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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. - PowerPoint PPT Presentation
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Distributed Stochastic Optimization via Correlated Scheduling
Michael J. NeelyUniversity of Southern California
http://www-bcf.usc.edu/~mjneely
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2
Fusion Center
Observation ω1(t)
Observation ω2(t)
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Distributed sensor reports
• 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]
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2Fusion Center
ω1(t)
ω2(t)
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Redundant reports do not increase utility.
Distributed sensor reports
• 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]
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2Fusion Center
Maximize: U
Subject to: P1 ≤ c
P2 ≤ c
ω1(t)
ω2(t)
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Main ideas for this example4
• 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
Agreeon plan 0 1 2 3
t4
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1) Coordinate on a plan before time 0.
2) Distributively implement plan after time 0.
Example “plans”
Agreeon plan 0 1 2 3
t4
Example plan: Sensor 1: • t=even Do not report.• t=odd Report if ω1(t)=1.Sensor 2: • t=even Report with prob p if ω2(t)=1 • t=odd: Do not report.
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Common source of randomness
Example: 1 slot = 1 dayEach person looks at Boston Globe every day:• If first letter is a “T” Plan 1• If first letter is an “S” Plan 2• Etc.
Day 1 Day 2
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Specific exampleAssume:• 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 probability θ2
Optimizing θ1, θ2 gives u = 4/9 ≈ 0.44444
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Approach 2: Correlated reportingPure 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.
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Approach 2: Correlated reportingX(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
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Summary of approaches
Independent reporting
Correlated reporting
Centralized reporting
0.47917
0.44444
0.5
Strategy u
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Summary of approaches
Independent reporting
Correlated reporting
Centralized reporting
0.47917
0.44444
0.5
Strategy u
It can be shown that this is optimal over all distributed strategies!
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General distributed optimization
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))
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Pure strategies
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|
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Optimality Theorem
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).
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LP and complexity reduction• 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)
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Discussion of Theorem 1Theorem 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.
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Online Dynamic ApproachWe want an algorithm that: • Operates online• Does not need π(ω) probabilities. • Can adapt when these probabilities
change.
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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• 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]
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Lyapunov optimization approach• 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]
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Virtual queue: If stable, then: Time average power ≤ c.
“arrivals” “service”
Separable problemsIf 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.
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Simulation (non-separable problem)
• 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
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Utility versus V parameter (V=1/ε)U
tility
V (recall V = 1/ε)
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Average power versus timeAv
erag
e po
wer
up
to ti
me
t
Time t
power constraint 1/3V=10
V=50
V=100
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Adaptation to non-ergodic changes25
Adaptation to non-ergodic changes26
Optimal utility for phase 2
Optimal utility for phases 1 and 3
Oscillates about the average constraint c
Conclusions• 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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