The best-arm identification (BAI) problem is one of the most fundamental problems in interactive machine learning, which has two flavors: the fixed-budget setting (FB) and the fixed-confidence setting (FC). For $K$-armed bandits with the unique best arm, the optimal sample complexities for both settings have been settled down, and they match up to logarithmic factors. This prompts an interesting research question about the generic, potentially structured BAI problems: Is FB harder than FC or the other way around? In this paper, we show that FB is no harder than FC up to logarithmic factors. We do this constructively: we propose a novel algorithm called FC2FB (fixed confidence to fixed budget), which is a meta algorithm that takes in an FC algorithm $\mathcal{A}$ and turn it into an FB algorithm. We prove that this FC2FB enjoys a sample complexity that matches, up to logarithmic factors, that of the sample complexity of $\mathcal{A}$. This means that the optimal FC sample complexity is an upper bound of the optimal FB sample complexity up to logarithmic factors. Our result not only reveals a fundamental relationship between FB and FC, but also has a significant implication: FC2FB, combined with existing state-of-the-art FC algorithms, leads to improved sample complexity for a number of FB problems.

翻译：最佳臂识别（BAI）问题是交互式机器学习中最基本的问题之一，其包含两种设定：固定预算（FB）与固定置信度（FC）。对于具有唯一最佳臂的 $K$ 臂赌博机问题，两种设定的最优样本复杂度均已确定，且二者在对数因子内匹配。这引出了一个关于通用、可能具有结构性的 BAI 问题的有趣研究课题：FB 是否比 FC 更难，抑或相反？本文证明，在对数因子内，FB 不劣于 FC。我们通过构造性方法实现这一点：提出了一种名为 FC2FB（固定置信度转固定预算）的新型算法，它是一种元算法，接收一个 FC 算法 $\mathcal{A}$ 并将其转换为一个 FB 算法。我们证明，该 FC2FB 算法享有的样本复杂度与算法 $\mathcal{A}$ 的样本复杂度在对数因子内匹配。这意味着最优的 FC 样本复杂度是对数因子内最优 FB 样本复杂度的上界。我们的结果不仅揭示了 FB 与 FC 之间的基本关系，还具有重要启示：FC2FB 与现有最先进的 FC 算法相结合，可为一系列 FB 问题带来改进的样本复杂度。