Targeted Variance Reduction: Robust Bayesian Optimization of Black-Box
  Simulators with Noise Parameters
John Joshua Miller Simon Mak
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.