We consider the measurement model $Y = AX,$ where $X$ and, hence, $Y$ are random variables and $A$ is an a priori known tall matrix. At each time instance, a sample of one of $Y$'s coordinates is available, and the goal is to estimate $\mu := \mathbb{E}[X]$ via these samples. However, the challenge is that a small but unknown subset of $Y$'s coordinates are controlled by adversaries with infinite power: they can return any real number each time they are queried for a sample. For such an adversarial setting, we propose the first asynchronous online algorithm that converges to $\mu$ almost surely. We prove this result using a novel differential inclusion based two-timescale analysis. Two key highlights of our proof include: (a) the use of a novel Lyapunov function for showing that $\mu$ is the unique global attractor for our algorithm's limiting dynamics, and (b) the use of martingale and stopping time theory to show that our algorithm's iterates are almost surely bounded.

翻译：我们考虑测量模型 $Y = AX$，其中 $X$ 与 $Y$ 均为随机变量，$A$ 是预先已知的高矩阵。在每个时间点，可获得 $Y$ 的一个坐标样本，目标是通过这些样本估计 $\mu := \mathbb{E}[X]$。然而，挑战在于 $Y$ 的一小部分未知坐标由具有无限能力的对手控制：每当被请求样本时，它们可以返回任意实数。针对此类对抗性设定，我们提出了首个异步在线算法，该算法几乎必然收敛于 $\mu$。我们通过基于微分包含的新型双时间尺度分析证明了这一结果。证明的两个关键亮点包括：(a) 使用新颖的李雅普诺夫函数证明 $\mu$ 是算法极限动态的唯一全局吸引子；(b) 利用鞅与停时理论证明算法迭代几乎必然有界。