Improved Regret and Contextual Linear Extension for Pandora's Box and Prophet Inequality (2505.18828v1)

Abstract: We study the Pandora's Box problem in an online learning setting with semi-bandit feedback. In each round, the learner sequentially pays to open up to $n$ boxes with unknown reward distributions, observes rewards upon opening, and decides when to stop. The utility of the learner is the maximum observed reward minus the cumulative cost of opened boxes, and the goal is to minimize regret defined as the gap between the cumulative expected utility and that of the optimal policy. We propose a new algorithm that achieves $\widetilde{O}(\sqrt{nT})$ regret after $T$ rounds, which improves the $\widetilde{O}(n\sqrt{T})$ bound of Agarwal et al. [2024] and matches the known lower bound up to logarithmic factors. To better capture real-life applications, we then extend our results to a natural but challenging contextual linear setting, where each box's expected reward is linear in some known but time-varying $d$-dimensional context and the noise distribution is fixed over time. We design an algorithm that learns both the linear function and the noise distributions, achieving $\widetilde{O}(nd\sqrt{T})$ regret. Finally, we show that our techniques also apply to the online Prophet Inequality problem, where the learner must decide immediately whether or not to accept a revealed reward. In both non-contextual and contextual settings, our approach achieves similar improvements and regret bounds.

Insights into Improved Regret and Contextual Linear Extension for Pandora's Box and Prophet Inequality

The research paper under discussion addresses advancements in two classical problems of stochastic optimization: Pandora’s Box and Prophet Inequality, particularly focusing on an online learning framework. In these problems, agents must strategically decide on sampling from distributions of unknown rewards with costs, aiming to minimize regret – a common metric in sequential decision-making tasks. The paper provides a nuanced analysis of these problems by incorporating semi-bandit feedback in an online learning setting, achieving strong theoretical guarantees.

Major Contributions and Numerical Results

This paper distinguishes itself primarily by improving existing regret bounds for Pandora’s Box as an online learning task. Classical algorithms had a regret of $\mathcal{O}(n\sqrt{T})$ where $n$ is the number of boxes and $T$ is the number of rounds. The researchers propose a new algorithm that decreases this regret to $\mathcal{O}(\sqrt{nT})$, aligning with the lower bound, thus ensuring near-optimal performance. Moreover, this algorithm adapts an innovative approach to crafting optimistic distributions and dynamically reallocating probability mass based on empirical observation, serving as a counter to the probability mass static allocation method used in previous work.

Furthermore, they extend their methodologies to a contextual linear setting — a more complex and realistic scenario where each box’s potential reward is a linear function of a $d$-dimensional context vector. Here, the paper achieves a regret bound of $\mathcal{O}(nd\sqrt{T})$. This stands in contrast to prior techniques, which only succeeded in obtaining a $(nT^{5/6})$ bound in a similar contextual setting. This is an outstanding improvement, especially effective when the context dimensions ($d$) grow slower than $T^{1/3}$, rendering this approach more efficient.

Finally, the paper applies these extended methods to the Prophet Inequality problem as well, achieving analogous improvement in regret bounds. This coherence across different but related optimization problems bolsters the robustness and versatility of the proposed techniques.

Theoretical and Practical Implications

The implications of this paper span both theoretical and practical domains. Theoretically, it closes significant gaps in the understanding of stochastic optimization under semi-bandit feedback, offering robust analytical tools and frameworks that address a central problem in algorithmic decision-making. The results especially shine in the contextual realms, which replicate real-world complexities more accurately than static models. Practically, these insights can drastically impact areas that rely on sequential decision-making under uncertainty, such as online markets, finance, and adaptive control systems in engineering.

Future Directions

The avenues this research opens are manifold. One immediate pursuit could be addressing the open question regarding the optimality of the $\mathcal{O}(nd\sqrt{T})$ bound in contextually linear settings. As our abilities to model decision-making scenarios involving complex contexts improve, analogous extensions of these techniques to non-linear or higher-order contextual models would be valuable. Another promising direction is the application of these algorithms in systems with non-stationary distributions or adversarial contexts, which would further juxtapose the stable distribution assumption underlying most current models.

In summary, this work represents a significant stride forward in understanding and solving Pandora’s Box and Prophet Inequality problems under a more general and challenging framework of online learning. The integration of contextual elements and the subsequent theoretical advancements naturally push the boundaries toward more universal and applicable optimization solutions.