Many scientific and industrial applications require the joint optimization of multiple, potentially competing objectives. Multi-objective Bayesian optimization (MOBO) is a sample-efficient framework for identifying Pareto-optimal solutions. At the heart of MOBO is the acquisition function, which determines the next candidate to evaluate by navigating the best compromises among the objectives. Acquisition functions that rely on integrating over the objective space scale poorly to a large number of objectives. In this paper, we show a natural connection between the non-dominated solutions and the highest multivariate rank, which coincides with the extreme level line of the joint cumulative distribution function (CDF). Motivated by this link, we propose the CDF indicator, a Pareto-compliant metric for evaluating the quality of approximate Pareto sets, that can complement the popular hypervolume indicator. We then introduce an acquisition function based on the CDF indicator, called BOtied. BOtied can be implemented efficiently with copulas, a statistical tool for modeling complex, high-dimensional distributions. Our experiments on a variety of synthetic and real-world experiments demonstrate that BOtied outperforms state-of-the-art MOBO algorithms while being computationally efficient for many objectives.

We propose a new score-based approach to generate 3D molecules represented as atomic densities on regular grids. First, we train a denoising neural network that learns to map from a smooth distribution of noisy molecules to the distribution of real molecules. Then, we follow the _neural empirical Bayes_ framework [Saremi and Hyvarinen, 2019] and generate molecules in two steps: (i) sample noisy density grids from a smooth distribution via underdamped Langevin Markov chain Monte Carlo, and (ii) recover the "clean" molecule by denoising the noisy grid with a single step. Our method, _VoxMol_, generates molecules in a fundamentally different way than the current state of the art (ie, diffusion models applied to atom point clouds). It differs in terms of the data representation, the noise model, the network architecture and the generative modeling algorithm. Our experiments show that VoxMol captures the distribution of drug-like molecules better than state of the art, while being faster to generate samples.