# Bregman meets L'evy: Stochastic mirror descent with heavy-tailed noise in continuous and discrete time

arXiv: https://arxiv.org/abs/2606.03769 · HTML: https://mlsanity.com/paper/2606.03769/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 121 references that are in the map.
- First posted on arXiv 2 Jun 2026.

## Builds on (most cited first, 1 of 1)

- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)

## Abstract (arXiv)

We study the robustness of stochastic mirror descent (SMD) under heavy-tailed noise, focusing on whether the method retains its convergence guarantees when run with infinite-variance stochastic gradient input. To address this question in a principled manner, we begin by introducing a continuous-time model of SMD as a stochastic differential equation (SDE) driven by a centered L'evy noise process with finite p-th order moments, 1 < p ≤ 2. This scheme -- which we call the L'evy mirror flow (LMF) -- arises naturally as the scaling limit of SMD in the presence of heavy-tailed noise. In particular, when p < 2 -- the heavy noise regime -- the trajectories of LMF generically exhibit jump discontinuities of arbitrary magnitude which, if frequent enough, lead to infinite variance. Nonetheless, despite this highly singular behavior, we show that LMF attains ε-optimality within O(ε-p/(p-1)) time in the convex case, and within tilde O(ε-1/(p-1)) time for (relatively) strongly convex objectives. These guarantees provide a transparent characterization of the impact of frequent long jumps on the convergence of the process, and percolate to a series of matching discrete-time guarantees for several variants of SMD under heavy-tailed noise.