Most of the existing federated multi-armed bandits (FMAB) designs are based on the presumption that clients will implement the specified design to collaborate with the server. In reality, however, it may not be possible to modify the client's existing protocols. To address this challenge, this work focuses on clients who always maximize their individual cumulative rewards, and introduces a novel idea of "reward teaching", where the server guides the clients towards global optimality through implicit local reward adjustments. Under this framework, the server faces two tightly coupled tasks of bandit learning and target teaching, whose combination is non-trivial and challenging. A phased approach, called Teaching-After-Learning (TAL), is first designed to encourage and discourage clients' explorations separately. General performance analyses of TAL are established when the clients' strategies satisfy certain mild requirements. With novel technical approaches developed to analyze the warm-start behaviors of bandit algorithms, particularized guarantees of TAL with clients running UCB or epsilon-greedy strategies are then obtained. These results demonstrate that TAL achieves logarithmic regrets while only incurring logarithmic adjustment costs, which is order-optimal w.r.t. a natural lower bound. As a further extension, the Teaching-While-Learning (TWL) algorithm is developed with the idea of successive arm elimination to break the non-adaptive phase separation in TAL. Rigorous analyses demonstrate that when facing clients with UCB1, TWL outperforms TAL in terms of the dependencies on sub-optimality gaps thanks to its adaptive design. Experimental results demonstrate the effectiveness and generality of the proposed algorithms.

翻译：现有联邦多臂赌博机（FMAB）设计大多基于客户将执行指定方案以与服务器协作的假设。然而在现实中，修改客户现有协议往往难以实现。为解决该挑战，本文聚焦于始终最大化个体累积收益的客户，并提出"奖励教学"这一新颖概念——服务器通过隐式局部奖励调整引导客户达成全局最优性。在该框架下，服务器面临赌博机学习与目标教学两大紧密耦合任务，其组合具有显著挑战性。本文首先设计分阶段方法"先学后教"（TAL），分别激励与抑制客户的探索行为。当客户策略满足特定温和条件时，建立了TAL的通用性能分析框架。通过发展分析赌博机算法冷启动行为的新技术手段，进一步获得了客户采用UCB或ε-贪心策略时TAL的精细化保证。结果表明，TAL在仅产生对数级调整成本的同时实现对数级遗憾，关于自然下界达到阶最优。作为拓展，基于连续臂消除思想提出"边学边教"（TWL）算法，突破了TAL中非自适应阶段分离的局限。严格分析表明，面对采用UCB1策略的客户时，TWL凭借其自适应设计在次优间隙的依赖关系上优于TAL。实验结果验证了所提算法的有效性与普适性。