A recurring task in research is iterative optimization of a process that can be evaluated only by conducting an experiment. Powerful algorithms for assisting this process exist, but they largely ignore the valuable knowledge of expert scientists. We consider a problem within this general scope, not aiming to automate the optimization but instead studying how to infer tacit expert knowledge. This complements the current literature focusing on how such information is used in the optimization process, paying little attention on how the information is obtained. We consider a new formulation where the expertise is inferred by passively observing a human solving an optimization problem, without requiring explicit elicitation techniques. Our solution leverages concepts from Bayesian optimization (BO) commonly used for automating the optimization, but now these tools are used as a theoretical model for the user behavior instead. We assume the expert solves the task approximately in the same manner as a BO algorithm would and solve what kind of prior knowledge about the target function is consistent with the sequence of choices they made. We introduce the problem and a concrete solution, and show that the recovered priors match the true priors in controlled simulated studies. We also empirically evaluate the robustness of the method against violations of the modeling assumptions and demonstrate it on real user data.

We study a class of optimization problems in the Wasserstein space (the space of probability measures) where the objective function is nonconvex along generalized geodesics. Specifically, the objective exhibits some difference-of-convex structure along these geodesics. The setting also encompasses sampling problems where the logarithm of the target distribution is difference-of-convex. We derive multiple convergence insights for a novel semi Forward-Backward Euler scheme under several nonconvex (and possibly nonsmooth) regimes. Notably, the semi Forward-Backward Euler is just a slight modification of the Forward-Backward Euler whose convergence is---to our knowledge---still unknown in our very general non-geodesically-convex setting.

Eliciting a high-dimensional probability distribution from an expert via noisy judgments is notoriously challenging, yet useful for many applications, such as prior elicitation and reward modeling. We introduce a method for eliciting the expert's belief density as a normalizing flow based solely on preferential questions such as comparing or ranking alternatives. This allows eliciting in principle arbitrarily flexible densities, but flow estimation is susceptible to the challenge of collapsing or diverging probability mass that makes it difficult in practice. We tackle this problem by introducing a novel functional prior for the flow, motivated by a decision-theoretic argument, and show empirically that the belief density can be inferred as the function-space maximum a posteriori estimate. We demonstrate our method by eliciting multivariate belief densities of simulated experts, including the prior belief of a general-purpose large language model over a real-world dataset.

Bayesian optimization (BO) is a powerful framework for optimizing black-box, expensive-to-evaluate functions. Over the past decade, many algorithms have been proposed to integrate cheaper, lower-fidelity approximations of the objective function into the optimization process, with the goal of converging towards the global optimum at a reduced cost. This task is generally referred to as multi-fidelity Bayesian optimization (MFBO). However, MFBO algorithms can lead to higher optimization costs than their vanilla BO counterparts, especially when the low-fidelity sources are poor approximations of the objective function, therefore defeating their purpose. To address this issue, we propose rMFBO (robust MFBO), a methodology to make any GP-based MFBO scheme robust to the addition of unreliable information sources. rMFBO comes with a theoretical guarantee that its performance can be bound to its vanilla BO analog, with high controllable probability. We demonstrate the effectiveness of the proposed methodology on a number of numerical benchmarks, outperforming earlier MFBO methods on unreliable sources. We expect rMFBO to be particularly useful to reliably include human experts with varying knowledge within BO processes.

Bayesian optimization is an effective method for finding extrema of a black-box function. We propose a new type of Bayesian optimization for learning user preferences in high-dimensional spaces. The central assumption is that the underlying objective function cannot be evaluated directly, but instead a minimizer along a projection can be queried, which we call a projective preferential query. The form of the query allows for feedback that is natural for a human to give, and which enables interaction. This is demonstrated in a user experiment in which the user feedback comes in the form of optimal position and orientation of a molecule adsorbing to a surface. We demonstrate that our framework is able to find a global minimum of a high-dimensional black-box function, which is an infeasible task for existing preferential Bayesian optimization frameworks that are based on pairwise comparisons.