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.

Prophet inequality concerns a basic optimal stopping problem and states that simple threshold stopping policies -- i.e., accepting the first reward larger than a certain threshold -- can achieve tight

In reinforcement learning (RL), agents sequentially interact with changing environments while aiming to maximize the obtained rewards.

A unifying theme in the design of intelligent agents is to efficiently optimize a policy based on what prior knowledge of the problem is available and what actions can be taken to learn more about it.

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