We consider a variant of sequential testing by betting where, at each time step, the statistician is presented with multiple data sources (arms) and obtains data by choosing one of the arms. We consider the composite global null hypothesis $\mathscr{P}$ that all arms are null in a certain sense (e.g. all dosages of a treatment are ineffective) and we are interested in rejecting $\mathscr{P}$ in favor of a composite alternative $\mathscr{Q}$ where at least one arm is non-null (e.g. there exists an effective treatment dosage). We posit an optimality desideratum that we describe informally as follows: even if several arms are non-null, we seek $e$-processes and sequential tests whose performance are as strong as the ones that have oracle knowledge about which arm generates the most evidence against $\mathscr{P}$. Formally, we generalize notions of log-optimality and expected rejection time optimality to more than one arm, obtaining matching lower and upper bounds for both. A key technical device in this optimality analysis is a modified upper-confidence-bound-like algorithm for unobservable but sufficiently "estimable" rewards. In the design of this algorithm, we derive nonasymptotic concentration inequalities for optimal wealth growth rates in the sense of Kelly [1956]. These may be of independent interest.

翻译：我们考虑一种基于赌注的序贯检验变体，其中在每个时间步长，统计学家面临多个数据源（臂），并通过选择其中一个臂获取数据。我们考虑复合全局零假设 $\mathscr{P}$，即所有臂在某种意义下均为零（例如，治疗的各个剂量均无效），并希望拒绝 $\mathscr{P}$ 以支持复合备择假设 $\mathscr{Q}$，其中至少有一个臂为非零（例如，存在有效的治疗剂量）。我们提出一个最优性需求，非正式描述如下：即使多个臂为非零，我们寻求的 $e$ 过程与序贯检验的性能应相当于具有先知知识（知晓哪个臂能生成对抗 $\mathscr{P}$ 的最强证据）的过程。形式上，我们将对数最优性和预期拒绝时间最优性的概念推广至多臂情形，并为其匹配相应的下界与上界。在该最优性分析中，一个关键技术工具是针对不可观测但足够“可估计”的奖励所设计的改进型上置信界算法。在该算法设计过程中，我们推导了基于Kelly [1956] 意义下的最优财富增长率的非渐近浓度不等式，这些结果可能具有独立的研究价值。