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Long-time asymptotics of noisy SVGD outside the population limit

Victor Priser, Pascal Bianchi, Adil Salim

TL;DR

This work establishes that the limit set of noisy SVGD for large is well-defined and characterize this limit set, showing that it approaches the target distribution as increases, and proves that noisy SVGD provably avoids the variance collapse observed for SVGD.

Abstract

Stein Variational Gradient Descent (SVGD) is a widely used sampling algorithm that has been successfully applied in several areas of Machine Learning. SVGD operates by iteratively moving a set of interacting particles (which represent the samples) to approximate the target distribution. Despite recent studies on the complexity of SVGD and its variants, their long-time asymptotic behavior (i.e., after numerous iterations ) is still not understood in the finite number of particles regime. We study the long-time asymptotic behavior of a noisy variant of SVGD. First, we establish that the limit set of noisy SVGD for large is well-defined. We then characterize this limit set, showing that it approaches the target distribution as increases. In particular, noisy SVGD provably avoids the variance collapse observed for SVGD. Our approach involves demonstrating that the trajectories of noisy SVGD closely resemble those described by a McKean-Vlasov process.

Long-time asymptotics of noisy SVGD outside the population limit

TL;DR

This work establishes that the limit set of noisy SVGD for large is well-defined and characterize this limit set, showing that it approaches the target distribution as increases, and proves that noisy SVGD provably avoids the variance collapse observed for SVGD.

Abstract

Stein Variational Gradient Descent (SVGD) is a widely used sampling algorithm that has been successfully applied in several areas of Machine Learning. SVGD operates by iteratively moving a set of interacting particles (which represent the samples) to approximate the target distribution. Despite recent studies on the complexity of SVGD and its variants, their long-time asymptotic behavior (i.e., after numerous iterations ) is still not understood in the finite number of particles regime. We study the long-time asymptotic behavior of a noisy variant of SVGD. First, we establish that the limit set of noisy SVGD for large is well-defined. We then characterize this limit set, showing that it approaches the target distribution as increases. In particular, noisy SVGD provably avoids the variance collapse observed for SVGD. Our approach involves demonstrating that the trajectories of noisy SVGD closely resemble those described by a McKean-Vlasov process.
Paper Structure (37 sections, 21 theorems, 133 equations, 2 figures, 1 algorithm)

This paper contains 37 sections, 21 theorems, 133 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related works
  3. Contributions
  4. Paper structure
  5. Background
  6. Notations
  7. Optimal transport
  8. Functional inequalities
  9. Noisy Stein Variational Gradient Descent
  10. Convergence results of noisy SVGD
  11. Limit set of noisy SVGD is well-defined
  12. Description of the limit set
  13. Long-time convergence of the empirical measure
  14. Overview of the convergence proof and dynamical behavior of noisy SVGD
  15. Interpolated process
  16. ...and 22 more sections

Key Result

Lemma 1

Let Assumptions hyp:algo and hyp:stab be satisfied. Assume $\lambda >0$. Then, $\sup_{k,n} {\mathbb E}\| X^{1,n}_k \|^4 <\infty$.

Figures (2)

  • Figure 1: Dimension-averaged Marginal Variance of SVGD and noisy SVGD at convergence for sampling from a standard Gaussian.
  • Figure 2: Dimension-averaged Marginal Variance of SVGD and noisy SVGD at convergence for sampling from a standard Gaussian.

Theorems & Definitions (33)

  • Definition 1: Logarithmic Sobolev Inequality
  • Definition 2: Distributional limit set
  • Lemma 1
  • Theorem 1
  • Definition 3
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Definition 4
  • Proposition 1
  • ...and 23 more