⚡ electrical engineering

Optimal Feedback Communication with Information Maximization and Distortion Minimization

This paper establishes conditions for achieving maximal mutual information in feedback communication and demonstrates that for symmetric discrete channels, the posterior matching scheme is the optimal strategy that simultaneously maximizes information transfer and minimizes estimation distortion.

Original authors: Aolin Xu

Published 2026-06-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Aolin Xu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to send a secret message (a real number, like a temperature reading) to a friend using a walkie-talkie that has a glitchy, noisy connection. You have a special advantage: after you speak, your friend immediately tells you what they heard, and you can use that information to decide what to say next. This is called feedback communication.

The paper by Aolin Xu tackles a tricky puzzle: How do you send this message in a way that does two things at once?

Maximize Information: Make sure your friend learns as much as possible about the secret number by the end of the conversation.
Minimize Distortion: Make sure that after every single sentence you say, your friend's best guess of the number is as accurate as possible right then and there.

Here is the breakdown of the paper's findings using simple analogies.

The Problem: The "Perfect Guess" Dilemma

Usually, in communication theory, we just care about getting the message right at the very end. But in real-time systems (like a robot controlling a drone), you need a good guess now, not just at the end.

The author asks: Can we design a speaking strategy that guarantees we get the maximum possible information and keeps the "guessing error" as low as possible at every single step?

The Solution: The "Posterior Matching" Strategy

The paper proves that for certain types of noisy channels (specifically symmetric ones, like a channel where errors happen randomly and equally), there is a "Golden Rule" for speaking. This rule is called Posterior Matching.

The Analogy: The Shrink-Wrapped Map
Imagine your secret number is a point hidden somewhere on a long, continuous map (from 0 to 1).

The Goal: You want to tell your friend which "district" of the map the point is in.
The Strategy:
1. Your friend has a current "belief" about where the point is (a probability map).
2. You look at this map and divide it into $k$ equal-sized districts (like slicing a pie into equal pieces).
3. You tell your friend which district the point is in.
4. Your friend updates their map to only look inside that specific district.
5. You repeat this process, constantly shrinking the search area.

The paper shows that this specific way of dividing the map (matching the current belief to the channel's capacity) is the only way to achieve both goals simultaneously for these specific channels.

Key Findings in Plain English

1. The "Sufficiency" of the Golden Rule
The paper first establishes that if you want to maximize the total information sent, you don't strictly need to use this "Posterior Matching" strategy. There are other ways to get the maximum total information.

2. The "Necessity" for Real-Time Accuracy
However, the paper's big discovery is that if you also want to minimize the error at every single step (not just at the end), then the "Posterior Matching" strategy becomes essential.

The Metaphor: Think of it like tuning a radio. You can turn the dial to get a clear signal at the end of the song (maximizing total info). But if you want the music to be clear throughout the whole song, you have to tune it in a very specific, continuous way. The paper proves that for symmetric channels, this specific tuning (Posterior Matching) is the only way to keep the music clear at every moment.

3. The "Regularization" Trick
The author introduces a clever mathematical trick. Usually, trying to minimize error at every step is a messy, impossible math problem. But by adding a "rule" that says "you must also maximize total information," the problem suddenly becomes solvable.

Analogy: It's like trying to find the shortest path through a maze. If you just look for the shortest path, it's a nightmare. But if you add a rule that "you must also visit every corner of the maze," the path actually becomes a straight, predictable line. The "information maximization" acts as a guide rail that makes the "error minimization" easy to solve.

Who Does This Apply To?

The paper specifically solves this for channels that are "symmetric" (where errors are random and fair), such as:

k-ary Symmetric Channels: Like a game where you guess a number, and sometimes the channel swaps it with another number randomly.
k-ary Erasure Channels: Like a game where sometimes the message gets lost entirely, but when it arrives, it's perfect.

Summary

The paper proves that for specific types of noisy communication lines, the famous Posterior Matching scheme isn't just a good idea; it is the optimal and essentially necessary method if you want to:

Send as much data as possible.
Keep the receiver's guess accurate at every single moment, not just at the end.

