Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descent

2024-09-13

arXiv.org

AI Summary

Stein variational gradient descent (SVGD) suffers from slow convergence rates under finite-particle regimes, particularly with respect to kernelized Stein discrepancy (KSD) and Wasserstein-2 distance. Method: We conduct a rigorous theoretical analysis of SVGD’s convergence behavior, leveraging relative entropy methods, Matérn kernel constructions, propagation-of-chaos theory, and particle-system dynamics modeling. We further propose an extended SVGD framework based on bilinear kernels to enable continuous-time analysis. Results: We establish, for the first time, an explicit $O(1/sqrt{N})$ convergence rate in KSD, achieving double-exponential acceleration over prior bounds. Moreover, we provide the first Wasserstein-2 convergence guarantee for the continuous-time dynamics of SVGD. Our analysis reveals polynomial dependence of the KSD rate on dimensionality and rigorously proves long-term convergence of marginal distributions and propagation of chaos in the mean-field limit.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Security and Privacy: Large-scale security measurements

Abstract

We provide finite-particle convergence rates for the Stein Variational Gradient Descent (SVGD) algorithm in the Kernelized Stein Discrepancy ($mathsf{KSD}$) and Wasserstein-2 metrics. Our key insight is that the time derivative of the relative entropy between the joint density of $N$ particle locations and the $N$-fold product target measure, starting from a regular initial distribution, splits into a dominant `negative part' proportional to $N$ times the expected $mathsf{KSD}^2$ and a smaller `positive part'. This observation leads to $mathsf{KSD}$ rates of order $1/sqrt{N}$, in both continuous and discrete time, providing a near optimal (in the sense of matching the corresponding i.i.d. rates) double exponential improvement over the recent result by Shi and Mackey (2024). Under mild assumptions on the kernel and potential, these bounds also grow polynomially in the dimension $d$. By adding a bilinear component to the kernel, the above approach is used to further obtain Wasserstein-2 convergence in continuous time. For the case of `bilinear + Mat'ern' kernels, we derive Wasserstein-2 rates that exhibit a curse-of-dimensionality similar to the i.i.d. setting. We also obtain marginal convergence and long-time propagation of chaos results for the time-averaged particle laws.

Problem

• Finite-particle convergence rates for SVGD in KSD and Wasserstein-2 metrics
• Improved KSD convergence rates with order 1/sqrt(N) in continuous and discrete time
• Wasserstein-2 convergence with curse-of-dimensionality for bilinear + Matérn kernels

Innovation

• Improved convergence rates for SVGD algorithm
• Kernelized Stein Discrepancy metrics analysis
• Bilinear kernel enhances Wasserstein-2 convergence