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On the mean-field limit for Stein variational gradient descent: stability and multilevel approximation

Simon Weissmann, Jakob Zech

TL;DR

This paper tackles the computational challenge of Bayesian inference with expensive likelihoods by developing a multilevel Stein variational gradient descent (ML-SVGD) method. By coupling interacting particle systems across multiple likelihood-approximation levels, the authors derive a mean-field stability framework and an estimator that leverages telescoping differences to reduce variance and computational cost. They prove explicit error bounds for the multilevel estimator and analyze complexity, showing significant speedups over single-level SVGD under realistic assumptions on level accuracy. A PDE-driven inverse problem demonstrates the practical gain, confirming the theoretical predictions and highlighting the method's potential for scalable Bayesian computation in expensive forward-model contexts.

Abstract

In this paper we propose and analyze a novel multilevel version of Stein variational gradient descent (SVGD). SVGD is a recent particle based variational inference method. For Bayesian inverse problems with computationally expensive likelihood evaluations, the method can become prohibitive as it requires to evolve a discrete dynamical system over many time steps, each of which requires likelihood evaluations at all particle locations. To address this, we introduce a multilevel variant that involves running several interacting particle dynamics in parallel corresponding to different approximation levels of the likelihood. By carefully tuning the number of particles at each level, we prove that a significant reduction in computational complexity can be achieved. As an application we provide a numerical experiment for a PDE driven inverse problem, which confirms the speed up suggested by our theoretical results.

On the mean-field limit for Stein variational gradient descent: stability and multilevel approximation

TL;DR

This paper tackles the computational challenge of Bayesian inference with expensive likelihoods by developing a multilevel Stein variational gradient descent (ML-SVGD) method. By coupling interacting particle systems across multiple likelihood-approximation levels, the authors derive a mean-field stability framework and an estimator that leverages telescoping differences to reduce variance and computational cost. They prove explicit error bounds for the multilevel estimator and analyze complexity, showing significant speedups over single-level SVGD under realistic assumptions on level accuracy. A PDE-driven inverse problem demonstrates the practical gain, confirming the theoretical predictions and highlighting the method's potential for scalable Bayesian computation in expensive forward-model contexts.

Abstract

In this paper we propose and analyze a novel multilevel version of Stein variational gradient descent (SVGD). SVGD is a recent particle based variational inference method. For Bayesian inverse problems with computationally expensive likelihood evaluations, the method can become prohibitive as it requires to evolve a discrete dynamical system over many time steps, each of which requires likelihood evaluations at all particle locations. To address this, we introduce a multilevel variant that involves running several interacting particle dynamics in parallel corresponding to different approximation levels of the likelihood. By carefully tuning the number of particles at each level, we prove that a significant reduction in computational complexity can be achieved. As an application we provide a numerical experiment for a PDE driven inverse problem, which confirms the speed up suggested by our theoretical results.
Paper Structure (21 sections, 12 theorems, 80 equations, 2 figures, 2 tables)

This paper contains 21 sections, 12 theorems, 80 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contributions:
  3. Preliminaries and notation
  4. Stein variational gradient descent
  5. Multilevel Stein Variational gradient descent
  6. Single level approximation
  7. Multilevel approximation
  8. Convergence analysis
  9. Main result
  10. Stability of the mean-field equation
  11. Variance reduction: Combined stability and mean-field limit
  12. Complexity analysis
  13. Numerical results
  14. Conclusions
  15. Illustration of the multilevel mean-field particle approximation
  16. ...and 6 more sections

Key Result

Lemma 2.2

Under Assumption ass:kernel2 there exists $\mathop{\mathrm{c_{lip}}}\nolimits<\infty$ depending on the constants in Assumption ass:kernel2, such that the mapping $(z,\rho)\mapsto P_\rho\nabla\log\left(\frac{\rho}{\pi}\right)$ is $\mathop{\mathrm{c_{lip}}}\nolimits$-Lipschitz in the sense

Figures (2)

  • Figure 1: Error convergence for $10$ iterations (left), $100$ iterations (middle) and $200$ iterations (right).
  • Figure 2: Illustration of the multilevel mean-field particle approximation.

Theorems & Definitions (21)

  • Lemma 2.2: Lemma 14 in NEURIPS20203202111c
  • Lemma 2.3: Lemma 13 in NEURIPS20203202111c
  • Proposition 2.4: Proposition 7 in NEURIPS20203202111c
  • Proposition 3.2
  • Remark 3.4
  • Remark 3.5
  • Theorem 4.1: Error bound
  • Theorem 4.2
  • Proposition 4.3
  • Lemma 4.4
  • ...and 11 more