Prophet Inequalities with Linear Correlations and Augmentations (2001.10600v2)

Abstract: In a classical online decision problem, a decision-maker who is trying to maximize her value inspects a sequence of arriving items to learn their values (drawn from known distributions), and decides when to stop the process by taking the current item. The goal is to prove a "prophet inequality": that she can do approximately as well as a prophet with foreknowledge of all the values. In this work, we investigate this problem when the values are allowed to be correlated. Since non-trivial guarantees are impossible for arbitrary correlations, we consider a natural "linear" correlation structure introduced by Bateni et al. [ESA 2015] as a generalization of the common-base value model of Chawla et al. [GEB 2015]. A key challenge is that threshold-based algorithms, which are commonly used for prophet inequalities, no longer guarantee good performance for linear correlations. We relate this roadblock to another "augmentations" challenge that might be of independent interest: many existing prophet inequality algorithms are not robust to slight increase in the values of the arriving items. We leverage this intuition to prove bounds (matching up to constant factors) that decay gracefully with the amount of correlation of the arriving items. We extend these results to the case of selecting multiple items by designing a new $(1+o(1))$ approximation ratio algorithm that is robust to augmentations.

• The paper introduces linear correlation models in prophet inequalities, broadening traditional independent value assumptions.
• The paper identifies the augmentation problem where minor value increments undermine standard threshold algorithms.
• The paper proposes a fixed threshold algorithm that guarantees a two-fold approximation, ensuring robust performance under adversarial conditions.

Overview of Prophet Inequalities with Linear Correlations and Augmentations

The paper addresses advancements in prophet inequalities, specifically focusing on online decision-making problems characterized by correlation structures among the arriving item values. Prophet inequalities aim to ensure that an online decision-maker can perform nearly as well as a prophet with knowledge of all item values in advance. This research expands on these inequalities by introducing correlation models and augmentations, particularly linear correlations.

Main Contributions and Theoretical Insights

1. Linear Correlation Structure:
   • The paper investigates prophet inequalities when the values of items are correlated in a linear manner. This framework extends the commonly used independence assumptions by allowing the values to be linear combinations of independent features. The linear correlation model is a generalization of previous models presented by Bateni et al. and Chawla et al.
2. Challenges with Linear Correlations:
   • Traditional threshold-based algorithms, effective in independent scenarios, do not hold up under linear correlations. The paper identifies the "augmentation" problem, highlighting that even slight increases in item values can compromise many existing algorithms.
3. Augmented Prophets Problem:
   • The authors introduce and solve the augmented prophets problem, proving that certain threshold rules remain robust even when item values are incremented adversarially. They demonstrate that a particular fixed threshold algorithm can achieve a two-fold approximation, ensuring competitive results even under adversarial augmentations.
4. Algorithmic Strategies and Results:
   • Leveraging intuition from the augmentation problem, the research derives bounds that gracefully degrade with increasing correlation in the items. The paper proposes a novel inclusion-threshold algorithm that achieves satisfactory approximation ratios, scaling with the level of correlation (captured by column and row sparsity parameters).
   • A key result is the $(1+o(1))$ approximation ratio algorithm for selecting multiple items, addressing robustness to augmentations effectively.
5. Implications and Lower Bounds:
   • The paper provides a comprehensive analysis showing that no online algorithm can surpass the approximation ratios determined by $\min\{s_{row}, s_{col}\}$, where these parameters denote sparsity limits of the linear dependency structure. This sets a foundational lower bound for the linear correlations scenario.

Implications and Future Directions

The implications of this research are multifaceted, particularly for both theoretical advancements in stopping-time problems and practical applications in areas such as mechanism design. By quantifying effectiveness under correlated environments, the paper opens avenues for designing algorithms in more complex, real-world applications where independent assumptions are untenable. This work lays a theoretical foundation for future exploration in robust online decision-making under correlation, suggesting potential for further refinement in algorithm design and expanded application in dynamic settings.

Future research might explore generalizations beyond linear correlations, considering more complex correlation structures and broader class dependencies. Additionally, extending these results to other variants of prophet inequalities that incorporate elements like limited randomness or partial information might offer valuable insights into their applicability across diversified fields. Overall, the advancements presented provide a significant step toward comprehensively understanding and leveraging correlation in optimal stopping-time problems.