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

2024-05-02

Neural Information Processing Systems

AI Summary

This work addresses efficient uniform sampling from high-dimensional convex bodies, providing strong convergence guarantees under Rényi divergence—including total variation (TV), Wasserstein-2 ($mathcal{W}_2$), Kullback–Leibler (KL), and $chi^2$ divergences. We propose a novel stochastic walk algorithm that models sampling through the lens of stochastic diffusion—a first in this context—and characterizes convergence rates via the functional isoperimetric constant of the target distribution, thereby departing from conventional polynomial mixing-time analysis. Theoretically, our algorithm achieves the optimal time complexity $O^*(n^2 R^2)$, where $n$ denotes dimensionality and $R$ the body’s diameter. Moreover, we derive unified, tight convergence bounds across the entire Rényi divergence family. To our knowledge, this is the first uniform sampling scheme attaining simultaneous optimality under multiple probability metrics.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Content-based information diffusion

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 target distribution.

Problem

• Develops new random walk for uniform sampling
• Achieves state-of-the-art runtime complexity guarantees
• Uses stochastic diffusion for convergence analysis

Innovation

• Algorithmic diffusion for convex body sampling
• Stochastic diffusion with functional isoperimetric analysis
• Achieves state-of-the-art runtime with Rényi divergence guarantees