Controlling Moments with Kernel Stein Discrepancies

• URL: http://arxiv.org/abs/2211.05408v5
• Date: Mon, 30 Sep 2024 10:54:28 GMT
• Title: Controlling Moments with Kernel Stein Discrepancies
• Authors: Heishiro Kanagawa, Alessandro Barp, Arthur Gretton, Lester Mackey,
• Abstract summary: Kernel Stein discrepancies (KSDs) measure the quality of a distributional approximation. We first show that standard KSDs used for weak convergence control fail to control moment convergence. We then provide sufficient conditions under which alternative diffusion KSDs control both moment and weak convergence.
• Score: 74.82363458321939
• License:
• Abstract: Kernel Stein discrepancies (KSDs) measure the quality of a distributional approximation and can be computed even when the target density has an intractable normalizing constant. Notable applications include the diagnosis of approximate MCMC samplers and goodness-of-fit tests for unnormalized statistical models. The present work analyzes the convergence control properties of KSDs. We first show that standard KSDs used for weak convergence control fail to control moment convergence. To address this limitation, we next provide sufficient conditions under which alternative diffusion KSDs control both moment and weak convergence. As an immediate consequence we develop, for each $q > 0$, the first KSDs known to exactly characterize $q$-Wasserstein convergence.

Related papers

• Convergence of Score-Based Discrete Diffusion Models: A Discrete-Time Analysis [56.442307356162864]
  We study the theoretical aspects of score-based discrete diffusion models under the Continuous Time Markov Chain (CTMC) framework. We introduce a discrete-time sampling algorithm in the general state space $[S]d$ that utilizes score estimators at predefined time points. Our convergence analysis employs a Girsanov-based method and establishes key properties of the discrete score function.
  arXiv  Detail & Related papers  (2024-10-03T09:07:13Z)
• Minimax Optimal Goodness-of-Fit Testing with Kernel Stein Discrepancy [13.429541377715298]
  We explore the minimax optimality of goodness-of-fit tests on general domains using the kernelized Stein discrepancy (KSD) The KSD framework offers a flexible approach for goodness-of-fit testing, avoiding strong distributional assumptions. We introduce an adaptive test capable of achieving minimax optimality up to a logarithmic factor by adapting to unknown parameters.
  arXiv  Detail & Related papers  (2024-04-12T07:06:12Z)
• Breaking the Heavy-Tailed Noise Barrier in Stochastic Optimization Problems [56.86067111855056]
  We consider clipped optimization problems with heavy-tailed noise with structured density. We show that it is possible to get faster rates of convergence than $mathcalO(K-(alpha - 1)/alpha)$, when the gradients have finite moments of order. We prove that the resulting estimates have negligible bias and controllable variance.
  arXiv  Detail & Related papers  (2023-11-07T17:39:17Z)
• KL Convergence Guarantees for Score diffusion models under minimal data assumptions [9.618473763561418]
  A notable challenge persists in the form of a lack of comprehensive quantitative results. This article focuses on score diffusion models with fixed step size stemming from the Ornstein-Uhlenbeck semigroup and its kinetic counterpart.
  arXiv  Detail & Related papers  (2023-08-23T16:31:08Z)
• Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models [49.81937966106691]
  We develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models. In contrast to prior works, our theory is developed based on an elementary yet versatile non-asymptotic approach.
  arXiv  Detail & Related papers  (2023-06-15T16:30:08Z)
• Using Perturbation to Improve Goodness-of-Fit Tests based on Kernelized Stein Discrepancy [3.78967502155084]
  Kernelized Stein discrepancy (KSD) is a score-based discrepancy widely used in goodness-of-fit tests. We show theoretically and empirically that the KSD test can suffer from low power when the target and the alternative distributions have the same well-separated modes but differ in mixing proportions.
  arXiv  Detail & Related papers  (2023-04-28T11:13:18Z)
• Targeted Separation and Convergence with Kernel Discrepancies [61.973643031360254]
  kernel-based discrepancy measures are required to (i) separate a target P from other probability measures or (ii) control weak convergence to P. In this article we derive new sufficient and necessary conditions to ensure (i) and (ii) For MMDs on separable metric spaces, we characterize those kernels that separate Bochner embeddable measures and introduce simple conditions for separating all measures with unbounded kernels.
  arXiv  Detail & Related papers  (2022-09-26T16:41:16Z)
• DR-DSGD: A Distributionally Robust Decentralized Learning Algorithm over Graphs [54.08445874064361]
  We propose to solve a regularized distributionally robust learning problem in the decentralized setting. By adding a Kullback-Liebler regularization function to the robust min-max optimization problem, the learning problem can be reduced to a modified robust problem. We show that our proposed algorithm can improve the worst distribution test accuracy by up to $10%$.
  arXiv  Detail & Related papers  (2022-08-29T18:01:42Z)
• KSD Aggregated Goodness-of-fit Test [38.45086141837479]
  We introduce a strategy to construct a test, called KSDAgg, which aggregates multiple tests with different kernels. We provide non-asymptotic guarantees on the power of KSDAgg. We find that KSDAgg outperforms other state-of-the-art adaptive KSD-based goodness-of-fit testing procedures.
  arXiv  Detail & Related papers  (2022-02-02T00:33:09Z)

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.