Simple Bayesian Algorithms for Best Arm Identification (1602.08448v4)

Abstract: This paper considers the optimal adaptive allocation of measurement effort for identifying the best among a finite set of options or designs. An experimenter sequentially chooses designs to measure and observes noisy signals of their quality with the goal of confidently identifying the best design after a small number of measurements. This paper proposes three simple and intuitive Bayesian algorithms for adaptively allocating measurement effort, and formalizes a sense in which these seemingly naive rules are the best possible. One proposal is top-two probability sampling, which computes the two designs with the highest posterior probability of being optimal, and then randomizes to select among these two. One is a variant of top-two sampling which considers not only the probability a design is optimal, but the expected amount by which its quality exceeds that of other designs. The final algorithm is a modified version of Thompson sampling that is tailored for identifying the best design. We prove that these simple algorithms satisfy a sharp optimality property. In a frequentist setting where the true quality of the designs is fixed, one hopes the posterior definitively identifies the optimal design, in the sense that that the posterior probability assigned to the event that some other design is optimal converges to zero as measurements are collected. We show that under the proposed algorithms this convergence occurs at an exponential rate, and the corresponding exponent is the best possible among all allocation

• The paper proposes three novel Bayesian algorithms (TTPS, TTVS, TTTS) that balance exploration and exploitation in best-arm identification tasks.
• It demonstrates that these methods achieve exponential convergence of the posterior, optimizing measurement allocation in multi-armed bandit settings.
• Empirical simulations confirm that the algorithms significantly reduce measurement requirements compared to traditional equal-allocation strategies.

An Expert Analysis of "Simple Bayesian Algorithms for Best-Arm Identification"

Daniel Russo's paper explores the problem of best-arm identification in multi-armed bandit settings, focusing on Bayesian methods for adaptively allocating measurement efforts. The paper is grounded in the context where an experimenter aims to identify the best design among several options by sequentially choosing models to observe and collecting noisy signals of their performance. The ultimate objective is to maximize the confidence in identifying the optimal design with the fewest observations possible.

Proposed Bayesian Algorithms

The paper introduces three Bayesian algorithms—Top-Two Probability Sampling (TTPS), Top-Two Value Sampling (TTVS), and a variant called Top-Two Thompson Sampling (TTTS). These methods are developed to effectively balance the exploration of different designs and the exploitation of promising ones.

1. Top-Two Probability Sampling (TTPS): This approach involves selecting the two designs with the highest posterior probabilities of being optimal and randomizing the next observation between them. This strategy inherently adapts to changes in measurement distributions and is structured to iteratively optimize the allocation of measurements.
2. Top-Two Value Sampling (TTVS): TTVS extends TTPS by considering not only the probability of a design being optimal but also the expected superiority of its quality. This is encapsulated in a value measure that integrates both these aspects, enhancing the decision-making process in selecting the best design.
3. Top-Two Thompson Sampling (TTTS): Building on the framework of Thompson Sampling, this variant introduces a top-two selection step to robustly manage exploration-exploitation trade-offs while ensuring a more diverse measurement allocation that doesn't just focus on a single estimated-best design.

Theoretical Foundations and Properties

A core result of the paper is the demonstration that these algorithms satisfy a sharp optimality property. Specifically, they are shown to facilitate an exponential rate of posterior convergence in scenarios where the true quality of designs is fixed. The convergence is claimed to occur at a rate characterized by an optimal exponent, which is the best possible across all allocation rules given the constraints of the problem setup.

The paper further discusses the sensitivity of the proposed methods to a tuning parameter, which plays a critical role in determining the allocation probability among the top-two selected designs. The robustness of the algorithms to this parameter setting is underscored, noting that even a default value results in performance close to the theoretical optimum.

Empirical Validation and Practical Implications

Simulation results presented align with theoretical findings, demonstrating that these methods can discern the best design with significantly fewer measurements compared to equal allocation strategies. In particular, TTTS and the other proposed methods efficiently allocate measurement efforts, correcting any over-exploration or under-exploration dynamically as the experiment proceeds.

Practically, these algorithms provide decision-makers with effective tools in various fields such as A/B testing, clinical trials, and engineering design optimizations. They are particularly valuable in settings characterized by high measurement costs or the need for rapid convergence to optimal decisions—common in online platforms and manufacturing processes.

Conclusions and Future Directions

Russo's work points to the broader potential for Bayesian algorithms in complex decision-making environments beyond conventional multi-armed bandit problems. The algorithms and the accompanying analytical techniques provide a robust framework for tackling related problems that require adaptive learning and decision-making under uncertainty.

Future research might explore simplifying the tuning mechanisms or extending the paradigm to more complex bandit structures, such as contextual bandits with large action spaces or settings with delayed feedback. These investigations will continue to stretch the limits and applicability of Bayesian methodologies in sequential decision-making tasks.