Multi-armed Bandits with Missing Outcome (2411.05661v1)

Abstract: While significant progress has been made in designing algorithms that minimize regret in online decision-making, real-world scenarios often introduce additional complexities, perhaps the most challenging of which is missing outcomes. Overlooking this aspect or simply assuming random missingness invariably leads to biased estimates of the rewards and may result in linear regret. Despite the practical relevance of this challenge, no rigorous methodology currently exists for systematically handling missingness, especially when the missingness mechanism is not random. In this paper, we address this gap in the context of multi-armed bandits (MAB) with missing outcomes by analyzing the impact of different missingness mechanisms on achievable regret bounds. We introduce algorithms that account for missingness under both missing at random (MAR) and missing not at random (MNAR) models. Through both analytical and simulation studies, we demonstrate the drastic improvements in decision-making by accounting for missingness in these settings.

• The paper presents theoretical regret bounds that quantify performance loss from missing outcomes under various missing data mechanisms.
• It develops modified UCB algorithms that incorporate auxiliary data and mediator variables to ensure unbiased reward estimates.
• The study enhances decision-making in fields like healthcare and advertising by effectively managing missing data in online algorithms.

An Analytical Exploration of Multi-Armed Bandits with Missing Outcomes

The paper "Multi-armed Bandits with Missing Outcome" addresses a critical gap in the analysis of multi-armed bandit (MAB) models by incorporating scenarios where observed outcomes are missing. This research is pivotal for online decision-making algorithms, which are facing persistent challenges due to missing data that often emerges in real-world applications. The authors have presented a theoretical analysis supported by tailored algorithms to handle various missingness mechanisms, including Missing Completely At Random (MCAR), Missing At Random (MAR), and Missing Not At Random (MNAR).

Key Contributions

The paper systematically explores the implications of missing outcomes in MAB settings, extending the traditional model to account for these missing observations. Two main contributions are outlined:

1. Theoretical Analysis and Regret Bounds: The research formalizes the impact of missing outcomes on achievable regret bounds in MAB scenarios. These bounds are crucial as they encapsulate the extent to which suboptimal decisions are made due to missing data. For each missing data mechanism considered, the authors present both lower and upper bounds on regret, ensuring a comprehensive understanding of potential performance deterioration.
2. Development of Upper Confidence Bound Algorithms: New versions of UCB algorithms are introduced to explicitly handle missing outcomes under different missingness assumptions. These algorithms leverage auxiliary data, a mediator variable when available, and are designed to maintain unbiased estimates in the presence of incomplete data.

Detailed Analysis

• MCAR Setting: The classical UCB algorithm is adapted for the MCAR setting, revealing that the worst-case regret remains logarithmic in terms of time horizon when the missingness mechanism is truly independent of both the action and the outcome. This confirms that missing data, when completely at random, can be effectively managed without incurring substantial loss in performance.
• MAR Setting: For missing outcomes that are MAR, the authors employ the auxiliary mediator variable to adjust estimates of expected rewards, showing significant reduction in regret compared to the use of classical methods that ignore the missing data structure. This adaptation is particularly crucial when the missingness correlates with observed or latent variables, as ignoring these factors leads to biased reward estimation.
• MNAR Scenario: Identifying the expected rewards is far more complex in MNAR cases, given the direct dependency between missingness and the unobserved reward outcomes. The paper adopts a strategy to solve integral equations to determine the inverse odds ratio, thus allowing unbiased estimation of reward expectations. Although computationally challenging, this approach demonstrates the feasibility of handling MNAR through well-defined assumptions and completeness conditions.

Practical and Theoretical Implications

Theoretical advancements presented in this research have profound implications for various applications such as healthcare, online advertising, and recommendation systems, where decisions rely on partially observed reward data. By accounting for the mechanism of missingness directly, the proposed solutions enhance decision-making processes, potentially leading to better outcomes in practice.

Moreover, the findings emphasize an important area of future exploration: the extension of these algorithms to settings involving continuous mediator variables and the further refinement of computational techniques for solving complex integral equations inherent to MNAR analysis.

Conclusion

In summary, this paper significantly contributes to the multi-armed bandit literature by addressing the consequences of missing outcomes—a scenario ubiquitous in practical applications but underexplored in theoretical studies. The introduction of algorithms tailored for different missingness mechanisms presents a robust framework that can be utilized and further refined for enhancing decision-making algorithms in uncertain environments. Future developments may extend these insights to more complex interactions and dependencies, bolstering their use across diverse problem domains.