Mean-Square Analysis of Discretized Itô Diffusions for Heavy-tailed Sampling
This work addresses sampling challenges for heavy-tailed distributions in statistics and machine learning, offering incremental improvements in complexity analysis under weaker assumptions.
The paper tackles the problem of sampling from heavy-tailed distributions by discretizing Itô diffusions, establishing iteration complexity for achieving ε-accuracy in the Wasserstein-2 metric under minimal assumptions of finite variance. It provides explicit upper bounds on moments and extends results to cases with only unnormalized density evaluations using Gaussian smoothing.
We analyze the complexity of sampling from a class of heavy-tailed distributions by discretizing a natural class of Itô diffusions associated with weighted Poincaré inequalities. Based on a mean-square analysis, we establish the iteration complexity for obtaining a sample whose distribution is $ε$ 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.