  The complexity of Pareto fronts imposes a great challenge on the convergence
analysis of multi-objective optimization methods. While most theoretical
convergence studies have addressed finite-set and/or discrete problems, others
have provided probabilistic guarantees, assumed a total order on the solutions,
or studied their asymptotic behaviour. In this paper, we revisit the
Tchebycheff weighted method in a hierarchical bandits setting and provide a
finite-time bound on the Pareto-compliant additive $\epsilon$-indicator. To the
best of our knowledge, this paper is one of few that establish a link between
weighted sum methods and quality indicators in finite time.
