1. 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

2. 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

3. 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;

4. Posterior Matching as a Solution

5. Bounding Error Probability

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

7. 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

8. Performance

9. Discussions