This paper revisits the problem of sampling and transmitting status updates through a channel with random delay under a sampling frequency constraint \cite{sun_17_tit}. We use the Age of Information (AoI) to characterize the status information freshness at the receiver. The goal is to design a sampling policy that can minimize the average AoI when the statistics of delay is unknown. We reformulate the problem as the optimization of a renewal-reward process, and propose an online sampling strategy based on the Robbins-Monro algorithm. We prove that the proposed algorithm satisfies the sampling frequency constraint. Moreover, when the transmission delay is bounded and its distribution is absolutely continuous, the average AoI obtained by the proposed algorithm converges to the minimum AoI when the number of samples $K$ goes to infinity with probability 1. We show that the optimality gap decays with rate $\mathcal{O}\left(\ln K/K\right)$, and the proposed algorithm is minimax rate optimal. Simulation results validate the performance of our proposed algorithm.

翻译：本文在取样频率限制 \ cite{sun_17_tit} 下重新审视了通过一个频道随机拖延取样和传送现状更新的问题。 我们使用信息时代来描述接收器的状态信息新鲜度。 目的是设计一个取样政策, 在无法了解延迟统计时可以将平均 AoI 最小化。 我们重新将问题描述为优化更新回报过程, 并提议基于 Robbins- Monro 算法的在线取样战略。 我们证明, 提议的算法满足了取样频率限制 。 此外, 当传输延迟被约束且分布绝对持续时, 提议的算法获得的平均 AoI 与最小的 AoI 一致, 当样本数量以美元到无限值, 概率为1 时, 我们显示最佳性差与 $\ mathcal{O ⁇ left (lnK/K\right) 的汇率发生衰减, 而提议的算法是最优化的。 模拟结果验证了我们提议的算法的性能 。