[Paper Review] Nash Region of the Linear Deterministic Interference Channel with Noisy Output Feedback

Víctor Quintero, Samir M. Perlaza+2 more|arXiv (Cornell University)|May 11, 2017

Wireless Communication Security Techniques16 references4 citations

TL;DR

This paper characterizes the $η$-Nash equilibrium ( $η$-NE) region for the two-user linear deterministic interference channel with noisy output feedback, showing that the R-HK-NOF coding scheme achieves all rate pairs in the intersection of the capacity region and an $η$-stable region for any $η > 0$. The key result is that random common randomness and feedback structure jointly enable stability against unilateral deviations, generalizing prior results without feedback and with perfect feedback.

ABSTRACT

In this paper, the $\\eta$-Nash equilibrium ($\\eta$-NE) region of the two-user linear deterministic interference channel (IC) with noisy channel-output feedback is characterized for all $\\eta > 0$. The $\\eta$-NE region, a subset of the capacity region, contains the set of all achievable information rate pairs that are stable in the sense of an $\\eta$-NE. More specifically, given an $\\eta$-NE coding scheme, there does not exist an alternative coding scheme for either transmitter-receiver pair that increases the individual rate by more than $\\eta$ bits per channel use. Existing results such as the $\\eta$-NE region of the linear deterministic IC without feedback and with perfect output feedback are obtained as particular cases of the result presented in this paper.

Motivation & Objective

• To characterize the $η$-Nash equilibrium ( $η$-NE) region in a two-user linear deterministic interference channel (LD-IC) with noisy output feedback.
• To establish conditions under which the R-HK-NOF coding scheme achieves stable rate pairs that are resilient to unilateral deviations by either transmitter-receiver pair.
• To unify and generalize prior results on $η$-NE regions in LD-ICs without feedback and with perfect feedback.
• To demonstrate that random common randomness ($\Omega_i$) plays a critical role in limiting rate improvement from unilateral deviations, enabling stability.

Proposed method

• The authors model the two-user linear deterministic interference channel with noisy output feedback using bit-pipe parameters ($\overrightarrow{n}_{ii}, n_{ji}, \overleftarrow{n}_{ii}$) to represent direct, cross-link, and feedback gains.
• They define the R-HK-NOF (Randomized Han-Kobayashi with Noisy Feedback) coding scheme, which uses common and private messages, along with random common randomness $\Omega_i$ known only to each transmitter-receiver pair.
• The system is analyzed using a rate-splitting approach where each transmitter splits its message into common and private parts, and feedback is modeled as a delayed, noisy version of the output via a shift matrix $\boldsymbol{S}$.
• The stability condition is formalized using the $\eta$-NE concept: no unilateral deviation can improve a user's rate by more than $\eta$ bits per channel use.
• Key inequalities are derived to bound the maximum rate improvement from deviation, relying on feedback parameters and the positive part operator $(\cdot)^+$.
• The proof establishes that the R-HK-NOF achieves all rate pairs in $\mathcal{C} \cap \mathcal{B}_\eta$, where $\mathcal{C}$ is the capacity region and $\mathcal{B}_\eta$ is the $\eta$-stable region.

Experimental results

Research questions

• RQ1 What is the set of all achievable rate pairs that are stable against unilateral deviations in a two-user LD-IC with noisy output feedback?
• RQ2 How does noisy feedback affect the stability of transmission schemes in interference channels, compared to no feedback or perfect feedback?
• RQ3 What role does random common randomness ($\Omega_i$) play in limiting the gain from unilateral deviations in the R-HK-NOF scheme?
• RQ4 Can the R-HK-NOF coding scheme achieve all rate pairs in the intersection of the capacity region and an $\eta$-stable region for any $\eta > 0$?
• RQ5 How do the feedback parameters ($\overleftarrow{n}_{ii}$) and interference levels ($n_{ji}$) jointly constrain the $\eta$-NE region?

Key findings

• The $\eta$-Nash equilibrium region for the LD-IC with noisy output feedback is fully characterized as the intersection of the capacity region $\mathcal{C}$ and a stability region $\mathcal{B}_\eta$, for any $\eta > 0$.
• The R-HK-NOF coding scheme achieves all rate pairs in $\mathcal{C} \cap \mathcal{B}_\eta$, making it an $\eta$-NE for all $\eta > 0$.
• A unilateral deviation by either transmitter-receiver pair can improve its rate by at most $\frac{2}{3}\eta$ bits per channel use, due to the limiting effect of the other pair's random common randomness $\Omega_j$.
• The result generalizes prior work: the $\eta$-NE region without feedback and with perfect feedback are special cases of this framework.
• The feedback delay is assumed to be one channel use, and the feedback signal is modeled as a delayed, degraded version of the output using a shift matrix $\boldsymbol{S}$.
• The stability is achieved not by power control or power control-like strategies, but by the strategic use of random common randomness $\Omega_i$ that constrains the benefit of deviation.