We study the prophet inequality, a fundamental problem in online decision-making and optimal stopping, in a practical setting where rewards are observed only through noisy realizations and reward distributions are unknown. At each stage, the decision-maker receives a noisy reward whose true value follows a linear model with an unknown latent parameter, and observes a feature vector drawn from a distribution. To address this challenge, we propose algorithms that integrate learning and decision-making via lower-confidence-bound (LCB) thresholding. In the i.i.d.\ setting, we establish that both an Explore-then-Decide strategy and an $\varepsilon$-Greedy variant achieve the sharp competitive ratio of $1 - 1/e$, under a mild condition on the optimal value. For non-identical distributions, we show that a competitive ratio of $1/2$ can be guaranteed against a relaxed benchmark. Moreover, with limited window access to past rewards, the tight ratio of $1/2$ against the optimal benchmark is achieved.

翻译：我们研究先知不等式这一在线决策与最优停时的基本问题，并将其置于奖励仅通过噪声实现可观测且奖励分布未知的实际场景中。在每个阶段，决策者会接收一个噪声奖励，其真实值服从带有未知潜在线性参数的线性模型，并观测到从分布中抽取的特征向量。为解决这一挑战，我们提出通过下置信界（LCB）阈值法融合学习与决策的算法。在独立同分布设置下，我们证明：在最优值的温和条件下，“先探索后决策”策略及$\varepsilon$-贪婪变体均能达到$1 - 1/e$的最优竞争比。对于非独立同分布情形，我们证明相对于松弛基准可保证$1/2$的竞争比。此外，在仅能有限窗口访问历史奖励的条件下，针对最优基准可实现严格紧的$1/2$竞争比。