Achievable Goals in Bayesian Multi-Objective Optimization
Résumé
We consider multi-objective optimization problems, min x∈Rd(f1(x), . . . , fm(x)), where the functions
are expensive to evaluate. In such a context, Bayesian methods relying on Gaussian Processes
(GP) [1], adapted to multi-objective problems [2] have allowed to approximate Pareto fronts in
a limited number of iterations.
In the current work, we assume that the Pareto front center has already been attained (typically
with the approach described in [3]) and that a computational budget remains. The goal is to
uncover of a broader central part of the Pareto front: the intersection of it with some region
to target, IR(see Fig. 1). IR
has however to be defined carefully: choosing it too wide, i.e.
too ambitious with regard to the remaining budget, will lead to a non converged approximation
front. Conversely, a suboptimal diversity of Pareto optimal solutions will be obtained if choosing
a too narrow area.
The GPs allow to forecast the future behavior of the algorithm: they are used in lieu of the
true functions to anticipate which inputs/outputs will be obtained when targeting growing
parts of the Pareto front. Virtual final Pareto fronts corresponding to a possible version of the
approximation front at the depletion of the budget are produced for each IR. A measure of
uncertainty is defined and applied to all of them to determine the optimal improvement region IR∗, balancing the size of the approximation front and the convergence to the Pareto front.
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