# Non-Asymptotic Analysis of Fractional Langevin Monte Carlo for Non-Convex Optimization

arXiv: https://arxiv.org/abs/1901.07487 · HTML: https://mlsanity.com/paper/1901.07487/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jan 2019.

## Built on by (most cited first, 8 of 8)

- [Eliminating Sharp Minima from SGD with Truncated Heavy-tailed Noise](https://mlsanity.com/paper/2102.04297/) (2021)
- [On the Sample Complexity and Metastability of Heavy-tailed Policy Search in Continuous Control](https://mlsanity.com/paper/2106.08414/) (2021)
- [Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithm](https://mlsanity.com/paper/2201.08349/) (2022)
- [A Separation in Heavy-Tailed Sampling: Gaussian vs. Stable Oracles for Proximal Samplers](https://mlsanity.com/paper/2405.16736/) (2024)
- [Implicit Compressibility of Overparametrized Neural Networks Trained with Heavy-Tailed SGD](https://mlsanity.com/paper/2306.08125/) (2023)
- [Mean-Square Analysis of Discretized Ito Diffusions for Heavy-tailed Sampling](https://mlsanity.com/paper/2303.00570/) (2023)
- [Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models](https://mlsanity.com/paper/2303.17109/) (2023)
- [Convergence, Sticking and Escape: Stochastic Dynamics Near Critical Points in SGD](https://mlsanity.com/paper/2505.18535/) (2025)

## Abstract (arXiv)

Recent studies on diffusion-based sampling methods have shown that Langevin Monte Carlo (LMC) algorithms can be beneficial for non-convex optimization, and rigorous theoretical guarantees have been proven for both asymptotic and finite-time regimes. Algorithmically, LMC-based algorithms resemble the well-known gradient descent (GD) algorithm, where the GD recursion is perturbed by an additive Gaussian noise whose variance has a particular form. Fractional Langevin Monte Carlo (FLMC) is a recently proposed extension of LMC, where the Gaussian noise is replaced by a heavy-tailed α-stable noise. As opposed to its Gaussian counterpart, these heavy-tailed perturbations can incur large jumps and it has been empirically demonstrated that the choice of α-stable noise can provide several advantages in modern machine learning problems, both in optimization and sampling contexts. However, as opposed to LMC, only asymptotic convergence properties of FLMC have been yet established. In this study, we analyze the non-asymptotic behavior of FLMC for non-convex optimization and prove finite-time bounds for its expected suboptimality. Our results show that the weak-error of FLMC increases faster than LMC, which suggests using smaller step-sizes in FLMC. We finally extend our results to the case where the exact gradients are replaced by stochastic gradients and show that similar results hold in this setting as well.