+1) = max[Qk(t) + Pk(t-D) - c, 0]

“service”

“arrivals”

Virtual queue: If stable, then:

Time average power ≤ c.

Separable problems

21

- If the utility and penalty functions are a separable sum of functions of individual variables (αn(t), ωn(t)), then:
- There is no optimality gap between centralized and distributed algorithms
- Problem complexity reduces from exponential to linear.

Simulation (non-separable problem)

22

- 3-user problem
- αn(t) in {0, 1} for n ={1, 2, 3}.
- ωn(t) in {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
- V=1/ε
- Get O(ε) guarantee to optimality
- Convergence time depends on 1/ε

Adaptation to non-ergodic changes

25

Adaptation to non-ergodic changes

26

Optimal utility for phase 2

Optimal utility for phases 1 and 3

Oscillates about the average constraint c

Conclusions

27

- Paper introduces correlated scheduling via common source of randomness.
- Common source of randomness is crucial for optimality in a distributed setting.
- Optimality gap between distributed and centralized problems (gap=0 for separable problems).
- Complexity reduction technique in paper.
- Online implementation via Lyapunov optimization.
- Online algorithm adapts to a changing environment.

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