Pareto Upper Confidence Bounds algorithms: An empirical study

Many real-world stochastic environments are inherently multi-objective environments with conflicting objectives. The multi-objective multi-armed bandits (MOMAB) are extensions of the classical, i.e. single objective, multi-armed bandits to reward vectors and multi-objective optimisation techniques are often required to design mechanisms with an efficient exploration / exploitation trade-off. In this paper, we propose the improved Pareto Upper Confidence Bound (iPUCB) algorithm that straightforwardly extends the single objective improved UCB algorithm to reward vectors by deleting the suboptimal arms. The goal of the improved Pareto UCB algorithm, i.e. iPUCB, is to identify the set of best arms, or the Pareto front, in a fixed budget of arm pulls. We experimentally compare the performance of the proposed Pareto upper confidence bound algorithm with the Pareto UCB1 algorithm and the Hoeffding race on a bi-objective example coming from an industrial control applications, i.e. the engagement of wet clutches. We propose a new regret metric based on the Kullback-Leibler divergence to measure the performance of a multi-objective multi-armed bandit algorithm. We show that iPUCB outperforms the other two tested algorithms on the given multi-objective environment.