Sampling and Estimating the Set of Pareto Optimal Solutions in Stochastic Multi-Objective Optimization
Résumé
Including uncertainty sources in multi-objective optimization allows more robust design
decisions at the cost of dealing with stochastic quantities of interest. The objectives
are then classically defined as the expectations of random quantities of interest, and
their estimation can be computationally expensive. Robbins-Monro-type algorithms are
an attractive alternative in this context, allowing for the minimization of the objectives
through noisy gradient updates. The stochastic multi-gradient algorithm (SMGDA)[1]
extends the Robbins-Monro approach to the multi-objective case. However, a bias in the
algorithm and the inherent noise in stochastic gradients cause the algorithm to converge
to only a subset of the whole Pareto front. Our contribution builds on the stochastic
multi-gradient algorithm to reliably estimate the whole Pareto front (objective space)
and Pareto optimal points (design space).
First, we reduce the bias of the stochastic multi-gradient calculation using an exponential
smoothing technique. Second, we simultaneously remove the remaining bias and promote
the exploration of the Pareto front by adding non-vanishing noise tangential to the front.
We prove that this algorithm generates samples in a concentrated set containing the whole
Pareto front. Finally, we estimate the set of Pareto optimal design points using only
the sequence generated during optimization. We also provide bootstrapped confidence
intervals using a nearest-neighbor model calibrated with a novel procedure based on the
hypervolume metric.
Our proposed method allows for the estimation of the whole of the Pareto front using
significantly fewer evaluations of the random quantities of interest when compared to a
direct sample-based estimation, which is valuable in the context of costly model evalua-
tions. We illustrate the efficacy of our approach with numerical examples in increasing
dimension and discuss how to apply the method to more complex problems.
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