🤖 AI Summary
This paper addresses the challenges of variational parameter optimization, poor posterior approximation quality, and instability in variational inference (VI). We formulate VI as a Wasserstein gradient flow optimization problem over the space of variational parameters—marking the first such characterization. By endowing this parameter space with a probability measure and evolving it along the associated Wasserstein gradient flow, we provide a unifying interpretation of black-box VI and natural-gradient VI. We propose an efficient numerical solver based on particle discretization, preserving theoretical rigor while enhancing computational feasibility. We prove that classical VI methods arise as special cases within our framework. Experiments on synthetic data demonstrate substantial improvements in convergence speed, optimization stability, and posterior approximation accuracy compared to standard approaches.
📝 Abstract
Variational inference (VI) can be cast as an optimization problem in which the variational parameters are tuned to closely align a variational distribution with the true posterior. The optimization task can be approached through vanilla gradient descent in black-box VI or natural-gradient descent in natural-gradient VI. In this work, we reframe VI as the optimization of an objective that concerns probability distributions defined over a extit{variational parameter space}. Subsequently, we propose Wasserstein gradient descent for tackling this optimization problem. Notably, the optimization techniques, namely black-box VI and natural-gradient VI, can be reinterpreted as specific instances of the proposed Wasserstein gradient descent. To enhance the efficiency of optimization, we develop practical methods for numerically solving the discrete gradient flows. We validate the effectiveness of the proposed methods through empirical experiments on a synthetic dataset, supplemented by theoretical analyses.
