Best-Arm Identification in Linear Bandits (1409.6110v2)

Abstract: We study the best-arm identification problem in linear bandit, where the rewards of the arms depend linearly on an unknown parameter $\theta^*$ and the objective is to return the arm with the largest reward. We characterize the complexity of the problem and introduce sample allocation strategies that pull arms to identify the best arm with a fixed confidence, while minimizing the sample budget. In particular, we show the importance of exploiting the global linear structure to improve the estimate of the reward of near-optimal arms. We analyze the proposed strategies and compare their empirical performance. Finally, as a by-product of our analysis, we point out the connection to the $G$-optimality criterion used in optimal experimental design.

• The paper analyzes the complexity of best-arm identification in linear bandits and proposes static and adaptive sample allocation strategies to minimize the sample budget while achieving high confidence.
• It introduces G-Allocation based on experimental design and XY-Allocation focusing on distinguishing critical directions, finding that XY-Allocation can be more efficient in empirical settings.
• The study demonstrates how exploiting the linear structure improves exploration efficiency and provides methods for efficient, precise decision-making in settings with correlated outcomes.

Best-Arm Identification in Linear Bandits: An Analytical Perspective

The paper "Best-Arm Identification in Linear Bandits" explores the nuanced problem of identifying the best arm in a linear bandit setting with a fixed confidence level. This paper is primarily concerned with sample allocation strategies designed to efficiently ascertain the optimal arm while minimizing the sample budget. The authors tackle this through a systematic analysis of the complexity inherent in the linear bandit problem, resulting in various proposed strategies to exploit the global linear structure of rewards to enhance the estimates of near-optimal arms.

Problem Formulation and Context

The linear bandit problem builds on the traditional multi-armed bandit (MAB) framework but introduces a correlation between arms by considering that the expected reward of each arm is a linear function of an unknown parameter vector $\theta^*$. In such a setting, pulling an arm provides information not just about the expected reward of that arm but also about others, indirectly leading to a more informative exploration process. The central challenge addressed is to identify the arm with the highest expected reward—the "best arm"—with high confidence while using as few samples as possible.

Complexity Characterization

The paper starts by characterizing the complexity of the best-arm identification in linear bandits via an oracle strategy. This oracle has access to $\theta^*$ and hence serves as a theoretical benchmark for the complexity analysis. The use of confidence sets in combination with empirical and theoretical gaps was key to forming stopping conditions for the allocation strategies.

The complexity $H_{\text{LB}}$ of best-arm identification is expressed in terms of the gaps between arm rewards and the degree of correlation among the arms, akin to the problem-dependent $H_{\text{MAB}}$ in multi-armed bandits. Importantly, the complexity quantification differs due to the linear dependence among arms, which introduces a dimensionality component absent in MAB.

Allocation Strategies

Two primary static allocation strategies are explored:

1. G-Allocation Strategy: This strategy draws from the G-optimality criterion in experimental design, minimizing the maximum prediction error across all arms. The strategy ensures a uniformly accurate estimate of $\theta^*$.
2. $\mathbf{XY}$-Allocation Strategy: It seeks to minimize the prediction error in the directions that are crucial for distinguishing between nearly optimal arms, offering an allocation strategy that dynamically prioritizes important directions.

Both strategies are backed by theoretical guarantees on sample complexity, with $\mathbf{XY}$-allocation sometimes outperforming G-allocation in empirical settings due to its focus on crucial discriminative directions.

Adaptive Strategies

Beyond static allocations, the paper proposes an $\mathbf{XY}$-Adaptive allocation strategy, which balances adaptivity with complexity by operating in distinct phases. This strategy updates its allocation focus based on previously gathered data, approaching the theoretical performance of the oracle without the full costs of adaptation traditionally seen in fully adaptive strategies.

Implications and Future Directions

The paper's findings illuminate the importance of understanding the linear structure inherent in the arms when strategizing for exploration, offering sophisticated yet theoretically substantiated methods to improve exploration efficiency in linear bandit settings. This work opens doors for further investigations into more complex linear bandit explorations under constraints such as limited budgets, large action spaces, or cases with heteroscedastic noise.

The results are pivotal in applications where decision-making needs to be both highly efficient and precise, given correlated outcomes across choices. Future developments could include enhancing these methods for online and contextual decision-making scenarios, thereby broadening the applicability of linear bandit models in real-world applications like personalized recommendations and adaptive experimentation in various domains.