In this paper, we study an online sampling problem of the Wiener process. The goal is to minimize the mean squared error (MSE) of the remote estimator under a sampling frequency constraint when the transmission delay distribution is unknown. The sampling problem is reformulated into a renewal reward optimization problem, and we propose an online sampling algorithm that can adaptively learn the optimal sampling policy through stochastic approximation. We show that the cumulative MSE regret grows with rate $\mathcal{O}(\ln k)$, where $k$ is the number of samples. Through Le Cam's two point method, we show that the worst-case cumulative MSE regret of any online sampling algorithm is lower bounded by $\Omega(\ln k)$. Hence, the proposed online sampling algorithm is minimax order-optimal. Finally, we validate the performance of the proposed algorithm via numerical simulations.
翻译：在本文中, 我们研究Wiener过程的在线抽样问题。 目标是在传输延迟分布未知的情况下, 在取样频率限制下, 最大限度地减少遥控测算器的平均正方差( MSE ) 。 取样问题被重塑为更新奖励优化问题, 我们提出一个在线抽样算法, 通过随机近似, 可以适应性地学习最佳取样政策 。 我们显示累积的MSE 遗憾会随着 $\ mathcal{ O} (\ ln k) 的汇率增长而增加, 美元是样本的数量。 我们通过 Le Cam 的两点方法, 显示任何在线取样算法中最坏的累计MSE 遗憾都受$\ Omega ( $) 的制约。 因此, 提议的在线抽样算法是微缩缩缩缩缩缩算法 。 最后, 我们通过数字模拟验证了提议的算法的性算法。
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