Bregman meets Lévy: Stochastic mirror descent with heavy-tailed noise in continuous and discrete time
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 Lévy noise process with finite $p$-th order moments, $1 < p \leq 2$. This scheme -- which we call the Lévy 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 ...
| Authors | Pierre-Louis Cauvin, Panayotis Mertikopoulos |
|---|---|
| Primary category | math.OC |
| Published | 2 June 2026 |
| Last updated at source | 2 June 2026 |
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