Particle-based energetic variational inference

We introduce a new variational inference (VI) framework, called energetic variational inference (EVI). It minimizes the VI objective function based on a prescribed energy-dissipation law. Using the EVI framework, we can derive many existing particle-based variational inference (ParVI) methods, including the popular Stein variational gradient descent (SVGD). More importantly, many new ParVI schemes can be created under this framework. For illustration, we propose a new particle-based EVI scheme, which performs the particle-based approximation of the density first and then uses the approximated density in the variational procedure, or “Approximation-then-Variation” for short. Thanks to this order of approximation and variation, the new scheme can maintain the variational structure at the particle level, and can significantly decrease the KL-divergence in each iteration. Numerical experiments show the proposed method outperforms some existing ParVI methods in terms of fidelity to the target distribution.