Designing multi-objective multi-armed bandits algorithms: A study

We propose an algorithmic framework for multi-objective multi-armed bandits with multiple rewards. Different partial order relationships from multi-objective optimization can be considered for a set of reward vectors, such as scalarization functions and Pareto search. A scalarization function transforms the multi-objective environment into a single objective environment and are a popular choice in multi-objective reinforcement learning. Scalarization techniques can be straightforwardly implemented into the current multi-armed bandit framework, but the efficiency of these algorithms depends very much on their type, linear or non-linear (e.g. Chebyshev), and their parameters. Using Pareto dominance order relationship allows to explore the multi-objective environment directly, however this can result in large sets of Pareto optimal solutions. In this paper we propose and evaluate the performance of multi-objective MABs using three regret metric criteria. The standard UCB1 is extended to scalarized multi-objective UCB1 and we propose a Pareto UCB1 algorithm. Both algorithms are proven to have a logarithmic upper bound for their expected regret. We also introduce a variant of the scalarized multi-objective UCB1 that removes online inefficient scalarizations in order to improve the algorithm's efficiency. These algorithms are experimentally compared on multi-objective Bernoulli distributions, Pareto UCB1 being the algorithm with the best empirical performance.