Ensemble-based gradient inference for particle methods in optimization and sampling

We propose an approach based on function evaluations and Bayesian inference to extract higher-order differential information of objective functions from a given ensemble of particles. Pointwise evaluation V(xi)i of some potential V in an ensemble xii contains implicit information about first or higher order derivatives, which can be made explicit with little computational effort (ensemble-based gradient inference -- EGI). We suggest to use this information for the improvement of established ensemble-based numerical methods for optimization and sampling such as Consensus-based optimization and Langevin-based samplers. Numerical studies indicate that the augmented algorithms are often superior to their gradient-free variants, in particular the augmented methods help the ensembles to escape their initial domain, to explore multimodal, non-Gaussian settings and to speed up the collapse at the end of optimization dynamics. The code for the numerical examples in this manuscript can be found in the paper's Github repository (https://github.com/MercuryBench/ensemble-based-gradient.git).