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The Stein-log-Sobolev inequality and the exponential rate of convergence for the continuous Stein variational gradient descent method

José A. Carrillo, Jakub Skrzeczkowski, Jethro Warnett

TL;DR

This work establishes a rigorous foundation for exponential convergence of the continuous Stein variational gradient descent by proving the Stein-log-Sobolev inequality (SLSI) for a broad class of kernels. A central innovation is interpreting the Stein-Fisher information as a duality pairing between $H^{-1}(\mathbb{R}^d)$ and $H^{1}(\mathbb{R}^d)$ and leveraging Fourier analysis to derive bounds via a kernel ansatz $K(x,y)=e^{V(x)-V_0(x)/2}\,k(x-y)\,e^{V(y)-V_0(y)/2}$. The paper constructs kernels whose Fourier transforms have quadratic decay and provides explicit constants for the SLSI, while also proving the existence of weak solutions to the mean-field Stein gradient flow with exponential decay toward equilibrium. It further delineates conditions under which SLSI can fail, highlighting the necessity of the proposed assumptions and offering guidance for kernel design in SVGD. Overall, the results deliver a rigorous, quantitative pathway to exponential convergence in continuous SVGD and clarify the role of kernel choice in achieving this behavior.

Abstract

The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the equilibrium. The main novelty in our approach is to interpret the Stein-Fisher information, also called the squared Stein discrepancy, as a duality pairing between $H^{-1}(\mathbb{R}^d)$ and $H^{1}(\mathbb{R}^d)$, which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions.

The Stein-log-Sobolev inequality and the exponential rate of convergence for the continuous Stein variational gradient descent method

TL;DR

This work establishes a rigorous foundation for exponential convergence of the continuous Stein variational gradient descent by proving the Stein-log-Sobolev inequality (SLSI) for a broad class of kernels. A central innovation is interpreting the Stein-Fisher information as a duality pairing between and and leveraging Fourier analysis to derive bounds via a kernel ansatz . The paper constructs kernels whose Fourier transforms have quadratic decay and provides explicit constants for the SLSI, while also proving the existence of weak solutions to the mean-field Stein gradient flow with exponential decay toward equilibrium. It further delineates conditions under which SLSI can fail, highlighting the necessity of the proposed assumptions and offering guidance for kernel design in SVGD. Overall, the results deliver a rigorous, quantitative pathway to exponential convergence in continuous SVGD and clarify the role of kernel choice in achieving this behavior.

Abstract

The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the equilibrium. The main novelty in our approach is to interpret the Stein-Fisher information, also called the squared Stein discrepancy, as a duality pairing between and , which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions.

Paper Structure

This paper contains 21 sections, 28 theorems, 272 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure of the paper
  4. Preliminaries and Notation
  5. Existence of Kernels that satisfy the Stein-log-Sobolev-Inequality
  6. Fisher information in Fourier variables
  7. Proof of Theorem \ref{['thm:existence_of_maternised_convolution_kernel_that_satisfy_slsi']} in 1D
  8. Sufficient condition on $\hat{k}$ and $q$
  9. Reconstructing kernels from weights
  10. Proof of the existence of kernel in higher dimensions
  11. Weak Solutions to the Stein Gradient Flow
  12. Uniform estimates for \ref{['thm:existence_of_distributional_solutions_to_mfsvgd']}
  13. Passing to the limit $R\rightarrow\infty$
  14. Necessary Properties of the Kernel
  15. Proof of Theorem \ref{['thm:conditions_for_slsi']}, Case \ref{['thm:failure_of_slsi_for_regular_convolution_kernels']}
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\rho_\infty$, $V$ be as in eq:general_prop_target_potentialV_f with $\mathcal{C}\in\mathbb{R}$, $\mu\in\mathbb{R}^{d}$ and $\Sigma\in \mathbb{R}^{d\times d}$ a strictly positive definite and symmetric matrix. Then for any kernel $k\in L^1(\mathbb{R}^{d})+L^2(\mathbb{R}^{d})$ that satisfies the

Figures (2)

  • Figure 1: The value $\frac{\alpha}{\hat{k}_{0,1}(\varepsilon)}\left(\frac{4 \pi \varepsilon}{p_{d/2,1}}\right)^2$ ($y$-axis, log-scale) vs the dimension $d$ ($x$-axis). The constraint \ref{['thm:existence of maternised convolution kernel that satisfy slsi:eq2']} is only satisfied for $d=1$.
  • Figure 2: The shape of the kernel frequency $\hat{k}_{0,d}$ with $\alpha=1$ and $\varepsilon=0.05, 0.1$ for different dimensions $d$. On the $x$ axis we have the radius $r$, and on the $y$ axis we have the value $\hat{k}_{0,d}(r)$. Contrary to the kernel which satisfies \ref{['eq:inequality_l2_lower_bound_formulated_H-1_H1']} in one dimension $\hat{k}_{0,d}(\xi) = \frac{1}{1+C\,|\xi|^2}$, the ones constructed for higher dimensions first increase and then decrease to 0 at infinity.

Theorems & Definitions (61)

  • Theorem 1.1: existence of kernels satisfying \ref{['eq:inequality_l2_lower_bound_formulated_H-1_H1']} and \ref{['eq:stein_log_sobolev_inequality']}
  • Remark 1.2
  • Theorem 1.3: existence of weak solutions to \ref{['eq:stein_gradient_descent_kullback_leibler']}
  • Theorem 1.4: Conditions for failure of \ref{['eq:stein_log_sobolev_inequality_general_alpha']}
  • Lemma 3.1
  • proof
  • Remark 3.2
  • proof : Proof of Theorem \ref{['thm:existence_of_maternised_convolution_kernel_that_satisfy_slsi']} in 1D
  • Remark 3.3
  • Lemma 3.4: sufficient conditions on $\hat{k}$ and $q$
  • ...and 51 more