Independent projections of diffusions: Gradient flows for variational inference and optimal mean field approximations

• URL: http://arxiv.org/abs/2309.13332v2
• Date: Wed, 09 Oct 2024 15:12:02 GMT
• Title: Independent projections of diffusions: Gradient flows for variational inference and optimal mean field approximations
• Authors: Daniel Lacker,
• Abstract summary: This paper presents a construction, called the emphindependent projection, which is optimal for two natural criteria. First, when the original diffusion is reversible with invariant measure $rho_*$, the independent projection serves as the Wasserstein gradient flow for the relative entropy $H(cdot,|,rho_*)$ constrained to the space of product measures. Second, among all processes with independent coordinates, the independent projection is shown to exhibit the slowest growth rate of path-space entropy relative to the original diffusion.
• Score: 0.0
• License:
• Abstract: What is the optimal way to approximate a high-dimensional diffusion process by one in which the coordinates are independent? This paper presents a construction, called the \emph{independent projection}, which is optimal for two natural criteria. First, when the original diffusion is reversible with invariant measure $\rho_*$, the independent projection serves as the Wasserstein gradient flow for the relative entropy $H(\cdot\,|\,\rho_*)$ constrained to the space of product measures. This is related to recent Langevin-based sampling schemes proposed in the statistical literature on mean field variational inference. In addition, we provide both qualitative and quantitative results on the long-time convergence of the independent projection, with quantitative results in the log-concave case derived via a new variant of the logarithmic Sobolev inequality. Second, among all processes with independent coordinates, the independent projection is shown to exhibit the slowest growth rate of path-space entropy relative to the original diffusion. This sheds new light on the classical McKean-Vlasov equation and recent variants proposed for non-exchangeable systems, which can be viewed as special cases of the independent projection.

Related papers

• Quasi-Bayes meets Vines [2.3124143670964448]
  We propose a different way to extend Quasi-Bayesian prediction to high dimensions through the use of Sklar's theorem. We show that our proposed Quasi-Bayesian Vine (QB-Vine) is a fully non-parametric density estimator with emphan analytical form.
  arXiv  Detail & Related papers  (2024-06-18T16:31:02Z)
• Space-Time Diffusion Bridge [0.4527270266697462]
  We introduce a novel method for generating new synthetic samples independent and identically distributed from real probability distributions. We use space-time mixing strategies that extend across temporal and spatial dimensions. We validate the efficacy of our space-time diffusion approach with numerical experiments.
  arXiv  Detail & Related papers  (2024-02-13T23:26:11Z)
• Distributed Markov Chain Monte Carlo Sampling based on the Alternating Direction Method of Multipliers [143.6249073384419]
  In this paper, we propose a distributed sampling scheme based on the alternating direction method of multipliers. We provide both theoretical guarantees of our algorithm's convergence and experimental evidence of its superiority to the state-of-the-art. In simulation, we deploy our algorithm on linear and logistic regression tasks and illustrate its fast convergence compared to existing gradient-based methods.
  arXiv  Detail & Related papers  (2024-01-29T02:08:40Z)
• Sampling and estimation on manifolds using the Langevin diffusion [45.57801520690309]
  Two estimators of linear functionals of $mu_phi $ based on the discretized Markov process are considered. Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion.
  arXiv  Detail & Related papers  (2023-12-22T18:01:11Z)
• Symmetric Mean-field Langevin Dynamics for Distributional Minimax Problems [78.96969465641024]
  We extend mean-field Langevin dynamics to minimax optimization over probability distributions for the first time with symmetric and provably convergent updates. We also study time and particle discretization regimes and prove a new uniform-in-time propagation of chaos result.
  arXiv  Detail & Related papers  (2023-12-02T13:01:29Z)
• Statistically Optimal Generative Modeling with Maximum Deviation from the Empirical Distribution [2.1146241717926664]
  We show that the Wasserstein GAN, constrained to left-invertible push-forward maps, generates distributions that avoid replication and significantly deviate from the empirical distribution. Our most important contribution provides a finite-sample lower bound on the Wasserstein-1 distance between the generative distribution and the empirical one. We also establish a finite-sample upper bound on the distance between the generative distribution and the true data-generating one.
  arXiv  Detail & Related papers  (2023-07-31T06:11:57Z)
• Efficient CDF Approximations for Normalizing Flows [64.60846767084877]
  We build upon the diffeomorphic properties of normalizing flows to estimate the cumulative distribution function (CDF) over a closed region. Our experiments on popular flow architectures and UCI datasets show a marked improvement in sample efficiency as compared to traditional estimators.
  arXiv  Detail & Related papers  (2022-02-23T06:11:49Z)
• Variational Refinement for Importance Sampling Using the Forward Kullback-Leibler Divergence [77.06203118175335]
  Variational Inference (VI) is a popular alternative to exact sampling in Bayesian inference. Importance sampling (IS) is often used to fine-tune and de-bias the estimates of approximate Bayesian inference procedures. We propose a novel combination of optimization and sampling techniques for approximate Bayesian inference.
  arXiv  Detail & Related papers  (2021-06-30T11:00:24Z)
• Variational Transport: A Convergent Particle-BasedAlgorithm for Distributional Optimization [106.70006655990176]
  A distributional optimization problem arises widely in machine learning and statistics. We propose a novel particle-based algorithm, dubbed as variational transport, which approximately performs Wasserstein gradient descent. We prove that when the objective function satisfies a functional version of the Polyak-Lojasiewicz (PL) (Polyak, 1963) and smoothness conditions, variational transport converges linearly.
  arXiv  Detail & Related papers  (2020-12-21T18:33:13Z)
• Exponential ergodicity of mirror-Langevin diffusions [16.012656579770827]
  We propose a class of diffusions called Newton-Langevin diffusions and prove that they converge to stationarity exponentially fast. We give an application to the problem of sampling from the uniform distribution on a convex body using a strategy inspired by interior-point methods.
  arXiv  Detail & Related papers  (2020-05-19T18:00:52Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.