It achieves this by using the requirement of "maximizing total data" as a mathematical tool to solve the much harder problem of "minimizing error at every step."

Technical Summary: Optimal Feedback Communication with Information Maximization and Distortion Minimization

Problem Statement
This paper addresses the problem of optimally transmitting a real-valued source $W$ through a channel with feedback over $n$ uses. The objective is twofold:

Information Maximization: Maximize the mutual information $I(W; Y^n)$ between the source and the sequence of all channel outputs $Y^n$ .
Distortion Minimization: At each step $t$ , minimize the Minimum Mean Squared Error (MMSE) of estimating the source $W$ from the channel outputs observed so far ( $Y^t$ ), subject to the constraint that the global mutual information remains maximized.

Formally, the authors seek a sequence of encoding functions $\phi_t: W \times Y^{t-1} \to X$ that maximizes $I(W; Y^n)$ while greedily minimizing $\text{mmse}(W|Y^t)$ at every step. The paper notes that without the information maximization constraint, the distortion minimization problem is generally intractable, even for noiseless channels.

Methodology and Theoretical Framework
The authors approach the problem by first establishing the conditions required for information maximization and then analyzing how these conditions constrain the distortion minimization problem.

Conditions for Information Maximization:
The paper derives a set of conditions sufficient for an encoder to achieve the maximal mutual information $nC$ (where $C$ is channel capacity):
- The encoding function $\phi_t$ must be deterministic.
- The marginal distribution of the channel input $X_t$ must be the capacity-achieving distribution $P^*_X$ .
- The input $X_t$ must be statistically independent of the past outputs $Y^{t-1}$ .
Furthermore, the authors prove that if the channel is input-identifiable (a condition where different input distributions map to linearly independent output distributions, satisfied by common models like BSC, BSC, and AWGN), these conditions are also necessary.
Optimization of Distortion:
To minimize the MMSE, the authors reformulate the problem as maximizing $E[E[W|Y^t]^2]$ . Under the assumption of an input-identifiable channel, they show that this expectation can be decomposed into a quadratic form involving a vector $b_{y^{t-1}}$ (representing conditional expectations of the source given inputs) and a channel-dependent matrix $K$ .

The optimization strategy relies on majorization theory. The authors demonstrate that for specific channel structures, the quadratic form is Schur-convex. Consequently, the optimal solution is achieved when the vector $b_{y^{t-1}}$ is "majorized" by other possible vectors. They prove that this majorization is achieved when the encoding function partitions the source space into contiguous intervals based on the conditional cumulative distribution function (CDF).

Key Results
The paper derives a specific solution for the joint information maximization and distortion minimization problem under the following conditions:

The channel is a Discrete Memoryless Channel (DMC).
The channel is input-identifiable.
The capacity-achieving input distribution $P^*_X$ is uniform.
The channel-induced matrix $K$ has equal diagonal and equal off-diagonal elements.

These conditions are satisfied by $k$ -ary symmetric channels and $k$ -ary erasure channels.

Under these conditions, the optimal encoding scheme is shown to be the Posterior Matching Scheme (specifically, $X_t = F^{*-1}_X(F_{W|Y^{t-1}}(W|Y^{t-1}))$ ). The paper establishes that:

While posterior matching is sufficient for information maximization alone, it is essentially necessary for achieving both information maximization and distortion minimization simultaneously.
Any other encoder solving this problem is merely a relabeling of the posterior matching output.

Significance and Claims
The paper claims to provide a new perspective on feedback communication by treating information maximization as a form of regularization for distortion minimization. This regularization renders the distortion minimization problem tractable, allowing for the derivation of an optimal solution that would otherwise be intractable.

The authors position this work as a bridge between information-theoretic feedback communication and control-oriented communication. They suggest that their findings justify the use of posterior matching in real-time applications where immediate state estimation is critical, such as networked control systems and human-computer interfaces, by showing that the scheme is not only capacity-achieving but also optimal for minimizing estimation error at every step.