This paper addresses the challenge of improving finite sample performance in Ranking and Selection by developing a Bahadur-Rao type expansion for the Probability of Correct Selection (PCS). While traditional large deviations approximations captures PCS behavior in the asymptotic regime, they can lack precision in finite sample settings. Our approach enhances PCS approximation under limited simulation budgets, providing more accurate characterization of optimal sampling ratios and optimality conditions dependent of budgets. Algorithmically, we propose a novel finite budget allocation (FCBA) policy, which sequentially estimates the optimality conditions and accordingly balances the sampling ratios. We illustrate numerically on toy examples that our FCBA policy achieves superior PCS performance compared to tested traditional methods. As an extension, we note that the non-monotonic PCS behavior described in the literature for low-confidence scenarios can be attributed to the negligence of simultaneous incorrect binary comparisons in PCS approximations. We provide a refined expansion and a tailored allocation strategy to handle low-confidence scenarios, addressing the non-monotonicity issue.

Backdoors on federated learning will be diluted by subsequent benign updates. This is reflected in the significant reduction of attack success rate as iterations increase, ultimately failing. We use a new metric to quantify the degree of this weakened backdoor effect, called attack persistence. Given that research to improve this performance has not been widely noted,we propose a Full Combination Backdoor Attack (FCBA) method. It aggregates more combined trigger information for a more complete backdoor pattern in the global model. Trained backdoored global model is more resilient to benign updates, leading to a higher attack success rate on the test set. We test on three datasets and evaluate with two models across various settings. FCBA's persistence outperforms SOTA federated learning backdoor attacks. On GTSRB, postattack 120 rounds, our attack success rate rose over 50% from baseline. The core code of our method is available at https://github.com/PhD-TaoLiu/FCBA.