Diverse Expected Improvement (DEI): Diverse Bayesian Optimization of Expensive Computer Simulators

The optimization of expensive black-box simulators arises in a myriad of modern scientific and engineering applications. Bayesian optimization provides an appealing solution, by leveraging a fitted surrogate model to guide the selection of subsequent simulator evaluations. In practice, however, the objective is often not to obtain a single good solution, but rather a ''basket'' of good solutions from which users can choose for downstream decision-making. This need arises in our motivating application for real-time control of internal combustion engines for flight propulsion, where a diverse set of control strategies is essential for stable flight control. There has been little work on this front for Bayesian optimization. We thus propose a new Diverse Expected Improvement (DEI) method that searches for diverse ''$\epsilon$-optimal'' solutions: locally-optimal solutions within a tolerance level $\epsilon > 0$ from a global optimum. We show that DEI yields a closed-form acquisition function under a Gaussian process surrogate model, which facilitates efficient sequential queries via automatic differentiation. This closed form further reveals a novel exploration-exploitation-diversity trade-off, which incorporates the desired diversity property within the well-known exploration-exploitation trade-off. We demonstrate the improvement of DEI over existing methods in a suite of numerical experiments, then explore the DEI in two applications on rover trajectory optimization and engine control for flight propulsion.

Targeted Variance Reduction: Robust Bayesian Optimization of Black-Box Simulators with Noise Parameters

The optimization of a black-box simulator over control parameters $\mathbf{x}$ arises in a myriad of scientific applications. In such applications, the simulator often takes the form $f(\mathbf{x},\boldsymbol{\theta})$, where $\boldsymbol{\theta}$ are parameters that are uncertain in practice. Robust optimization aims to optimize the objective $\mathbb{E}[f(\mathbf{x},\boldsymbol{\Theta})]$, where $\boldsymbol{\Theta} \sim \mathcal{P}$ is a random variable that models uncertainty on $\boldsymbol{\theta}$. For this, existing black-box methods typically employ a two-stage approach for selecting the next point $(\mathbf{x},\boldsymbol{\theta})$, where $\mathbf{x}$ and $\boldsymbol{\theta}$ are optimized separately via different acquisition functions. As such, these approaches do not employ a joint acquisition over $(\mathbf{x},\boldsymbol{\theta})$, and thus may fail to fully exploit control-to-noise interactions for effective robust optimization. To address this, we propose a new Bayesian optimization method called Targeted Variance Reduction (TVR). The TVR leverages a novel joint acquisition function over $(\mathbf{x},\boldsymbol{\theta})$, which targets variance reduction on the objective within the desired region of improvement. Under a Gaussian process surrogate on $f$, the TVR acquisition can be evaluated in closed form, and reveals an insightful exploration-exploitation-precision trade-off for robust black-box optimization. The TVR can further accommodate a broad class of non-Gaussian distributions on $\mathcal{P}$ via a careful integration of normalizing flows. We demonstrate the improved performance of TVR over the state-of-the-art in a suite of numerical experiments and an application to the robust design of automobile brake discs under operational uncertainty.