Stein's Lemma for the Reparameterization Trick with Exponential Family Mixtures

• URL: http://arxiv.org/abs/1910.13398v3
• Date: Sat, 01 Feb 2025 02:58:31 GMT
• Title: Stein's Lemma for the Reparameterization Trick with Exponential Family Mixtures
• Authors: Wu Lin, Mohammad Emtiyaz Khan, Mark Schmidt,
• Abstract summary: Stein's lemma plays an essential role in Stein's method. We extend Stein's lemma to exponential-family mixture distributions.
• Score: 23.941042092067338
• License:
• Abstract: Stein's method (Stein, 1973; 1981) is a powerful tool for statistical applications and has significantly impacted machine learning. Stein's lemma plays an essential role in Stein's method. Previous applications of Stein's lemma either required strong technical assumptions or were limited to Gaussian distributions with restricted covariance structures. In this work, we extend Stein's lemma to exponential-family mixture distributions, including Gaussian distributions with full covariance structures. Our generalization enables us to establish a connection between Stein's lemma and the reparameterization trick to derive gradients of expectations of a large class of functions under weak assumptions. Using this connection, we can derive many new reparameterizable gradient identities that go beyond the reach of existing works. For example, we give gradient identities when the expectation is taken with respect to Student's t-distribution, skew Gaussian, exponentially modified Gaussian, and normal inverse Gaussian.

Related papers

• A Stein Gradient Descent Approach for Doubly Intractable Distributions [5.63014864822787]
  We propose a novel Monte Carlo Stein variational gradient descent (MC-SVGD) approach for inference for doubly intractable distributions. The proposed method achieves substantial computational gains over existing algorithms, while providing comparable inferential performance for the posterior distributions.
  arXiv  Detail & Related papers  (2024-10-28T13:42:27Z)
• Tempered Calculus for ML: Application to Hyperbolic Model Embedding [70.61101116794549]
  Most mathematical distortions used in ML are fundamentally integral in nature. In this paper, we unveil a grounded theory and tools which can help improve these distortions to better cope with ML requirements. We show how to apply it to a problem that has recently gained traction in ML: hyperbolic embeddings with a "cheap" and accurate encoding along the hyperbolic vsean scale.
  arXiv  Detail & Related papers  (2024-02-06T17:21:06Z)
• Compressed and distributed least-squares regression: convergence rates with applications to Federated Learning [9.31522898261934]
  We investigate the impact of compression on gradient algorithms for machine learning. We highlight differences in terms of convergence rates between several unbiased compression operators. We extend our results to the case of federated learning.
  arXiv  Detail & Related papers  (2023-08-02T18:02:00Z)
• A Finite-Particle Convergence Rate for Stein Variational Gradient Descent [47.6818454221125]
  We provide the first finite-particle convergence rate for Stein variational descent gradient (SVGD) Our explicit, non-asymptotic proof strategy will serve as a template for future refinements.
  arXiv  Detail & Related papers  (2022-11-17T17:50:39Z)
• High Probability Bounds for a Class of Nonconvex Algorithms with AdaGrad Stepsize [55.0090961425708]
  We propose a new, simplified high probability analysis of AdaGrad for smooth, non- probability problems. We present our analysis in a modular way and obtain a complementary $mathcal O (1 / TT)$ convergence rate in the deterministic setting. To the best of our knowledge, this is the first high probability for AdaGrad with a truly adaptive scheme, i.e., completely oblivious to the knowledge of smoothness.
  arXiv  Detail & Related papers  (2022-04-06T13:50:33Z)
• The Schr\"odinger Bridge between Gaussian Measures has a Closed Form [101.79851806388699]
  We focus on the dynamic formulation of OT, also known as the Schr"odinger bridge (SB) problem. In this paper, we provide closed-form expressions for SBs between Gaussian measures.
  arXiv  Detail & Related papers  (2022-02-11T15:59:01Z)
• Complexity Analysis of Stein Variational Gradient Descent Under Talagrand's Inequality T1 [12.848239550098697]
  We study the complexity of Stein Variational Gradient Descent (SVGD), which is an algorithm to sample from $pi(x) propto exp(-Fx)) Our key assumption is that the target distribution satisfies the inequality's inequality T1.
  arXiv  Detail & Related papers  (2021-06-06T09:51:32Z)
• Stein Variational Gradient Descent: many-particle and long-time asymptotics [0.0]
  Stein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. We develop the cotangent space construction for the Stein geometry, prove its basic properties, and determine the large-deviation functional governing the many-particle limit. We identify the Stein-Fisher information as its leading order contribution in the long-time and many-particle regime.
  arXiv  Detail & Related papers  (2021-02-25T16:03:04Z)
• 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)
• A diffusion approach to Stein's method on Riemannian manifolds [65.36007959755302]
  We exploit the relationship between the generator of a diffusion on $mathbf M$ with target invariant measure and its characterising Stein operator. We derive Stein factors, which bound the solution to the Stein equation and its derivatives. We imply that the bounds for $mathbb Rm$ remain valid when $mathbf M$ is a flat manifold.
  arXiv  Detail & Related papers  (2020-03-25T17:03:58Z)

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.