# Mean-Square Analysis of Discretized It\^o Diffusions for Heavy-tailed Sampling

## TL;DR

This paper investigates the complexity of sampling from heavy-tailed distributions using discretized Itô diffusions, providing mean-square analysis and explicit iteration bounds under minimal variance assumptions.

## Contribution

It introduces a mean-square analysis framework for heavy-tailed sampling via discretized Itô diffusions, with explicit iteration complexity bounds and gradient estimation methods.

## Key findings

- Established iteration complexity bounds for heavy-tailed distributions
- Derived explicit moment bounds for heavy-tailed targets
- Extended results to gradient estimation via Gaussian smoothing

## Abstract

We analyze the complexity of sampling from a class of heavy-tailed distributions by discretizing a natural class of It\^o diffusions associated with weighted Poincar\'e inequalities. Based on a mean-square analysis, we establish the iteration complexity for obtaining a sample whose distribution is $\epsilon$ close to the target distribution in the Wasserstein-2 metric. In this paper, our results take the mean-square analysis to its limits, i.e., we invariably only require that the target density has finite variance, the minimal requirement for a mean-square analysis. To obtain explicit estimates, we compute upper bounds on certain moments associated with heavy-tailed targets under various assumptions. We also provide similar iteration complexity results for the case where only function evaluations of the unnormalized target density are available by estimating the gradients using a Gaussian smoothing technique. We provide illustrative examples based on the multivariate $t$-distribution.