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# Distributed Stochastic Optimization via Correlated Scheduling - PowerPoint PPT Presentation

Distributed Stochastic Optimization via Correlated Scheduling. 1. Fusion Center. Observation ω 1 (t). 1. Observation ω 2 (t). 2. Michael J. Neely University of Southern California http://www-bcf.usc.edu/~mjneely. Distributed sensor reports. 2. ω 1 (t). 1. Fusion Center.

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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

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.

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

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.

5

Agree

on plan

t

0

1

2

4

3

Coordinate on a plan before time 0.

Distributively implement plan after time 0.

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.

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.

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

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.

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

11

u

Strategy

Independent reporting

Correlated reporting

Centralized reporting

0.44444

0.47917

0.5

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!

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))

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|

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).

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)

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.

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.

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.

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

23

Utility

V (recall V = 1/ε)

24

V=100

V=50

Average power up to time t

V=10

power constraint 1/3

Time t

25

26

Optimal utility for phase 2

Optimal utility for phases 1 and 3

Oscillates about the average constraint c

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.