Feedback Communication Over the BSC with Sparse Feedback times and Causal Encoding

TL;DR

The paper addresses reducing feedback frequency in posterior matching over a BSC with noiseless feedback, seeking to maintain high rates at short blocklengths. It introduces a sparse-feedback scheme built on a non-binary, look-ahead partitioning approach and a Weighted Median Absolute Difference rule to ensure all rate and stability constraints hold while transmitting in packets of size D_l. The authors prove that the same rate bound established for sequential transmission remains achievable under sparse feedback, and they validate the approach with simulations showing substantial feedback sparsity (average D_l around 5–6 bits) without sacrificing rate, albeit with higher computational complexity. This work advances practical posterior-matching implementations by reducing feedback overhead while preserving near-optimal performance.

Abstract

Posterior matching uses variable-length encoding of the message controlled by noiseless feedback of the received symbols to achieve high rates for short average blocklengths. Traditionally, the feedback of a received symbol occurs before the next symbol is transmitted. The transmitter optimizes the next symbol transmission with full knowledge of every past received symbol. To move posterior matching closer to practical communication, this paper seeks to constrain how often feedback can be sent back to the transmitter. We focus on reducing the frequency of the feedback while still maintaining the high rates that posterior matching achieves with feedback after every symbol. As it turns out, the frequency of the feedback can be reduced significantly with no noticeable reduction in rate.

Figures (5)

• Figure 1: System diagram of a BSC with full, noiseless feedback. At sparse times $t=s_1,s_2,\dots, s_\eta$ transmit a block size $D_{l}$
• Figure 2: Feedback sparseness vs. message size $K$ of the "look-ahead" algorithm. The curves show average feedback packet size $\mathsf{E}[D_l]$ vs. $K$ for channels with capacity $C=0.50$ and $C=0.75$. The dashed line $--\Delta$ is the overall $\mathsf{E}[D_l]$, the dotted line $\cdot \cdot \Diamond$ excludes the systematic block $D_1=K$ and the solid line $-\circ$ is the performance for only non-systematic transmissions where $\rho_i(y^t) < 0.5 \, \forall i \in \Omega$, which is the target region of the "look-ahead" algorithm.
• Figure 3: Rate vs message size $K$ for the look-ahead algorithm for two channels with capacities $C=0.50$ and $C=0.75$ shown with the horizontal solid blue lines. The rate performance of the "look-ahead algorithm" is shown with the brown solid lines with dots. The green solid line with dots is for a non-sparse algorithm in antonini2023. The orange dashed curve is the rate lower bound $K/\mathsf{E}[\tau]$ for for systematic transmission from \ref{['eq: stopping time bound']} and the yellow dash dot line is the lower bound from \ref{['eq: stopping time optimized bound']} for uniform input distribution.
• Figure 4: Run-time complexity of the "look-ahead" algorithm vs. $K$, in average time per $1000$ symbols for channels with capacity $C=0.50$ and $C=0.75$.
• Figure 5: Rate performance vs. channel $p$ of the look-ahead algorithm for $K=32$. The solid solid dark blue curve shows the channel capacity. The "look-ahead algorithm" curve is the brown solid line $-\circ$. The green solid line $-\circ$ is for the non-sparse algorithm in antonini2023. The orange line dash is the rate lower bound $K/\mathsf{E}[\tau]$ for systematic transmission using \ref{['eq: stopping time bound']} and the yellow dash line is the lower bound from \ref{['eq: stopping time optimized bound']} for uniform input distribution.