Bayesian Multi-objective Optimization with Noisy Evaluations using the Knowledge Gradient

Slides presented at the PGMO Days 2019, held the 3rd and 4th December 2019 at EDF Lab Paris-Saclay. Abstract: We consider the problem of multi-objective optimization in the case where each objective is a stochastic black box that provides noisy evaluation results. More precisely, let \(f_1, \ldots f_q\) be \(q\) real-valued objective functions defined on a search domain \(\mathbb{X} \subset \mathbb{R}^d\), and assume that, for each \(x\in\mathbb{X}\), we can observe a noisy version of the objectives: \(Z_1 = f_1(x) + \varepsilon_1\), ..., \(Z_q = f_q(x) + \varepsilon_q\), where the \(\varepsilon_i\)s are zero-mean random variables. Our objective is to estimate the Pareto-optimal solutions of the problem: \(\begin{equation} \label{eq:1} \min f_1,\,\ldots,\,f_q\,. \end{equation}\) We adopt a Bayesian optimization approach, which is a classical approach when the affordable number of evaluations is severely limited. In essence, Bayesian optimization consists in choosing a probabilistic model for the outputs \(Z_i\) and defining a sampling criterion to select evaluation points in the search domain \(\mathbb{X}\). Here, we propose to discuss the extension of the Knowledge Gradient approach for solving the multi-objective problem