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Age Optimal Sampling for Unreliable Channels under Unknown Channel Statistics

Hongyi He, Haoyue Tang, Jiayu Pan, Jintao Wang, Jian Song, Leandros Tassiulas

TL;DR

This work studies minimizing the Age of Information (AoI) in a status-update system where a sensor transmits through an unreliable forward channel and receives delayed, error-free feedback via a backward channel. By reformulating the AoI objective as a stochastic-approximation problem, the authors develop an online Robbins-Monro-based algorithm to learn the AoI-optimal sampling threshold without knowing the delay statistics, and enhance it with momentum-based variance reduction. They prove almost-sure convergence of the learned threshold to the optimal value and establish sublinear regret growth with lower bounds that imply minimax-order optimality; they also provide a detailed convergence-rate analysis and variance-reduction benefits. Simulations under heavy-tailed delays confirm that the online policy outperforms fixed-threshold and zero-wait baselines, and that momentum improves stability and speed of convergence. Overall, the approach delivers provable, practical online learning for timely updates in uncertain two-way communication settings, with explicit performance guarantees.

Abstract

In this paper, we study a system in which a sensor forwards status updates to a receiver through an error-prone channel, while the receiver sends the transmission results back to the sensor via a reliable channel. Both channels are subject to random delays. To evaluate the timeliness of the status information at the receiver, we use the Age of Information (AoI) metric. The objective is to design a sampling policy that minimizes the expected time-average AoI, even when the channel statistics (e.g., delay distributions) are unknown. We first review the threshold structure of the optimal offline policy under known channel statistics and then reformulate the design of the online algorithm as a stochastic approximation problem. We propose a Robbins-Monro algorithm to solve this problem and demonstrate that the optimal threshold can be approximated almost surely. Moreover, we prove that the cumulative AoI regret of the online algorithm increases with rate $\mathcal{O}(\ln K)$, where $K$ is the number of successful transmissions. In addition, our algorithm is shown to be minimax order optimal, in the sense that for any online learning algorithm, the cumulative AoI regret up to the $K$-th successful transmissions grows with the rate at least $Ω(\ln K)$ in the worst case delay distribution. Finally, we improve the stability of the proposed online learning algorithm through a momentum-based stochastic gradient descent algorithm. Simulation results validate the performance of our proposed algorithm.

Age Optimal Sampling for Unreliable Channels under Unknown Channel Statistics

TL;DR

This work studies minimizing the Age of Information (AoI) in a status-update system where a sensor transmits through an unreliable forward channel and receives delayed, error-free feedback via a backward channel. By reformulating the AoI objective as a stochastic-approximation problem, the authors develop an online Robbins-Monro-based algorithm to learn the AoI-optimal sampling threshold without knowing the delay statistics, and enhance it with momentum-based variance reduction. They prove almost-sure convergence of the learned threshold to the optimal value and establish sublinear regret growth with lower bounds that imply minimax-order optimality; they also provide a detailed convergence-rate analysis and variance-reduction benefits. Simulations under heavy-tailed delays confirm that the online policy outperforms fixed-threshold and zero-wait baselines, and that momentum improves stability and speed of convergence. Overall, the approach delivers provable, practical online learning for timely updates in uncertain two-way communication settings, with explicit performance guarantees.

Abstract

In this paper, we study a system in which a sensor forwards status updates to a receiver through an error-prone channel, while the receiver sends the transmission results back to the sensor via a reliable channel. Both channels are subject to random delays. To evaluate the timeliness of the status information at the receiver, we use the Age of Information (AoI) metric. The objective is to design a sampling policy that minimizes the expected time-average AoI, even when the channel statistics (e.g., delay distributions) are unknown. We first review the threshold structure of the optimal offline policy under known channel statistics and then reformulate the design of the online algorithm as a stochastic approximation problem. We propose a Robbins-Monro algorithm to solve this problem and demonstrate that the optimal threshold can be approximated almost surely. Moreover, we prove that the cumulative AoI regret of the online algorithm increases with rate , where is the number of successful transmissions. In addition, our algorithm is shown to be minimax order optimal, in the sense that for any online learning algorithm, the cumulative AoI regret up to the -th successful transmissions grows with the rate at least in the worst case delay distribution. Finally, we improve the stability of the proposed online learning algorithm through a momentum-based stochastic gradient descent algorithm. Simulation results validate the performance of our proposed algorithm.

Paper Structure

This paper contains 38 sections, 12 theorems, 125 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. System Model
  4. Age of Information
  5. Problem Formulation
  6. Problem Resolution
  7. A Stochastic Approximation Perspective
  8. An Online Algorithm
  9. Theoretical Analysis
  10. Momentum-based Variance Reduce
  11. Simulations
  12. Simulation Settings
  13. Sampling without Frequency Constraint
  14. Sampling with Frequency Constraint
  15. Momentum-Based Variance Reduction
  16. ...and 23 more sections

Key Result

Corollary 1

pan2023optimal A policy $\pi = \{W_{k,j}\}$ is stationary and deterministic if each waiting time is selected by $W_{k,j}=w(D_{k, j}^F, D_{k, j}^B)$, where $w:\mathbb{R}\times\mathbb{R}\mapsto\{\mathbb{R}^+\cup 0\}$ is a deterministic function from the previous forward and backward transmission delay

Figures (7)

  • Figure 1: System Model
  • Figure 2: AoI Evolution
  • Figure 3: The expected time-average AoI evolution under log-normal(1, 1.8) without frequency constraint
  • Figure 4: The expected time-average AoI evolution under log-normal(1,1.8) with frequency constraint $f_{\text{max}} = \frac{1}{5(\overline{D^B}+\overline{D^F})}$
  • Figure 5: Performance under different value of $V$
  • ...and 2 more figures

Theorems & Definitions (17)

  • Remark 1
  • Corollary 1
  • Proposition 1
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Theorem 3
  • Remark 3
  • Claim 1
  • ...and 7 more