On the Complexity of Best Arm Identification in Multi-Armed Bandit Models (1407.4443v2)

Abstract: The stochastic multi-armed bandit model is a simple abstraction that has proven useful in many different contexts in statistics and machine learning. Whereas the achievable limit in terms of regret minimization is now well known, our aim is to contribute to a better understanding of the performance in terms of identifying the m best arms. We introduce generic notions of complexity for the two dominant frameworks considered in the literature: fixed-budget and fixed-confidence settings. In the fixed-confidence setting, we provide the first known distribution-dependent lower bound on the complexity that involves information-theoretic quantities and holds when m is larger than 1 under general assumptions. In the specific case of two armed-bandits, we derive refined lower bounds in both the fixed-confidence and fixed-budget settings, along with matching algorithms for Gaussian and Bernoulli bandit models. These results show in particular that the complexity of the fixed-budget setting may be smaller than the complexity of the fixed-confidence setting, contradicting the familiar behavior observed when testing fully specified alternatives. In addition, we also provide improved sequential stopping rules that have guaranteed error probabilities and shorter average running times. The proofs rely on two technical results that are of independent interest : a deviation lemma for self-normalized sums (Lemma 19) and a novel change of measure inequality for bandit models (Lemma 1).

• The paper introduces new theoretical lower bounds for sample complexity in both fixed-confidence and fixed-budget bandit settings.
• It proposes optimal algorithms such as the α-Elimination and SGLRT for Gaussian and Bernoulli models, showcasing practical efficacy.
• The research highlights the trade-offs between sequential and batch testing, guiding the design of adaptive decision-making strategies.

Complexity of Best-Arm Identification in Multi-Armed Bandit Models

The paper "On the Complexity of Best-Arm Identification in Multi-Armed Bandit Models" by Kaufmann, Cappé, and Garivier investigates the intricacies of identifying the best-performing arms within the stochastic multi-armed bandit framework. The authors propose new theoretical lower bounds and establish matching algorithms, addressing both fixed-budget and fixed-confidence settings in the context of statistical learning and machine learning applications.

The multi-armed bandit problem is a classical modeling approach in machine learning and statistics, representing scenarios where an agent repeatedly selects from multiple actions (arms) and receives stochastic rewards. The core objective is to identify the best-performing arm(s), balancing the trade-off between exploration (gathering information about the arms) and exploitation (choosing the best-known arm).

Fixed-Confidence and Fixed-Budget Settings

The paper primarily focuses on two settings:

• Fixed-Confidence Setting: The goal is to guarantee that with high confidence (at least $1-\delta$), the identified set of best arms $\hat{S}_m$ includes the actual best arms $\mathcal{S}^*_m$. The challenge here is to minimize the expected number of samples required, $\mathbb{E}[\tau_\delta]$, where $\tau_\delta$ is a stopping time.
• Fixed-Budget Setting: Here, the number of samples $t$ is predetermined, and the objective is to minimize the probability of error $$p_t(\nu) = \mathbb{P}(\hat{S}_m \neq \mathcal{S}^*_m)$$.

Main Contributions

Theoretical Lower Bounds

1. Generic Lower Bound in Fixed-Confidence Setting: The authors derive a generic lower bound for the sample complexity in the fixed-confidence setting based on information-theoretic quantities. This bound applies to general classes of bandit models parameterized by exponential families.
2. Two-Armed Bandits: Specific to two-armed bandit setups, the paper offers refined lower bounds. These bounds show that the complexity for identifying the best arm using fixed-budget strategies can be less than that for fixed-confidence strategies, challenging previous intuitions.
3. Bounds on $m$-Best Identifying Complexity: For more than two arms, the authors provide near-tight lower bounds for identifying the $m$-best arms in terms of Kullback-Leibler (KL) divergences.

Optimal and Near-Optimal Algorithms

1. Gaussian and Bernoulli Bandit Models: For Gaussian bandits with known variances, the authors introduce the $\alpha$-Elimination algorithm, which is proven to be optimal. For Bernoulli bandit models, they propose the Sequential Generalized Likelihood Ratio Test (SGLRT) algorithm that uses uniform sampling.
2. Comparative Performance: Empirical evaluations demonstrate the efficacy of these algorithms across different confidence levels and error probabilities, emphasizing their practical relevance.

Practical and Theoretical Implications

The results have significant implications for designing bandit algorithms in both theory and practice:

• Algorithm Design: The derived lower bounds provide a benchmark for evaluating the performance of any bandit algorithm. They guide the development of algorithms that can achieve near-optimal performance.
• Sequential vs. Batch Testing: The findings reveal nuanced differences between sequential (fixed-confidence) and batch (fixed-budget) testing strategies, underlining scenarios where one approach may be preferable over the other.
• Application Scope: Beyond theoretical contributions, the proposed methods have practical applications in areas like clinical trials, adaptive A/B testing in web optimization, and adaptive experimental designs.

Future Directions

The findings open several avenues for future research:

• Generalization to Unknown Variances and Non-parametric Models: Extending the results to scenarios where the arm distributions are non-parametric or have unknown variances.
• Multi-Arm Settings: Deepening the understanding and providing tighter bounds for multi-arm setups with $K > 2$ and $m \geq 1$.
• Adaptive Strategies: Developing adaptive strategies that can dynamically balance exploration and exploitation based on real-time performance metrics.

In conclusion, the paper by Kaufmann, Cappé, and Garivier makes substantial contributions to the theory of best-arm identification in multi-armed bandits. By providing rigorous lower bounds and practical algorithms, the research enhances both the understanding and the application of bandit models in identifying optimal decisions under uncertainty.