Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descent (2409.08469v3)

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 smallerpositive 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.

• The paper introduces a novel method connecting the joint density of particles with relative entropy, splitting its time derivative into dominant negative and minor positive components.
• It establishes an improved O(1/√N) convergence rate for the Kernel Stein Discrepancy under mild kernel and potential assumptions.
• The study extends its approach to derive Wasserstein-2 convergence rates and demonstrates marginal convergence, implying propagation of chaos.

Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descent

The paper, "Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descent," authored by Krishnakumar Balasubramanian, Sayan Banerjee, and Promit Ghosal, provides significant advancements in the understanding of the Stein Variational Gradient Descent (SVGD) algorithm. This algorithm is pivotal for sampling from complex, high-dimensional probability distributions, and the paper presents novel contributions primarily in the field of finite-particle convergence rates.

The primary focus is on deriving rates of convergence of SVGD in both the Kernel Stein Discrepancy (KSD) and Wasserstein-2 ($W_2$) metrics. These rates are crucial for practical applications and theoretical understanding as they quantify how quickly and accurately the algorithm can approximate the target distribution with a finite number of particles.

Key Insights and Contributions

The authors present several contributions to the field:

1. New Technique for Finite-Particle Convergence:
   • They analyze the joint density of $N$ particle locations and connect its relative entropy with the $N$-fold product target measure $\pi^{\otimes N}$.
   • This novel approach allows them to split the time derivative of the relative entropy into a dominant negative component proportional to $N$ times the expected KSD$^2$ and a much smaller positive component.
2. KSD Convergence Rates:
   • The paper establishes KSD convergence rates of order $1/\sqrt{N}$, which is a significant improvement over previous results that relied on double-exponential bounds in time.
   • Under mild assumptions on the kernel and potential, these KSD bounds grow linearly with the dimension $d$.
3. Wasserstein-2 Convergence:
   • By introducing a bilinear component to the SVGD kernel, they extend their approach to obtain $W_2$ convergence rates.
   • The derived $W_2$ rates for certain kernels, like the Matérn kernel, reveal the curse of dimensionality similar to the independent and identically distributed (i.i.d.) setting.
4. Marginal Convergence and Propagation of Chaos:
   • The paper also shows that under exchangeable initial conditions, the time-averaged particle laws converge weakly to the target distribution, which implies propagation of chaos in long-time regimes.

Implications and Future Directions

These results have significant implications for both the theoretical and practical aspects of SVGD:

• Theoretical Insights:
  • The new approach directly connects the joint density evolution of particles with their empirical distribution, providing a deeper understanding of the SVGD dynamics.
  • The optimal $O(1/\sqrt{N})$ rates for KSD highlight the algorithm's efficiency in approximating complex distributions, which is crucial for high-dimensional problems.
• Practical Applications:
  • The improved convergence rates can lead to more efficient implementations of SVGD in various fields such as machine learning, Bayesian inference, and applied mathematics.
  • Understanding the $W_2$ bounds, despite their dimensionality dependence, provides valuable insights for using SVGD in practical scenarios where Wasserstein distances are relevant.

Conclusion

This paper offers substantial advancements in the analysis of the Stein Variational Gradient Descent algorithm, particularly in finite-particle settings. The improved KSD rates and the novel connections drawn between the joint particle density and empirical measures stand out as key contributions. These findings are expected to impact both theoretical research and practical applications in high-dimensional sampling and inference tasks. Future developments may focus on exploring these techniques further, potentially extending the results to broader classes of potentials and kernel functions. This work opens new avenues for understanding particle-based variational methods and their applications in complex data-driven fields.