# A Separation in Heavy-Tailed Sampling: Gaussian vs. Stable Oracles for Proximal Samplers

arXiv: https://arxiv.org/abs/2405.16736 · HTML: https://mlsanity.com/paper/2405.16736/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 64 references that are in the map.
- First posted on arXiv 27 May 2024.

## Builds on (most cited first, 30 of 36)

- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [Structured Logconcave Sampling with a Restricted Gaussian Oracle](https://mlsanity.com/paper/2010.03106/) (2020)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [Improved analysis for a proximal algorithm for sampling](https://mlsanity.com/paper/2202.06386/) (2022)
- [Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo](https://mlsanity.com/paper/2202.05214/) (2022)
- [On sampling from a log-concave density using kinetic Langevin diffusions](https://mlsanity.com/paper/1807.09382/) (2018)
- [Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm](https://mlsanity.com/paper/2012.12810/) (2020)
- [On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness](https://mlsanity.com/paper/2005.13097/) (2020)
- [Learning Halfspaces with Massart Noise Under Structured Distributions](https://mlsanity.com/paper/2002.05632/) (2020)
- [Global Non-convex Optimization with Discretized Diffusions](https://mlsanity.com/paper/1810.12361/) (2018)
- [Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Sampling](https://mlsanity.com/paper/2109.13055/) (2021)
- [Convergence of Langevin Monte Carlo in Chi-Squared and Renyi Divergence](https://mlsanity.com/paper/2007.11612/) (2020)
- [Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity](https://mlsanity.com/paper/1907.11331/) (2019)
- [Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond](https://mlsanity.com/paper/1906.07868/) (2019)
- [Fractional Underdamped Langevin Dynamics: Retargeting SGD with Momentum under Heavy-Tailed Gradient Noise](https://mlsanity.com/paper/2002.05685/) (2020)
- [Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequality](https://mlsanity.com/paper/2303.03589/) (2023)
- [Improved dimension dependence of a proximal algorithm for sampling](https://mlsanity.com/paper/2302.10081/) (2023)
- [Sqrt(d) Dimension Dependence of Langevin Monte Carlo](https://mlsanity.com/paper/2109.03839/) (2021)
- [On the Ergodicity, Bias and Asymptotic Normality of Randomized Midpoint Sampling Method](https://mlsanity.com/paper/2011.03176/) (2020)
- [Logsmooth Gradient Concentration and Tighter Runtimes for Metropolized Hamiltonian Monte Carlo](https://mlsanity.com/paper/2002.04121/) (2020)
- [Lower Bounds on Metropolized Sampling Methods for Well-Conditioned Distributions](https://mlsanity.com/paper/2106.05480/) (2021)
- [The query complexity of sampling from strongly log-concave distributions in one dimension](https://mlsanity.com/paper/2105.14163/) (2021)
- [Estimating Normalizing Constants for Log-Concave Distributions: Algorithms and Lower Bounds](https://mlsanity.com/paper/1911.03043/) (2019)
- [Fisher information lower bounds for sampling](https://mlsanity.com/paper/2210.02482/) (2022)
- [Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler](https://mlsanity.com/paper/2302.06085/) (2023)
- [Non-Asymptotic Analysis of Fractional Langevin Monte Carlo for Non-Convex Optimization](https://mlsanity.com/paper/1901.07487/) (2019)

## Built on by (most cited first, 6 of 6)

- [On the query complexity of sampling from non-log-concave distributions](https://mlsanity.com/paper/2502.06200/) (2025)
- [Restricted Spectral Gap Decomposition for Simulated Tempering Targeting Mixture Distributions](https://mlsanity.com/paper/2505.15059/) (2025)
- [R'enyi Differential Privacy for Heavy-Tailed SDEs via Fractional Poincar'e Inequalities](https://mlsanity.com/paper/2511.15634/) (2025)
- [Functional Stochastic Localization](https://mlsanity.com/paper/2602.03999/) (2026)
- [Local Geometric Mixing via Dobrushin Contraction with Applications to Diffusion Path Monte Carlo and the Proximal Sampler](https://mlsanity.com/paper/2609.28338/) (2026)
- [Reflected Proximal Dynamics for Accelerated High-Accuracy Log-Concave Sampling](https://mlsanity.com/paper/2610.16248/) (2026)

## Abstract (arXiv)

We study the complexity of heavy-tailed sampling and present a separation result in terms of obtaining high-accuracy versus low-accuracy guarantees i.e., samplers that require only O(log(1/varepsilon)) versus Omega(poly(1/varepsilon)) iterations to output a sample which is varepsilon-close to the target in chi2-divergence. Our results are presented for proximal samplers that are based on Gaussian versus stable oracles. We show that proximal samplers based on the Gaussian oracle have a fundamental barrier in that they necessarily achieve only low-accuracy guarantees when sampling from a class of heavy-tailed targets. In contrast, proximal samplers based on the stable oracle exhibit high-accuracy guarantees, thereby overcoming the aforementioned limitation. We also prove lower bounds for samplers under the stable oracle and show that our upper bounds cannot be fundamentally improved.