Zhirui Chen

Zhirui Chen, P. N. Karthik, Vincent Y. F. Tan et al.

We study best arm identification in a federated multi-armed bandit setting with a central server and multiple clients, when each client has access to a {\em subset} of arms and each arm yields independent Gaussian observations. The goal is to identify the best arm of each client subject to an upper bound on the error probability; here, the best arm is one that has the largest {\em average} value of the means averaged across all clients having access to the arm. Our interest is in the asymptotics as the error probability vanishes. We provide an asymptotic lower bound on the growth rate of the expected stopping time of any algorithm. Furthermore, we show that for any algorithm whose upper bound on the expected stopping time matches with the lower bound up to a multiplicative constant ({\em almost-optimal} algorithm), the ratio of any two consecutive communication time instants must be {\em bounded}, a result that is of independent interest. We thereby infer that an algorithm can communicate no more sparsely than at exponential time instants in order to be almost-optimal. For the class of almost-optimal algorithms, we present the first-of-its-kind asymptotic lower bound on the expected number of {\em communication rounds} until stoppage. We propose a novel algorithm that communicates at exponential time instants, and demonstrate that it is asymptotically almost-optimal.

Zhirui Chen, P. N. Karthik, Yeow Meng Chee et al.

We consider a multi-armed bandit setting with finitely many arms, in which each arm yields an $M$-dimensional vector reward upon selection. We assume that the reward of each dimension (a.k.a. {\em objective}) is generated independently of the others. The best arm of any given objective is the arm with the largest component of mean corresponding to the objective. The end goal is to identify the best arm of {\em every} objective in the shortest (expected) time subject to an upper bound on the probability of error (i.e., fixed-confidence regime). We establish a problem-dependent lower bound on the limiting growth rate of the expected stopping time, in the limit of vanishing error probabilities. This lower bound, we show, is characterised by a max-min optimisation problem that is computationally expensive to solve at each time step. We propose an algorithm that uses the novel idea of {\em surrogate proportions} to sample the arms at each time step, eliminating the need to solve the max-min optimisation problem at each step. We demonstrate theoretically that our algorithm is asymptotically optimal. In addition, we provide extensive empirical studies to substantiate the efficiency of our algorithm. While existing works on pure exploration with multi-objective multi-armed bandits predominantly focus on {\em Pareto frontier identification}, our work fills the gap in the literature by conducting a formal investigation of the multi-objective best arm identification problem.

Zhirui Chen, P. N. Karthik, Yeow Meng Chee et al.

We study best arm identification (BAI) in linear bandits in the fixed-budget regime under differential privacy constraints, when the arm rewards are supported on the unit interval. Given a finite budget $T$ and a privacy parameter $\varepsilon>0$, the goal is to minimise the error probability in finding the arm with the largest mean after $T$ sampling rounds, subject to the constraint that the policy of the decision maker satisfies a certain {\em $\varepsilon$-differential privacy} ($\varepsilon$-DP) constraint. We construct a policy satisfying the $\varepsilon$-DP constraint (called {\sc DP-BAI}) by proposing the principle of {\em maximum absolute determinants}, and derive an upper bound on its error probability. Furthermore, we derive a minimax lower bound on the error probability, and demonstrate that the lower and the upper bounds decay exponentially in $T$, with exponents in the two bounds matching order-wise in (a) the sub-optimality gaps of the arms, (b) $\varepsilon$, and (c) the problem complexity that is expressible as the sum of two terms, one characterising the complexity of standard fixed-budget BAI (without privacy constraints), and the other accounting for the $\varepsilon$-DP constraint. Additionally, we present some auxiliary results that contribute to the derivation of the lower bound on the error probability. These results, we posit, may be of independent interest and could prove instrumental in proving lower bounds on error probabilities in several other bandit problems. Whereas prior works provide results for BAI in the fixed-budget regime without privacy constraints or in the fixed-confidence regime with privacy constraints, our work fills the gap in the literature by providing the results for BAI in the fixed-budget regime under the $\varepsilon$-DP constraint.