Relaxing noise assumptions in tuning‑free upper bounds to expected smoothness

Establish tuning‑free optimization guarantees for the algorithms analyzed in this paper (including DoG and DoWG in the convex smooth/Lipschitz settings and the restarted SGD variant in the nonconvex smooth setting) under noise models characterized by expected smoothness of the stochastic gradients, rather than assuming bounded or sub‑Gaussian gradient noise norms.

Background

The paper’s positive results rely on relatively strong noise assumptions: bounded almost‑sure noise or sub‑Gaussian norm of the gradient noise. In the conclusion, the authors explicitly raise the question of extending these guarantees to weaker, widely‑used stochastic models such as expected smoothness (e.g., as formalized by Gower et al., 2019).

Resolving this would broaden the applicability of tuning‑free optimization beyond the current restrictive assumptions and potentially align the theory with practical stochastic settings.