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Optimal Diffusion Processes

Saber Jafarizadeh

TL;DR

The paper addresses designing diffusion processes with a fixed stationary distribution $\pi(x)$ and fixed average variance $\widehat{\sigma}^{2}/2$ that converge most rapidly to equilibrium. It casts the problem as a variational spectral-gap optimization, solving a Rayleigh quotient via a min-max framework to show the optimum occurs with a linear drift and a nontrivial variance function, yielding $\lambda_1 = \widehat{\sigma}^{2}/[2 (m_2 - m_1^{2})]$ and $\phi_1(x)=(x-m_1)/\sqrt{m_2-m_1^{2}}$. The results imply a concavity of the optimal relaxation time in the stationary distribution and establish that all Pearson hypergeometric-type diffusions with variance degree up to 2 are optimal; the paper also provides explicit optimal parameters for higher-degree polynomial variance and non-polynomial variance (hyperexponential) cases. Collectively, these findings offer a principled method for designing diffusion samplers with maximal convergence speed under distributional and variance constraints, and illustrate the theory with concrete Pearson, polynomial, and non-polynomial examples.

Abstract

Of stochastic differential equations, diffusion processes have been adopted in numerous applications, as more relevant and flexible models. This paper studies diffusion processes in a different setting, where for a given stationary distribution and average variance, it seeks the diffusion process with optimal convergence rate. It is shown that the optimal drift function is a linear function and the convergence rate of the stochastic process is bounded by the ratio of the average variance to the variance of the stationary distribution. Furthermore, the concavity of the optimal relaxation time as a function of the stationary distribution has been proven, and it is shown that all Pearson diffusion processes of the Hypergeometric type with polynomial functions of at most degree two as the variance functions are optimal.

Optimal Diffusion Processes

TL;DR

The paper addresses designing diffusion processes with a fixed stationary distribution and fixed average variance that converge most rapidly to equilibrium. It casts the problem as a variational spectral-gap optimization, solving a Rayleigh quotient via a min-max framework to show the optimum occurs with a linear drift and a nontrivial variance function, yielding and . The results imply a concavity of the optimal relaxation time in the stationary distribution and establish that all Pearson hypergeometric-type diffusions with variance degree up to 2 are optimal; the paper also provides explicit optimal parameters for higher-degree polynomial variance and non-polynomial variance (hyperexponential) cases. Collectively, these findings offer a principled method for designing diffusion samplers with maximal convergence speed under distributional and variance constraints, and illustrate the theory with concrete Pearson, polynomial, and non-polynomial examples.

Abstract

Of stochastic differential equations, diffusion processes have been adopted in numerous applications, as more relevant and flexible models. This paper studies diffusion processes in a different setting, where for a given stationary distribution and average variance, it seeks the diffusion process with optimal convergence rate. It is shown that the optimal drift function is a linear function and the convergence rate of the stochastic process is bounded by the ratio of the average variance to the variance of the stationary distribution. Furthermore, the concavity of the optimal relaxation time as a function of the stationary distribution has been proven, and it is shown that all Pearson diffusion processes of the Hypergeometric type with polynomial functions of at most degree two as the variance functions are optimal.
Paper Structure (12 sections, 4 theorems, 67 equations, 1 table)

This paper contains 12 sections, 4 theorems, 67 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fokker-Plank Equation
  4. Optimal Diffusion Processes
  5. Main Results
  6. Examples of Optimal Diffusion Processes
  7. Pearson Class of Diffusion Processes
  8. Diffusion Processes with Polynomial Variance Functions of Degree Higher than Two
  9. Diffusion Processes with Non-Polynomial Variance Function
  10. Conclusions
  11. Optimal Results (Solution)
  12. Hypergeometric Function

Key Result

Theorem 1

For given stationary distribution $\pi(x)$ and average variance $\widehat{\sigma}^{2}/2$, the optimal results corresponding to the diffusion process with fastest convergence rate are as below, where $\lambda_{1}$ and $\tau = 1 / \lambda_{1}$ are the optimal convergence rate and the optimal relaxation time of the diffusion process. $\phi_1(x)$ is the optimal eigenfunction corresponding to the opti

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • Proof 1
  • Lemma 2
  • Proof 2
  • Lemma 3
  • Proof 3