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Maximal-Capacity Discrete Memoryless Channel Identification

Maximilian Egger, Rawad Bitar, Antonia Wachter-Zeh, Deniz Gündüz, Nir Weinberger

TL;DR

A gap-elimination algorithm termed BestChanID is proposed, which is oblivious to the capacity-achieving input distributions, and is guaranteed to output the DMC with the largest capacity, with a desired confidence, and two additional algorithms NaiveChanSel and MedianChanEl are presented.

Abstract

The problem of identifying the channel with the highest capacity among several discrete memoryless channels (DMCs) is considered. The problem is cast as a pure-exploration multi-armed bandit problem, which follows the practical use of training sequences to sense the communication channel statistics. A capacity estimator is proposed and tight confidence bounds on the estimator error are derived. Based on this capacity estimator, a gap-elimination algorithm termed BestChanID is proposed, which is oblivious to the capacity-achieving input distribution and is guaranteed to output the DMC with the largest capacity, with a desired confidence. Furthermore, two additional algorithms NaiveChanSel and MedianChanEl, that output with certain confidence a DMC with capacity close to the maximal, are introduced. Each of those algorithms is beneficial in a different regime and can be used as a subroutine in BestChanID. The sample complexity of all algorithms is analyzed as a function of the desired confidence parameter, the number of channels, and the channels' input and output alphabet sizes. The cost of best channel identification is shown to scale quadratically with the alphabet size, and a fundamental lower bound for the required number of channel senses to identify the best channel with a certain confidence is derived.

Maximal-Capacity Discrete Memoryless Channel Identification

TL;DR

A gap-elimination algorithm termed BestChanID is proposed, which is oblivious to the capacity-achieving input distributions, and is guaranteed to output the DMC with the largest capacity, with a desired confidence, and two additional algorithms NaiveChanSel and MedianChanEl are presented.

Abstract

The problem of identifying the channel with the highest capacity among several discrete memoryless channels (DMCs) is considered. The problem is cast as a pure-exploration multi-armed bandit problem, which follows the practical use of training sequences to sense the communication channel statistics. A capacity estimator is proposed and tight confidence bounds on the estimator error are derived. Based on this capacity estimator, a gap-elimination algorithm termed BestChanID is proposed, which is oblivious to the capacity-achieving input distribution and is guaranteed to output the DMC with the largest capacity, with a desired confidence. Furthermore, two additional algorithms NaiveChanSel and MedianChanEl, that output with certain confidence a DMC with capacity close to the maximal, are introduced. Each of those algorithms is beneficial in a different regime and can be used as a subroutine in BestChanID. The sample complexity of all algorithms is analyzed as a function of the desired confidence parameter, the number of channels, and the channels' input and output alphabet sizes. The cost of best channel identification is shown to scale quadratically with the alphabet size, and a fundamental lower bound for the required number of channel senses to identify the best channel with a certain confidence is derived.
Paper Structure (17 sections, 18 theorems, 82 equations, 2 figures, 3 algorithms)

This paper contains 17 sections, 18 theorems, 82 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Outline
  4. Preliminaries and System Model
  5. Capacity Estimator and A Confidence Interval
  6. Best Channel Identification
  7. Approximate Best Channel Identification
  8. An Information-Theoretic Lower Bound
  9. Conclusion
  10. Proof of \ref{['lemma:kl_bound']}
  11. Proof of \ref{['prop: Concentration for capacity']}
  12. Proof of \ref{['thm:req_samples']}
  13. Proof of \ref{['thm:correctness_complexity']}
  14. Proof of \ref{['lemma:tr']}
  15. Proof of \ref{['lemma:trs']}
  16. ...and 2 more sections

Key Result

Lemma 1

For any DMC $V \colon{\cal X}\to{\cal Y}$, it holds that

Figures (2)

  • Figure 1: We simulate BestChanID with two different numbers $k \in \{10, 20\}$ of randomly generated binary DMCs (solid and dashed lines) and minimum suboptimality caps $\Delta_{\min} = \min_{j \in [k] \setminus j^\star} \Delta_{j}\in \{0.08, 0.13\}$ (blue and orange) for different values of $\delta$ between $0.5$ and $1$.
  • Figure 2: Comparison of the proposed ($\varepsilon, \delta$)-PAC algorithms tested on different parameter sets ($\varepsilon, \delta$) with $10$ binary DMCs with random transition matrices $W_{j}, j \in [10]$. Solid lines mean ($\varepsilon=0.1, \delta=0.1$), dashed dotted lines ($\varepsilon=0.1, \delta=0.7$) and dashed lines ($\varepsilon=0.3, \delta=0.1$).

Theorems & Definitions (49)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof : Sketch of Proof
  • Remark 1
  • Lemma 3
  • proof
  • Theorem 1
  • ...and 39 more