Semi-Bandit Learning for Monotone Stochastic Optimization (2312.15427v1)

Abstract: Stochastic optimization is a widely used approach for optimization under uncertainty, where uncertain input parameters are modeled by random variables. Exact or approximation algorithms have been obtained for several fundamental problems in this area. However, a significant limitation of this approach is that it requires full knowledge of the underlying probability distributions. Can we still get good (approximation) algorithms if these distributions are unknown, and the algorithm needs to learn them through repeated interactions? In this paper, we resolve this question for a large class of "monotone" stochastic problems, by providing a generic online learning algorithm with $\sqrt{T \log T}$ regret relative to the best approximation algorithm (under known distributions). Importantly, our online algorithm works in a semi-bandit setting, where in each period, the algorithm only observes samples from the r.v.s that were actually probed. Our framework applies to several fundamental problems in stochastic optimization such as prophet inequality, Pandora's box, stochastic knapsack, stochastic matchings and stochastic submodular optimization.

Semi-Bandit Learning for Monotone Stochastic Optimization

The paper "Semi-Bandit Learning for Monotone Stochastic Optimization" addresses a fundamental question in stochastic optimization: how can effective algorithms be designed when underlying probability distributions of stochastic inputs are unknown? Unlike traditional methods that assume full distributional knowledge, this paper focuses on scenarios requiring algorithms to learn these distributions through repeated interactions. Specifically, the authors develop an online learning framework tailored for a class of problems termed "monotone" stochastic problems, offering a novel semi-bandit setting that allows for more practical learning when only partial feedback is available.

Key Contributions

The core contribution of this research is the development of an online learning algorithm that demonstrates a regret bound of $\sqrt{T \log T}$ relative to the best-known approximation algorithm when probability distributions are known. This is significant as it means that despite the absence of full distributional knowledge, the proposed approach asymptotically achieves close to optimal performance. The versatility of the framework is demonstrated across several canonical problems in stochastic optimization, such as stochastic knapsack, stochastic matchings, and prophet inequalities.

The paper lays out a general procedure for transforming offline approximation algorithms into online learning algorithms suitable for unknown distributions. This transformation hinges critically on a designed method to construct "optimistic" empirical distributions that stochastically dominate the true unknown distributions, a principle grounded in the notion of optimism in the face of uncertainty.

Regret Analysis

A primary feature of this work is a detailed regret analysis. The regret, a measure of the performance difference between the algorithm and an oracle with full distributional knowledge, is shown to scale optimally with $T$, the number of rounds. The authors employ a clever analytical technique which identifies the semi-bandit settings' unique characteristics, leveraging the probability that a particular item is probed to optimize exploration versus exploitation dynamically.

Practical Implications

The results have broad applications in fields where decision-making under uncertainty is crucial and full feedback is impractical. The domains of online advertising, adaptive K-armed bandit problems, and economic models where acquiring full information incurs costs or delays are particularly relevant. The emphasis on semi-bandit feedback provides a pragmatic angle, making the algorithms applicable to real-world systems where only partial data is accessible during learning phases.

Future Directions

The paper opens several avenues for future exploration. Potential improvements include developing broader classes of stochastic problems beyond the monotone constraints while maintaining efficient regret bounds. Another direction could involve refining the empirical distribution estimates used in constructing the learning strategy to further improve computational performance and scalability.

In summary, the semi-bandit learning framework for monotone stochastic optimization stands as a robust contribution to the field of online learning, offering promising pathways for efficient decision-making in uncertain environments. Theoretically, it narrows the gulf between full-information algorithms and those with restricted feedback, framing a compelling narrative for further inquiry and innovation in adaptive learning systems.