In this paper, we present a flow-based method for global optimization of continuous Sobolev functions, called Stein Boltzmann Sampling (SBS). SBS initializes uniformly a number of particles representing candidate solutions, then uses the Stein Variational Gradient Descent (SVGD) algorithm to sequentially and deterministically move those particles in order to approximate a target distribution whose mass is concentrated around promising areas of the domain of the optimized function. The target is chosen to be a properly parametrized Boltzmann distribution. For the purpose of global optimization, we adapt the generic SVGD theoretical framework allowing to address more general target distributions over a compact subset of $\mathbb{R}^d$, and we prove SBS's asymptotic convergence. In addition to the main SBS algorithm, we present two variants: the SBS-PF that includes a particle filtering strategy, and the SBS-HYBRID one that uses SBS or SBS-PF as a continuation after other particle- or distribution-based optimization methods. A detailed comparison with state-of-the-art methods on benchmark functions demonstrates that SBS and its variants are highly competitive, while the combination of the two variants provides the best trade-off between accuracy and computational cost.

翻译：本文提出了一种基于流的连续索博列夫函数全局优化方法，称为斯坦玻尔兹曼采样。SBS首先均匀初始化一批表示候选解的粒子，随后利用斯坦变分梯度下降算法对这些粒子进行顺序且确定性的移动，以逼近一个目标分布，该分布的质量集中在被优化函数定义域内有前景的区域。目标分布被选取为适当参数化的玻尔兹曼分布。针对全局优化的目的，我们改进了通用的SVGD理论框架，使其能够处理定义在$\mathbb{R}^d$的紧子集上的更一般的目标分布，并证明了SBS的渐近收敛性。除了主SBS算法外，我们还提出了两种变体：包含粒子滤波策略的SBS-PF，以及将SBS或SBS-PF作为其他基于粒子或分布的优化方法之后续步骤的SBS-HYBRID。在基准函数上与最先进方法的详细比较表明，SBS及其变体具有很强的竞争力，而两种变体的结合在精度与计算成本之间提供了最佳权衡。