Tilted Matching for Feedback Channels B. Nakiboglu, L. Zheng END-OF-PHASE GOAL COMMUNITY CHALLENGE ACHIEVEMENT DESCRIPTION STATUS QUO NEW INSIGHTS Feedback is an efficient way for error correcting, but often used for ACK/NACK and retransmissions Using feedback to guide FEC has only limited examples Performance metric for Dynamic coding is missing Break away from uniform increment, allow coding to be considered as a dynamically changing optimization New performance metric and the resulting coding schemes for dynamic problems MAIN RESULT: Tilted a posteriori matching achieves the best error exponent HOW IT WORKS: Smooth upper bound to error prob. Make sure at each time t, conditioned on any history, the above metric decreases by a multiplicative factor; Match tilted a posteriori distribution to the desired input distribution. AP tilting  By finding the a posteriori matching scheme with the optimal error exponent, we expose the limitation of error exponent optimal FB coding The dynamic aspect of FEC coding, which is crucial in understanding dynamic information exchange requires new formulation Uniform Belief Increment Limits Performance

Forward Error Correction with Feedback Feedback does not increase capacity of DMC, but is often used to improve reliability, especially with variable length codes, where feedback signals are used to initiate retransmissions; What is hidden behind variable length codes? Retransmission costs ignored in average transmission time; Forward transmission does not utilize feedback; Do not try to recover from a partial error; Dynamic forward error protection codes: communicating with a moving target; Feedback for “incrementally tuned” transmissions Noise variance reduction in AWGN channel, Schalkwijk & Kailath Posterior matching, Shayevitz & Feder

Dynamic View of Coding Problems Communication as “steering” the belief at the receiver towards to right decision, Coleman; Decision regions correspond to reward functions at the end of the block, but needs to be smoothed; Encoding function does not depend on the correct message, and solves multiple parallel problems; Randomness comes from the channel; With feedbacks or noisy feedbacks, encoding can depend on the current belief at the receiver, thus have a dynamic target;

Posterior Matching as a Solution

Bounding Error Probability

The Dynamic Coding Problem The a posterior distribution Define At the end of the block, Uniform Progress Assumption: Resulting exponential bound: Pe

Uniform Progress Assumption and Tilted Matching Assumption of uniform progress is natural for channel without feedback, but sub-optimal when feedback is available; The optimal encoding is posterior matching of This achieves the best known error exponent achieves sphere packing bound for high rates

Performance

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