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 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 uncertain budgets.

翻译：本文考虑在线局部错误发现率（FDR）控制问题，即在顺序进行多重检验时，以最大化总期望发现数为目标。我们将该问题建模为一个包含接受/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，并附加随机预算补充的不确定性。首先针对一般到达分布，提出一种简单策略，其遗憾值达到$O(\sqrt{T})$量级。我们进一步证明该遗憾率在一般情况下无法改进。随后聚焦离散到达分布，发现现有在线资源分配文献中的许多重解启发式方法，虽然在规范设置下能实现有界损失，但可能产生$\Omega(\sqrt{T})$甚至$\Omega(T)$的遗憾值。基于规范策略往往过于乐观而过度接受到达的观察，我们提出一种包含预算缓冲的新型策略。研究表明，仅需增加对数尺度的缓冲即可将遗憾值从$\Omega(\sqrt{T})$甚至$\Omega(T)$降低至$O(\ln^2 T)$。数值实验验证了理论结论。该建模框架可推广至本文研究问题之外的更广泛场景，研究结果强调了在预算不确定的在线资源分配问题中，有效策略设计需在避免错误接受与减少错误拒绝之间取得平衡。