Concentration analysis of multivariate elliptic diffusion processes

• URL: http://arxiv.org/abs/2206.03329v1
• Date: Tue, 7 Jun 2022 14:15:05 GMT
• Title: Concentration analysis of multivariate elliptic diffusion processes
• Authors: Cathrine Aeckerle-Willems, Claudia Strauch and Lukas Trottner
• Abstract summary: We prove concentration inequalities and associated PAC bounds for continuous- and discrete-time additive functionals. Our analysis relies on an approach via the Poisson equation allowing us to consider a very broad class of subexponentially ergodic processes.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove concentration inequalities and associated PAC bounds for continuous- and discrete-time additive functionals for possibly unbounded functions of multivariate, nonreversible diffusion processes. Our analysis relies on an approach via the Poisson equation allowing us to consider a very broad class of subexponentially ergodic processes. These results add to existing concentration inequalities for additive functionals of diffusion processes which have so far been only available for either bounded functions or for unbounded functions of processes from a significantly smaller class. We demonstrate the power of these exponential inequalities by two examples of very different areas. Considering a possibly high-dimensional parametric nonlinear drift model under sparsity constraints, we apply the continuous-time concentration results to validate the restricted eigenvalue condition for Lasso estimation, which is fundamental for the derivation of oracle inequalities. The results for discrete additive functionals are used to investigate the unadjusted Langevin MCMC algorithm for sampling of moderately heavy-tailed densities $\pi$. In particular, we provide PAC bounds for the sample Monte Carlo estimator of integrals $\pi(f)$ for polynomially growing functions $f$ that quantify sufficient sample and step sizes for approximation within a prescribed margin with high probability.

Related papers

• Kinetic Interacting Particle Langevin Monte Carlo [0.0]
  This paper introduces and analyses interacting underdamped Langevin algorithms, for statistical inference in latent variable models. We propose a diffusion process that evolves jointly in the space of parameters and latent variables. We provide two explicit discretisations of this diffusion as practical algorithms to estimate parameters of statistical models.
  arXiv  Detail & Related papers  (2024-07-08T09:52:46Z)
• Flow matching achieves almost minimax optimal convergence [50.38891696297888]
  Flow matching (FM) has gained significant attention as a simulation-free generative model. This paper discusses the convergence properties of FM for large sample size under the $p$-Wasserstein distance. We establish that FM can achieve an almost minimax optimal convergence rate for $1 leq p leq 2$, presenting the first theoretical evidence that FM can reach convergence rates comparable to those of diffusion models.
  arXiv  Detail & Related papers  (2024-05-31T14:54:51Z)
• Logistic-beta processes for dependent random probabilities with beta marginals [58.91121576998588]
  We propose a novel process called the logistic-beta process, whose logistic transformation yields a process with common beta marginals. It can model dependence on both discrete and continuous domains, such as space or time, and has a flexible dependence structure through correlation kernels. We illustrate the benefits through nonparametric binary regression and conditional density estimation examples, both in simulation studies and in a pregnancy outcome application.
  arXiv  Detail & Related papers  (2024-02-10T21:41:32Z)
• Noise-Free Sampling Algorithms via Regularized Wasserstein Proximals [3.4240632942024685]
  We consider the problem of sampling from a distribution governed by a potential function. This work proposes an explicit score based MCMC method that is deterministic, resulting in a deterministic evolution for particles.
  arXiv  Detail & Related papers  (2023-08-28T23:51:33Z)
• Interacting Particle Langevin Algorithm for Maximum Marginal Likelihood Estimation [2.53740603524637]
  We develop a class of interacting particle systems for implementing a maximum marginal likelihood estimation procedure. In particular, we prove that the parameter marginal of the stationary measure of this diffusion has the form of a Gibbs measure. Using a particular rescaling, we then prove geometric ergodicity of this system and bound the discretisation error. in a manner that is uniform in time and does not increase with the number of particles.
  arXiv  Detail & Related papers  (2023-03-23T16:50:08Z)
• Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation [59.45669299295436]
  We propose a Monte Carlo PDE solver for training unsupervised neural solvers. We use the PDEs' probabilistic representation, which regards macroscopic phenomena as ensembles of random particles. Our experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency.
  arXiv  Detail & Related papers  (2023-02-10T08:05:19Z)
• Mathematical analysis of singularities in the diffusion model under the submanifold assumption [0.0]
  We show that the analytical mean drift function in DDPM and the score function in SGMally blow up in the final stages of the sampling process for singular data distributions. We derive a new target function and associated loss, which remains bounded even for singular data distributions.
  arXiv  Detail & Related papers  (2023-01-19T05:13:03Z)
• Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency [53.90687548731265]
  We study optimal procedures for estimating a linear functional based on observational data. For any convex and symmetric function class $mathcalF$, we derive a non-asymptotic local minimax bound on the mean-squared error.
  arXiv  Detail & Related papers  (2023-01-16T02:57:37Z)
• Off-policy estimation of linear functionals: Non-asymptotic theory for semi-parametric efficiency [59.48096489854697]
  The problem of estimating a linear functional based on observational data is canonical in both the causal inference and bandit literatures. We prove non-asymptotic upper bounds on the mean-squared error of such procedures. We establish its instance-dependent optimality in finite samples via matching non-asymptotic local minimax lower bounds.
  arXiv  Detail & Related papers  (2022-09-26T23:50:55Z)
• A blob method method for inhomogeneous diffusion with applications to multi-agent control and sampling [0.6562256987706128]
  We develop a deterministic particle method for the weighted porous medium equation (WPME) and prove its convergence on bounded time intervals. Our method has natural applications to multi-agent coverage algorithms and sampling probability measures.
  arXiv  Detail & Related papers  (2022-02-25T19:49:05Z)
• 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)

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.