Stein Boltzmann Sampling: A Variational Approach for Global Optimization

Abstract

In this paper, we present a deterministic particle-based method for global optimization of continuous Sobolev functions, called \emph{Stein Boltzmann Sampling} (SBS). SBS initializes uniformly a number of particles representing candidate solutions, then uses the \emph{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.