Online Conversion under Horizon Uncertainty: From Competitive Analysis to Learning-Augmented Algorithms

Abstract

Online allocation problems involve making sequential decisions under incomplete information, where inputs are revealed incrementally over time. A prominent subclass of online allocation problems is the one-way online conversion problem (a.k.a. one-way trading), which focuses on selling (or buying) a single type of divisible resource under dynamically changing prices. Decision-making in online conversion requires balancing immediate reward against the potential for better future opportunities. A critical yet less-explored aspect of online conversion is the impact of time horizon uncertainty, which determines the duration over which decisions are made. The horizon may be predetermined, revealed partway, or entirely unknown, introducing layers of complexity that significantly influence conversion strategies. Additionally, practical constraints such as box constraints, which limit the maximum allowable trade per step, further complicate the decision-making process and demand more nuanced algorithmic approaches. Despite progress in addressing online conversion problems, significant research gaps remain. Existing studies often focus on unconstrained settings or assume complete knowledge of the horizon. Few works explore the combined effects of horizon uncertainty and practical constraints like box constraints on algorithm design and performance. Additionally, leveraging horizon predictions to enhance performance has been underexplored in this context. This thesis addresses these gaps by making two main contributions. First, we propose a unified algorithm to address three models of horizon uncertainty—known horizon, notification at a specific step, and unknown horizon—under both constrained and unconstrained settings. Through competitive analysis, the algorithm achieves tight guarantees, demonstrating optimal performance across all scenarios. Second, the thesis extends the unified algorithm to incorporate horizon predictions, introducing a learning-augmented algorithm that bridges the gap between worst-case and average-case performance. By balancing robustness under adversarial conditions with consistency when predictions are accurate, the algorithm exhibits adaptability in uncertain environments. Together, these contributions advance the theoretical foundation of online conversion and provide practical insights for applications where horizon uncertainty and constraints like box constraints play a critical role