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Channel Coding with Mean and Variance Cost Constraints

Adeel Mahmood, Aaron B. Wagner

TL;DR

The paper introduces a new mean-variance cost constraint $(\Gamma,V)$ for channel coding over discrete memoryless channels, proving a strong converse and a finite, optimal second-order coding rate. It characterizes the second-order performance through a Gaussian-approximation framework using the $\mathcal{K}(r,V)$ function and a three-point minimizer, providing matching non-feedback converse and achievability results. Importantly, it shows that feedback via a novel timid/bold scheme strictly improves the second-order rate for any $V>0$, extending the applicability of such strategies beyond previous settings. The results deepen our understanding of how power constraints with both average and variability considerations influence finite-blocklength performance and demonstrate practical gains from feedback in cost-constrained channel coding.

Abstract

We consider channel coding for discrete memoryless channels (DMCs) with a novel cost constraint that constrains both the mean and the variance of the cost of the codewords. We show that the maximum (asymptotically) achievable rate under the new cost formulation is equal to the capacity-cost function; in particular, the strong converse holds. We further characterize the optimal second-order coding rate of these cost-constrained codes; in particular, the optimal second-order coding rate is finite. We then show that the second-order coding performance is strictly improved with feedback using a new variation of timid/bold coding, significantly broadening the applicability of timid/bold coding schemes from unconstrained compound-dispersion channels to all cost-constrained channels. Equivalent results on the minimum average probability of error are also given.

Channel Coding with Mean and Variance Cost Constraints

TL;DR

The paper introduces a new mean-variance cost constraint for channel coding over discrete memoryless channels, proving a strong converse and a finite, optimal second-order coding rate. It characterizes the second-order performance through a Gaussian-approximation framework using the function and a three-point minimizer, providing matching non-feedback converse and achievability results. Importantly, it shows that feedback via a novel timid/bold scheme strictly improves the second-order rate for any , extending the applicability of such strategies beyond previous settings. The results deepen our understanding of how power constraints with both average and variability considerations influence finite-blocklength performance and demonstrate practical gains from feedback in cost-constrained channel coding.

Abstract

We consider channel coding for discrete memoryless channels (DMCs) with a novel cost constraint that constrains both the mean and the variance of the cost of the codewords. We show that the maximum (asymptotically) achievable rate under the new cost formulation is equal to the capacity-cost function; in particular, the strong converse holds. We further characterize the optimal second-order coding rate of these cost-constrained codes; in particular, the optimal second-order coding rate is finite. We then show that the second-order coding performance is strictly improved with feedback using a new variation of timid/bold coding, significantly broadening the applicability of timid/bold coding schemes from unconstrained compound-dispersion channels to all cost-constrained channels. Equivalent results on the minimum average probability of error are also given.
Paper Structure (15 sections, 18 theorems, 250 equations)

This paper contains 15 sections, 18 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Non-Feedback Converse
  5. Non-Feedback Achievability
  6. Feedback Improves the SOCR
  7. Supplementary Lemmas
  8. Proof of non-feedback converse
  9. Proof of non-feedback Achievability
  10. Proof of Feedback Achievability
  11. Statement and Proof of Lemma \ref{['lemcon3t']}
  12. Proof of Lemma \ref{['Kfunctionproperties']}
  13. Proof of Lemma \ref{['fqq']}
  14. Proof of Lemma \ref{['mostgenconv']}
  15. Proof of Lemma \ref{['combinedlemmas']}

Key Result

Proposition 1

Let $P^*$ be a solution to $(main_form)$ and $Q^* = P^*W$. For $\Gamma \in (\Gamma_0, \Gamma^*)$, we have $c(P^*) = \Gamma$ and for every $a \in \mathcal{A}$, where $C'(\Gamma) > 0$.

Theorems & Definitions (31)

  • Proposition 1
  • Proposition 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 1: 9099482
  • Remark 1
  • Example 1
  • ...and 21 more