Distributed Stochastic Optimization via Correlated Scheduling

1 / 27

# 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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Distributed Stochastic Optimization via Correlated Scheduling' - keaira

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

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
Utility versus V parameter (V=1/ε)

23

Utility

V (recall V = 1/ε)

Average power versus time

24

V=100

V=50

Average power up to time t

V=10

power constraint 1/3

Time t

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.