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 $\{V(x^i)\}_i$ of some potential $V$ in an ensemble $\{x^i\}_i$ 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).

翻译：我们提出一种基于函数评估与贝叶斯推理的方法，用于从给定粒子集成中提取目标函数的高阶微分信息。对势函数$V$在集成$\{x^i\}_i$中的逐点评估$\{V(x^i)\}_i$隐含着关于一阶或高阶导数的信息，通过少量计算即可显式提取（基于集成的梯度推理——EGI）。我们建议将此信息用于改进共识优化和朗之万采样器等现有基于集成的数值优化与采样方法。数值研究表明，增强算法通常优于其无梯度变体，尤其在帮助粒子集成逃离初始域、探索多模态非高斯场景以及加速优化动力学末期收敛方面表现突出。本文数值示例的代码可在论文的GitHub仓库（https://github.com/MercuryBench/ensemble-based-gradient.git）中获取。