We consider the problem of online local false discovery rate (FDR) control where multiple tests are conducted sequentially, with the goal of maximizing the total expected number of discoveries. We formulate the problem as an online resource allocation problem with accept/reject decisions, which from a high level can be viewed as an online knapsack problem, with the additional uncertainty of exogenous random budget replenishment. We start with general arrival distributions and propose a simple policy that achieves a $O(\sqrt{T})$ regret. We complement the result by showing that such regret rate is in general not improvable. We then shift our focus to discrete arrival distributions. We find that many existing re-solving heuristics in the online resource allocation literature, albeit achieve bounded loss in canonical settings, may incur a $\Omega(\sqrt{T})$ or even a $\Omega(T)$ regret. With the observation that canonical policies tend to be too optimistic and over accept arrivals, we propose a novel policy that incorporates budget buffers. We show that small additional logarithmic buffers suffice to reduce the regret from $\Omega(\sqrt{T})$ or even $\Omega(T)$ to $O(\ln^2 T)$. Numerical experiments are conducted to validate our theoretical findings. Our formulation may have wider applications beyond the problem considered in this paper, and our results emphasize how effective policies should be designed to reach a balance between circumventing wrong accept and reducing wrong reject in online resource allocation problems with exogenous budget replenishment.

翻译：我们考虑在线局部错误发现率（FDR）控制问题，其中多个检验按序进行，目标为最大化期望总发现数。我们将该问题建模为具有接受/拒绝决策的在线资源分配问题，从高层次看可视为在线背包问题，并额外考虑了外生随机预算补充的不确定性。首先针对一般到达分布，我们提出一种简单策略，实现$O(\sqrt{T})$的遗憾值，并通过证明表明该遗憾率通常不可改进来补充此结果。随后聚焦离散到达分布，发现现有在线资源分配文献中的多种重求解启发式算法，即便在标准设定下能实现有界损失，也可能导致$\Omega(\sqrt{T})$甚至$\Omega(T)$的遗憾值。基于标准策略过于乐观而过度接受到达的观察，我们提出一种纳入预算缓冲的新策略，证明仅需额外对数阶缓冲即可将遗憾值从$\Omega(\sqrt{T})$甚至$\Omega(T)$降至$O(\ln^2 T)$。通过数值实验验证理论结果。本文的建模框架可能具有超越当前问题的更广泛应用，结果揭示了在设计有效策略时，如何在在线资源分配问题中平衡规避错误接受与减少错误拒绝以应对外生预算补充的重要性。