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PAC Learnability for Reliable Communication over Discrete Memoryless Channels

Jiakun Liu, Wenyi Zhang, H. Vincent Poor

TL;DR

The paper addresses reliable communication over unknown discrete memoryless channels by framing decoding-metric selection and code-rate design as a PAC-learning problem. It shows that naive plug-in methods fail with finite data, and introduces the alpha-virtual-sample algorithm to produce decoding metrics with provably high LM-rate performance, enabling rates approaching the channel mutual information $I(p,w)$. Building on this, the VSEE scheme combines virtual-sample decoding with entropy-based mutual information estimation to yield learnable code rates $R$ satisfying $I(p,w)-\epsilon \le R\le I_{\mathrm{LM}}(p,w,K)$ with high probability, thereby establishing PAC learnability of DMCs. Empirical evaluations demonstrate practical training sizes sufficing for reliable metric selection and rate estimation, supporting the viability of data-driven decoding in channel uncertainty and guiding future work on input-distribution learning and tighter finite-sample bounds.

Abstract

In practical communication systems, knowledge of channel models is often absent, and consequently, transceivers need be designed based on empirical data. In this work, we study data-driven approaches to reliably choosing decoding metrics and code rates that facilitate reliable communication over unknown discrete memoryless channels (DMCs). Our analysis is inspired by the PAC (probably approximately correct) learning theory and does not rely on any assumptions on the statistical characteristics of DMCs. We show that a naive plug-in algorithm for choosing decoding metrics is likely to fail for finite training sets. We propose an alternative algorithm called the virtual sample algorithm and establish a non-asymptotic lower bound on its performance. The virtual sample algorithm is then used as a building block for constructing a learning algorithm that chooses a decoding metric and a code rate using which a transmitter and a receiver can reliably communicate at a rate arbitrarily close to the channel mutual information. Therefore, we conclude that DMCs are PAC learnable.

PAC Learnability for Reliable Communication over Discrete Memoryless Channels

TL;DR

The paper addresses reliable communication over unknown discrete memoryless channels by framing decoding-metric selection and code-rate design as a PAC-learning problem. It shows that naive plug-in methods fail with finite data, and introduces the alpha-virtual-sample algorithm to produce decoding metrics with provably high LM-rate performance, enabling rates approaching the channel mutual information . Building on this, the VSEE scheme combines virtual-sample decoding with entropy-based mutual information estimation to yield learnable code rates satisfying with high probability, thereby establishing PAC learnability of DMCs. Empirical evaluations demonstrate practical training sizes sufficing for reliable metric selection and rate estimation, supporting the viability of data-driven decoding in channel uncertainty and guiding future work on input-distribution learning and tighter finite-sample bounds.

Abstract

In practical communication systems, knowledge of channel models is often absent, and consequently, transceivers need be designed based on empirical data. In this work, we study data-driven approaches to reliably choosing decoding metrics and code rates that facilitate reliable communication over unknown discrete memoryless channels (DMCs). Our analysis is inspired by the PAC (probably approximately correct) learning theory and does not rely on any assumptions on the statistical characteristics of DMCs. We show that a naive plug-in algorithm for choosing decoding metrics is likely to fail for finite training sets. We propose an alternative algorithm called the virtual sample algorithm and establish a non-asymptotic lower bound on its performance. The virtual sample algorithm is then used as a building block for constructing a learning algorithm that chooses a decoding metric and a code rate using which a transmitter and a receiver can reliably communicate at a rate arbitrarily close to the channel mutual information. Therefore, we conclude that DMCs are PAC learnable.
Paper Structure (20 sections, 8 theorems, 62 equations, 4 figures, 1 algorithm)

This paper contains 20 sections, 8 theorems, 62 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Communication Model
  4. Problem Formulation
  5. Theoretical Results
  6. Empirical Evaluations of Algorithms
  7. Virtual Sample Algorithm
  8. VSEE Algorithm
  9. Proof Sketch of the Theorems
  10. Conclusion
  11. Lemmas Used in Proofs of the Theorems in Section \ref{['s:results']}
  12. Proof of Lemma \ref{['lem:sketch.c0lmr']}
  13. Proof of Lemma \ref{['lem:sketch.p0m']}
  14. Proof of Lemma \ref{['lem:sketch.chlmr']}
  15. Proof of Lemma \ref{['lem:lemmaproofs.eeerror']}
  16. ...and 5 more sections

Key Result

Theorem 1

If $| \mathcal{X} | \ge 2$, $| \mathcal{Y} | \ge 2$, $0 < \epsilon < 1$, and $0 < \delta < 1 / 2$, then for every $n \in \mathbb{Z}_{> 0}$, there exist a PMF $p$ on $\mathcal{X}$ and a transition function $w$ from $\mathcal{X}$ to $\mathcal{Y}$ satisfying

Figures (4)

  • Figure 1: A learning algorithm accepts $n$ i.i.d. input-output pairs of the DMC as input and returns a decoding metric $k$ and a code rate $r$, which are used by an encoder and a decoder.
  • Figure 2: The CDFs of $I_{\mathrm{LM}} ( p , w , K_{\mathrm{PI}} )$ and $I_{\mathrm{LM}} ( p ,w , K_{\mathrm{VS}} )$.
  • Figure 3: The probabilities of \ref{['e:evaluations.vsa.success']} for $\alpha \in \{ 0.5 , 0.5325 , 0.55 , 0.6 , \cdots , 0.95 \}$.
  • Figure 4: The realizations of $( I_{\mathrm{LM}} ( p , w , K ) , R )$ in 1000 independent experiments. The relation \ref{['e:evaluations.vseea.success']} is satisfied if and only if the blue dot representing the realization lies in the region surrounded by the red lines.

Theorems & Definitions (11)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Theorem 2
  • Definition 3
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 1 more