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Minimizing Queue Length Regret for Arbitrarily Varying Channels

G Krishnakumar, Abhishek Sinha

TL;DR

This work tackles online channel scheduling with $N$ arbitrarily varying channels under an oblivious adversary, aiming to minimize worst-case queue length regret without stability assumptions. By representing the queue via the Lindley recursion and showing the regret bound reduces to a weakly adaptive MAB problem that controls performance over all sub-intervals, the authors design an algorithm built on Online Gradient Descent plus exploration. They prove a high-probability regret bound of $\tilde{O}(\sqrt{N}\,T^{3/4})$ uniformly over sub-intervals, which implies the same bound for the queue length regret. Notably, the results hold even under non-stationary arrivals and non-stable queues, offering robust guarantees for QoS in adversarial and changing wireless environments. A key open question remains closing the gap to the lower bound $\Omega(\sqrt{NT})$ and extending the framework to broader channel and traffic models.

Abstract

We consider an online channel scheduling problem for a single transmitter-receiver pair equipped with $N$ arbitrarily varying wireless channels. The transmission rates of the channels might be non-stationary and could be controlled by an oblivious adversary. At every slot, incoming data arrives at an infinite-capacity data queue located at the transmitter. A scheduler, which is oblivious to the current channel rates, selects one of the $N$ channels for transmission. At the end of the slot, the scheduler only gets to know the transmission rate of the selected channel. The objective is to minimize the queue length regret, defined as the difference between the queue length at some time $T$ achieved by an online policy and the queue length obtained by always transmitting over the single best channel in hindsight. We propose a weakly adaptive Multi-Armed Bandit (MAB) algorithm for minimizing the queue length regret in this setup. Unlike previous works, we do not make any stability assumptions about the queue or the arrival process. Hence, our result holds even when the queueing process is unstable. Our main observation is that the queue length regret can be upper bounded by the regret of a MAB policy that competes against the best channel in hindsight uniformly over all sub-intervals of $[T]$. As a technical contribution of independent interest, we then propose a weakly adaptive adversarial MAB policy which achieves $\tilde{O}(\sqrt{N}T^{\frac{3}{4}})$ regret with high probability, implying the same bound for queue length regret.

Minimizing Queue Length Regret for Arbitrarily Varying Channels

TL;DR

This work tackles online channel scheduling with arbitrarily varying channels under an oblivious adversary, aiming to minimize worst-case queue length regret without stability assumptions. By representing the queue via the Lindley recursion and showing the regret bound reduces to a weakly adaptive MAB problem that controls performance over all sub-intervals, the authors design an algorithm built on Online Gradient Descent plus exploration. They prove a high-probability regret bound of uniformly over sub-intervals, which implies the same bound for the queue length regret. Notably, the results hold even under non-stationary arrivals and non-stable queues, offering robust guarantees for QoS in adversarial and changing wireless environments. A key open question remains closing the gap to the lower bound and extending the framework to broader channel and traffic models.

Abstract

We consider an online channel scheduling problem for a single transmitter-receiver pair equipped with arbitrarily varying wireless channels. The transmission rates of the channels might be non-stationary and could be controlled by an oblivious adversary. At every slot, incoming data arrives at an infinite-capacity data queue located at the transmitter. A scheduler, which is oblivious to the current channel rates, selects one of the channels for transmission. At the end of the slot, the scheduler only gets to know the transmission rate of the selected channel. The objective is to minimize the queue length regret, defined as the difference between the queue length at some time achieved by an online policy and the queue length obtained by always transmitting over the single best channel in hindsight. We propose a weakly adaptive Multi-Armed Bandit (MAB) algorithm for minimizing the queue length regret in this setup. Unlike previous works, we do not make any stability assumptions about the queue or the arrival process. Hence, our result holds even when the queueing process is unstable. Our main observation is that the queue length regret can be upper bounded by the regret of a MAB policy that competes against the best channel in hindsight uniformly over all sub-intervals of . As a technical contribution of independent interest, we then propose a weakly adaptive adversarial MAB policy which achieves regret with high probability, implying the same bound for queue length regret.
Paper Structure (7 sections, 3 theorems, 28 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 7 sections, 3 theorems, 28 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. An Online Scheduling Algorithm
  4. A Weakly Adaptive Algorithm for Adversarial Bandits
  5. Numerical Simulations
  6. Conclusion
  7. Appendix

Key Result

Theorem 1

The Online Gradient Descent (OGD) policy for the expert problem in the full information setting, described in Algorithm ogd.f is weakly adaptive and achieves the following bound for every sub-interval $I \subseteq [T]$:

Figures (2)

  • Figure 1: Schematic of the scheduling problem with $N$ arbitrary varying channels.
  • Figure 2: Comparison of queue length regret of Algorithm \ref{['mab']} with krishnasamy2016regret, stahlbuhk2021learning in a non-stationary environment where $T$ is divided into $m=7$ blocks, highlighted with alternating backgrounds.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Lemma 1