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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.