Core Concepts

A new diffusion-based algorithm, called In-and-Out, achieves state-of-the-art runtime complexity for uniformly sampling high-dimensional convex bodies, with stronger guarantees on the output than previously known, including in Rényi divergence.

Abstract

The paper presents a new random walk algorithm called In-and-Out for uniformly sampling high-dimensional convex bodies. The key insights are:
The algorithm can be viewed through the lens of stochastic analysis, with the two steps of each iteration corresponding to alternating forward and backward heat flows. This perspective allows the authors to use tools from functional analysis to analyze the convergence rate.
The analysis shows that the mixing guarantee is determined by the isoperimetric constants (Poincaré or log-Sobolev) of the target uniform distribution, rather than the specific details of the random walk. This leads to stronger guarantees in Rényi divergence, which implies bounds on other well-known divergences like TV, W2, KL, and χ2.
For isotropic convex bodies, the algorithm achieves state-of-the-art runtime complexity, matching the best known results for previous samplers like Ball walk and Speedy walk, but with the stronger Rényi divergence guarantees.
The analysis also extends to non-convex bodies, with the number of proper steps (accepted proposals) bounded by the isoperimetric constants, without requiring convexity.
The key technical contributions are: (1) Establishing contraction results for the forward and backward heat flows, even for the non-smooth uniform distribution; (2) Showing that the number of rejections in the algorithm can be controlled using local conductance and the convexity of the body.

Stats

The uniform distribution over a convex body K has Poincaré constant CPI(πK) ≲ ∥Cov(πK)∥op log d and log-Sobolev constant CLSI(πK) ≲ D^2, where D is the diameter of K. If πK is isotropic, then CPI(πK) ≲ log d and CLSI(πK) ≲ D.

Quotes

"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 Rényi divergence (which implies TV, W2, KL, χ2)."
"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."

Key Insights Distilled From

by Yunbum Kook,... at **arxiv.org** 05-03-2024

Deeper Inquiries

The In-and-Out algorithm can be extended to sample from more general classes of distributions beyond uniform distributions over convex bodies by adapting the rejection sampling step to accommodate different target distributions. The key idea is to modify the proposal distribution in the rejection sampling step to match the target distribution. This can be achieved by adjusting the acceptance criteria based on the target distribution's properties. For example, if the target distribution is not uniform, the proposal distribution in the rejection sampling step can be tailored to better match the target distribution's shape and characteristics. By incorporating the specific properties of the target distribution into the sampling process, the algorithm can effectively sample from a wider range of distributions.

The diffusion-based approach used in the In-and-Out algorithm can be applied to other constrained sampling problems beyond convex bodies by leveraging the concept of stochastic diffusion processes and heat flows. The key is to model the sampling process as a continuous-time diffusion process and analyze its convergence properties using techniques from stochastic analysis. By formulating the sampling problem in terms of diffusion processes, it becomes possible to apply similar principles to other constrained sampling problems. This approach allows for a more nuanced understanding of the sampling process and can lead to efficient algorithms for sampling from a variety of constrained distributions.

The stronger Rényi divergence guarantees provided by the In-and-Out algorithm have potential applications in various fields beyond traditional use cases for convex body sampling. Some potential applications include:
Differential Privacy: Rényi divergence is a key metric in differential privacy, a field that focuses on protecting sensitive information in data analysis. The guarantees provided by the In-and-Out algorithm can enhance the privacy guarantees of differential privacy mechanisms by ensuring stronger divergence bounds.
Machine Learning: Rényi divergence is used in various machine learning tasks, such as clustering, classification, and generative modeling. The guarantees offered by the In-and-Out algorithm can improve the performance and robustness of machine learning models by providing more accurate divergence estimates.
Statistical Inference: Rényi divergence is a useful tool in statistical inference for comparing probability distributions. The guarantees from the In-and-Out algorithm can enhance the accuracy and reliability of statistical analyses by providing tighter bounds on divergence measures.
Overall, the Rényi divergence guarantees from the In-and-Out algorithm have broad applications in fields that rely on accurate and efficient sampling methods and divergence metrics.

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