Zachary Jones, PhD student, Platon Team
Title: Stochastic Tangential Pareto Dynamics for Pareto Front Estimation and Uncertainty Quantification
Abstract:
The framework of stochastic multi-objective programming allows for the inclusion of uncertainties in multi-objective optimization problems at the cost of transforming the set of objectives into a set of expectations of random quantities.Of the family of Robins-Monro algorithms, the stochastic multi-gradient algorithm (SMGDA) gives a solution to these types of problems without ever having to calculate the expected values of the objectives or their gradients.However, a bias in the algorithm causes it to converge to only a subset of the whole Pareto front, limiting its use.
To sample the whole Pareto front, we reduce the bias of the stochastic multi-gradient calculation using an exponential smoothing technique, and promote the exploration of the Pareto front by adding non-vanishing noise tangential to the front.We probabilistically estimate and rank members of the Pareto set, using only the sequence generated during optimization, while also providing 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 few evaluations of the random quantities of interest, which is valuable in the context of costly model evaluations.
We prove that our algorithm, stochastic tangential pareto dynamics (STPD), generates samples in a concentrated set containing the whole Pareto front, and we illustrate the efficacy of our approach with numerical examples in increasing dimension.
