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Using Feedback in MANETs: a Control Perspective
Todd P. [email protected] of Illinois
DARPA ITMANET
Current Uses of Feedback
Theory•Feedback modeled noiseless•Point-to-point: capacity unchanged •Significantly improved error exponents•Reduction in complexity
•MANETs: Enlargement of capacity region
Current Uses of Feedback
PracticeFeedback is noisy, used primarily for•Robustness to channel uncertainty•Estimation of channel parameters•ARQ-style communication w/ erasures
Current Uses of Feedback
PracticeFeedback is noisy, used primarily for•Robustness to channel uncertainty•Estimation of channel parameters•ARQ-style communication w/ erasures
But: Burnashev-style “forward error correction+ARQ” schemes are extremely fragile w/ noisy feedback (Kim, Lapidoth, Weissman 07)
•Instantiate network feedback control algorithms for MANETs•Develop iterative practical schemes for noisy feedback?•Coding w/ feedback over statistically unknown channels?•Develop fundamental limits of error exponents with feedback w/ fixed block length
Applicability of Feedback in MANETs
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0 0.25 0.50 0.75 1.00
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....011010]1,0[ W
Communication w/ Noiseless Feedback
0 0.25 0.50 0.75 1.00
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Communication w/ Noiseless Feedback
Given an encoder’s Tx strategy, decoding is almost trivial (Baye’s rule)
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Communication w/ Noiseless Feedback
Given an encoder’s Tx strategy, decoding is almost trivial (Baye’s rule)How do we select a (recursive) encoder
strategy for an arbitrary memoryless channel?
A Control Interpretation of the Dynamics of the PosteriorColeman ’09: “A Stochastic Control Approach to ‘Posterior Matching’-style Feedback Communication Schemes”
A Control Interpretation of the Dynamics of the PosteriorColeman ’09: “A Stochastic Control Approach to ‘Posterior Matching’-style Feedback Communication Schemes”
Fk-1
Controller
Z-1
P(Fk|Fk-1, uk)uk Fk
reference signalFw
*
A Control Interpretation of the Dynamics of the PosteriorColeman ’09: “A Stochastic Control Viewpoint on ‘Posterior Matching’-style Feedback Communication Schemes”
Xk
Fk
Fk+1
Fw*
D(F w* ||F k+1)
D(F w* ||F k)
Reward at any stage k is the reduction in
“distance” to target
Stochastic Control: RewardColeman ’09
Maximum Long-Term Average RewardColeman ’09
Maximum Long-Term Average Reward
(1),(2) hold w/ equality if:• a) Y’s all independent• b) Each Xi drawn
according to P*(x)
Coleman ’09
Maximum Long-Term Average Reward
(1),(2) hold w/ equality if:• a) Y’s all independent• b) Each Xi drawn
according to P*(x)
• Horstein ’63 (BSC)• Schalwijk-Kailath ’66 (AWGN)• Shayevitz-Feder ‘07, ‘08 (DMC)
Coleman ’09
The Posterior Matching Scheme: an Optimal Solution
• Next input indep of everything decoder has seen so far, with capacity-achieving marginal distribution
• No forward error correction. Adapt on the fly.
Coleman ’09
Posterior matching scheme
The Posterior Matching Scheme: an Optimal SolutionColeman ’09
• Next input indep of everything decoder has seen so far, with capacity-achieving marginal distribution
• No forward error correction. Adapt on the fly.
Posterior matching scheme
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Implications for Demonstrating Achievable Rates
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1
Coleman ’09
Lyapunov Function
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Posterior matching scheme:
Coleman ’09
Lyapunov Function (cont’d)
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1
Coleman ’09
ControlTheory
InformationTheory
Symbiotic Relationship
Converse Thms Give Upper Bounds on Average Long-Term Rewards for StochasticControl Problem
Coleman ’09: “A Stochastic Control Viewpoint on ‘Posterior Matching’-style Feedback Communication Schemes”
ControlTheory
InformationTheory
Symbiotic Relationship
Converse Thms Give Upper Bounds on Average Long-Term Rewards for StochasticControl Problem
KL Divergence Lyapunov functions guarantee all rates achievable
Coleman ’09: “A Stochastic Control Viewpoint on ‘Posterior Matching’-style Feedback Communication Schemes”
Research Results with This Methodology•Interpret feedback communication encoder design as stochastic control of posterior towards certainty•Converse theorems specify fundamental performance bounds on a stochastic control problem related to controlling posterior.• An optimal policy implies the existence of a Lyapunov function, which is in essence a KL divergence •Lyapunov function directly implies achievability for all R < C Coleman ’09
Research Results with This Methodology
Gorantla and Coleman ‘09: Encoders that achieve El Gamal 78: “Physically degraded broadcast channels w/ feedback“ capacity region in an iterative fashion w/ low complexity
•Interpret feedback communication encoder design as stochastic control of posterior towards certainty•Converse theorems specify fundamental performance bounds on a stochastic control problem related to controlling posterior.• An optimal policy implies the existence of a Lyapunov function, which is in essence a KL divergence •Lyapunov function directly implies achievability for all R < C Coleman ’09
New Important Directions this Approach Enables
ControlTheory
Information
Theory
•Develop iterative low-complexity encoders/decoders for noisy feedback? Partially Observed Markov Decision Process
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New Important Directions this Approach Enables
ControlTheory
Information
Theory
•Develop iterative low-complexity encoders/decoders for noisy feedback? Partially Observed Markov Decision Process•Optimal coding w/ feedback over statistically unknown channels? Reinforcement learning from control literature
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111
New Important Directions this Approach Enables
ControlTheory
Information
Theory
•Develop iterative low-complexity encoders/decoders for noisy feedback? Partially Observed Markov Decision Process•Optimal coding w/ feedback over statistically unknown channels? Reinforcement learning from control literature•Develop fundamental limits of error exponents with feedback w/ fixed block length Lyapunov function enables a fundamental Martingale condition
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111
New Important Directions this Approach Enables
ControlTheory
Information
Theory
•Develop iterative low-complexity encoders/decoders for noisy feedback? Partially Observed Markov Decision Process•Optimal coding w/ feedback over statistically unknown channels? Reinforcement learning from control literature•Develop fundamental limits of error exponents with feedback w/ fixed block length Lyapunov function enables a fundamental Martingale condition•Also: stochastic control approach provides a rubric to check tightness of converses via structure of optimal solution
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