In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies

• URL: http://arxiv.org/abs/2405.01425v2
• Date: Wed, 20 Nov 2024 19:01:42 GMT
• Title: In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies
• Authors: Yunbum Kook, Santosh S. Vempala, Matthew S. Zhang,
• Abstract summary: We present a new random walk for uniformly sampling high-dimensional convex bodies. It achieves state-of-the-art runtime complexity with stronger guarantees on the output.
• Score: 7.70133333709347
• License:
• Abstract: We present a new random walk for uniformly sampling high-dimensional convex bodies. It achieves state-of-the-art runtime complexity with stronger guarantees on the output than previously known, namely in R\'enyi divergence (which implies TV, $\mathcal{W}_2$, KL, $\chi^2$). The proof departs from known approaches for polytime algorithms for the problem -- we utilize a stochastic diffusion perspective to show contraction to the target distribution with the rate of convergence determined by functional isoperimetric constants of the stationary density.

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)
• Harmonic Path Integral Diffusion [0.4527270266697462]
  We present a novel approach for sampling from a continuous multivariate probability distribution, which may either be explicitly known (up to a normalization factor) or represented via empirical samples. Our method constructs a time-dependent bridge from a delta function centered at the origin of the state space at $t=0$, transforming it into the target distribution at $t=1$. We contrast these algorithms with other sampling methods, particularly simulated and path integral sampling, highlighting their advantages in terms of analytical control, accuracy, and computational efficiency.
  arXiv  Detail & Related papers  (2024-09-23T16:20:21Z)
• 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)
• 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)
• Adaptive Annealed Importance Sampling with Constant Rate Progress [68.8204255655161]
  Annealed Importance Sampling (AIS) synthesizes weighted samples from an intractable distribution. We propose the Constant Rate AIS algorithm and its efficient implementation for $alpha$-divergences.
  arXiv  Detail & Related papers  (2023-06-27T08:15:28Z)
• A Geometric Perspective on Diffusion Models [57.27857591493788]
  We inspect the ODE-based sampling of a popular variance-exploding SDE. We establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm.
  arXiv  Detail & Related papers  (2023-05-31T15:33:16Z)
• Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling [34.66940399825547]
  This paper characterizes the mixing time of the Langevin Algorithm to its stationary distribution. We introduce a technique from the differential privacy literature to the sampling literature.
  arXiv  Detail & Related papers  (2022-10-16T05:11:16Z)
• Utilising the CLT Structure in Stochastic Gradient based Sampling : Improved Analysis and Faster Algorithms [14.174806471635403]
  We consider approximations of sampling algorithms, such as Gradient Langevin Dynamics (SGLD) and the Random Batch Method (RBM) for Interacting Particle Dynamcs (IPD) We observe that the noise introduced by the approximation is nearly Gaussian due to the Central Limit Theorem (CLT) while the driving Brownian motion is exactly Gaussian. We harness this structure to absorb the approximation error inside the diffusion process, and obtain improved convergence guarantees for these algorithms.
  arXiv  Detail & Related papers  (2022-06-08T10:17:40Z)
• 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)
• Mean-Square Analysis with An Application to Optimal Dimension Dependence of Langevin Monte Carlo [60.785586069299356]
  This work provides a general framework for the non-asymotic analysis of sampling error in 2-Wasserstein distance. Our theoretical analysis is further validated by numerical experiments.
  arXiv  Detail & Related papers  (2021-09-08T18:00:05Z)
• Stochastic Saddle-Point Optimization for Wasserstein Barycenters [69.68068088508505]
  We consider the populationimation barycenter problem for random probability measures supported on a finite set of points and generated by an online stream of data. We employ the structure of the problem and obtain a convex-concave saddle-point reformulation of this problem. In the setting when the distribution of random probability measures is discrete, we propose an optimization algorithm and estimate its complexity.
  arXiv  Detail & Related papers  (2020-06-11T19:40:38Z)

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.