No-regret Algorithms for Multi-task Bayesian Optimization
This work addresses the problem of efficient multi-objective optimization for scenarios where tasks share similarities, offering rigorous regret guarantees, though it is incremental in extending existing Bayesian optimization methods.
The paper tackles multi-objective Bayesian optimization by modeling inter-task dependencies with multi-task kernels, developing two algorithms with worst-case regret bounds that explicitly capture task similarities, and demonstrates advantages in synthetic and real-life benchmarks.
We consider multi-objective optimization (MOO) of an unknown vector-valued function in the non-parametric Bayesian optimization (BO) setting, with the aim being to learn points on the Pareto front of the objectives. Most existing BO algorithms do not model the fact that the multiple objectives, or equivalently, tasks can share similarities, and even the few that do lack rigorous, finite-time regret guarantees that capture explicitly inter-task structure. In this work, we address this problem by modelling inter-task dependencies using a multi-task kernel and develop two novel BO algorithms based on random scalarizations of the objectives. Our algorithms employ vector-valued kernel regression as a stepping stone and belong to the upper confidence bound class of algorithms. Under a smoothness assumption that the unknown vector-valued function is an element of the reproducing kernel Hilbert space associated with the multi-task kernel, we derive worst-case regret bounds for our algorithms that explicitly capture the similarities between tasks. We numerically benchmark our algorithms on both synthetic and real-life MOO problems, and show the advantages offered by learning with multi-task kernels.