Multiple Identifications in Multi-Armed Bandits (1205.3181v1)

Abstract: We study the problem of identifying the top $m$ arms in a multi-armed bandit game. Our proposed solution relies on a new algorithm based on successive rejects of the seemingly bad arms, and successive accepts of the good ones. This algorithmic contribution allows to tackle other multiple identifications settings that were previously out of reach. In particular we show that this idea of successive accepts and rejects applies to the multi-bandit best arm identification problem.

Multiple Identifications in Multi-Armed Bandits

The paper "Multiple Identifications in Multi-Armed Bandits" introduces a novel algorithmic approach to tackling the problem of identifying the top m arms in a multi-armed bandit (MAB) setting. The key contribution of the paper is the introduction of the Successive Accepts and Rejects (SAR) algorithm, which extends the established Successive Rejects (SR) algorithm to accommodate the challenge of selecting multiple top-performing arms within a given constraint on evaluations.

The Methodological Advances

The SAR algorithm is specifically tailored to identify the top m arms from K unknown distributions within a fixed evaluation budget. By introducing a mechanism of successive acceptance and rejection of arms, the SAR algorithm provides an elegant solution that adapts to the dynamic stochastic environment typical of MAB problems. The underpinning principle is to classify arms continually, promoting the promising candidates while eliminating underperformers based on empirical estimates of their mean rewards.

Numerical Analysis and Performance

The theoretical analysis is supported by robust numerical simulations, indicating that the SAR algorithm can achieve superior performance compared to traditional methods such as uniform sampling and a naively adapted SR algorithm. It is especially noteworthy that SR, when used for identifying multiple best arms, often underperforms compared to SAR and even simple uniform sampling strategies. This highlights the distinctive challenges posed by multi-arm identification versus finding a single best arm.

Complexity and Theoretical Insights

The authors propose extended complexity measures, $H^{\langle m \rangle}$ and $H^{[M]}$, to quantify the difficulty of identifying the top m arms or solving multi-bandit instances, respectively. These metrics generalize the notion of arm selection complexity, originally formulated for single arm identification, to encompass scenarios where multiple selections or simultaneous problems are tackled. The theoretical contribution includes deriving bounds on the evaluation budget necessary for accurate identification, thus providing insights into the fundamental limits of MAB algorithms under resource constraints.

Practical and Theoretical Implications

Practically, the SAR algorithm and the associated complexity analysis have implications for applications in which multiple optimal choices need to be made in uncertain settings, such as hyperparameter optimization in machine learning or decision-making in financial portfolios. Theoretically, the insights could drive future research into adaptive algorithms that can dynamically adjust strategies based on evolving complexity measures without requiring a priori knowledge about the problem structure.

Future Directions

While the SAR algorithm presents significant advancements for the problem domain, potential future research could explore extensions to more complex evaluation settings, including non-stationary environments or the incorporation of domain-specific constraints. Moreover, the development of entirely parameter-free MAB strategies remains a fertile avenue for investigation, promising further reduction in computational overhead and increased flexibility.

Overall, this paper provides substantial advancements for the field of multi-armed bandits, offering both practical algorithmic solutions and deep theoretical insights. It positions SAR as a pivotal tool in the toolkit of researchers and practitioners working with stochastic decision processes.