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Improved bounds on the interactive capacity via error pattern analysis

Mudit Aggarwal, Manuj Mukherjee

TL;DR

This work advances the study of interactive capacity by deriving a tighter lower bound for the binary erasure channel: C_I(ε) ≥ 0.104·C_Sh(ε). The authors achieve this through a carefully designed simulation protocol with a reduced alphabet and, crucially, a novel error-pattern analysis that bounds the likelihood of harmful erasure patterns via a Markov-reward framework and concentration inequalities. They further extend the bound to all erasure probabilities using a repetition-based channel-reduction argument, culminating in a stronger bound that improves over the prior BKOS21 results by approximately a factor of 1.75. The methods not only tighten the specific result for BEC(ε) but also suggest a broader utility of error-pattern analysis for other stochastic-noise communication problems, with potential extensions to additional channel models.

Abstract

Any interactive protocol between a pair of parties can be reliably simulated in the presence of noise with a multiplicative overhead on the number of rounds (Schulman 1996). The reciprocal of the best (least) overhead is called the interactive capacity of the noisy channel. In this work, we present lower bounds on the interactive capacity of the binary erasure channel. Our lower bound improves the best known bound due to Ben-Yishai et al. 2021 by roughly a factor of 1.75. The improvement is due to a tighter analysis of the correctness of the simulation protocol using error pattern analysis. More precisely, instead of using the well-known technique of bounding the least number of erasures needed to make the simulation fail, we identify and bound the probability of specific erasure patterns causing simulation failure. We remark that error pattern analysis can be useful in solving other problems involving stochastic noise, such as bounding the interactive capacity of different channels.

Improved bounds on the interactive capacity via error pattern analysis

TL;DR

This work advances the study of interactive capacity by deriving a tighter lower bound for the binary erasure channel: C_I(ε) ≥ 0.104·C_Sh(ε). The authors achieve this through a carefully designed simulation protocol with a reduced alphabet and, crucially, a novel error-pattern analysis that bounds the likelihood of harmful erasure patterns via a Markov-reward framework and concentration inequalities. They further extend the bound to all erasure probabilities using a repetition-based channel-reduction argument, culminating in a stronger bound that improves over the prior BKOS21 results by approximately a factor of 1.75. The methods not only tighten the specific result for BEC(ε) but also suggest a broader utility of error-pattern analysis for other stochastic-noise communication problems, with potential extensions to additional channel models.

Abstract

Any interactive protocol between a pair of parties can be reliably simulated in the presence of noise with a multiplicative overhead on the number of rounds (Schulman 1996). The reciprocal of the best (least) overhead is called the interactive capacity of the noisy channel. In this work, we present lower bounds on the interactive capacity of the binary erasure channel. Our lower bound improves the best known bound due to Ben-Yishai et al. 2021 by roughly a factor of 1.75. The improvement is due to a tighter analysis of the correctness of the simulation protocol using error pattern analysis. More precisely, instead of using the well-known technique of bounding the least number of erasures needed to make the simulation fail, we identify and bound the probability of specific erasure patterns causing simulation failure. We remark that error pattern analysis can be useful in solving other problems involving stochastic noise, such as bounding the interactive capacity of different channels.
Paper Structure (12 sections, 17 theorems, 26 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 12 sections, 17 theorems, 26 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Results
  3. Techniques
  4. Related works
  5. Notation and organization
  6. Problem description and results
  7. A simulation protocol for the erasure channel
  8. Proof of Theorem \ref{['th:strong']}
  9. Progress analysis of Algorithm \ref{['alg:sim']}
  10. Error-pattern analysis
  11. Channel reduction and proof of Theorem \ref{['th:main']}
  12. Conclusion

Key Result

Theorem 1

A binary erasure channel with erasure probability $\epsilon$ satisfies where $\text{C}_{\text{I}}(\epsilon)$ and $\text{C}_{\text{Sh}}(\epsilon)$ respectively denote the interactive and Shannon capacities of $\text{BEC}(\epsilon)$.

Figures (1)

  • Figure 1: State transition diagram of the Markov reward-on-edges process $(Z_i,R_i),i\geq 1$. The edges are labelled by the tuple (probability of transition, reward).

Theorems & Definitions (18)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Corollary 4
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 8 more