Pareto-Optimality, Smoothness, and Stochasticity in Learning-Augmented One-Max-Search

One-max search is a classic problem in online decision-making, in which a trader acts on a sequence of revealed prices and accepts one of them irrevocably to maximise its profit. The problem has been studied both in probabilistic and in worst-case settings, notably through competitive analysis, and more recently in learning-augmented settings in which the trader has access to a prediction on the sequence. However, existing approaches either lack smoothness, or do not achieve optimal worst-case guarantees: they do not attain the best possible trade-off between the consistency and the robustness of the algorithm. We close this gap by presenting the first algorithm that simultaneously achieves both of these important objectives. Furthermore, we show how to leverage the obtained smoothness to provide an analysis of one-max search in stochastic learning-augmented settings which capture randomness in both the observed prices and the prediction.

Consider a trader who owns a single item and observes its price evolving over time. Each day, the trader sees the current price and must decide whether to sell the item or wait, aiming to maximize the final profit. This decision problem, known as One-Max Search, is challenging because future prices are unknown.With recent progress in machine learning and forecasting, it is now possible to obtain predictions about future prices. While these predictions can offer valuable guidance, they are often imperfect, and the trader has no reliable way to assess their accuracy in advance. Prior research has developed algorithms that leverage such predictions, performing well when the forecast is accurate and remaining robust when it is not. However, these algorithms are highly sensitive to small prediction errors, which can lead to significant performance degradation.In this work, we introduce a new algorithm that retains the strengths of earlier approaches while improving the dependence on prediction accuracy. Additionally, our method demonstrates enhanced performance in settings where both prices and predictions exhibit stochastic behavior.