arXiv:2611.03201 · v1

On the Convergence of Vanilla Adam under Heavy-Tailed Noise and Index Sampling

首次提交 2026-11-02 22:26

版本更新 2026-11-02 22:26

雷达观测 2026-11-04 23:41

摘要

Understanding how stochastic gradient noise affects the convergence of Adam is an important problem in stochastic optimization, particularly when the noise exhibits heavy-tailed behavior. In this paper, we investigate the convergence of vanilla Adam in both the expected risk and the empirical risk minimization, focusing on the effects of heavy-tailed noise and its relationship to index sampling. For the expected risk minimization, we establish convergence guarantees under Hölder smoothness and bounded $\mathfrak{p}$-th moment assumptions, allowing stochastic gradient noise with infinite variance. Our analysis provides explicit upper bounds on the expected squared gradient norm and shows that the iteration-dependent convergence term decays more slowly as $\mathfrak{p}$ decreases. For the empirical risk minimization, we establish asymptotic and non-asymptotic convergence guarantees and demonstrate that heavy-tailed population noise can deteriorate convergence bounds through dataset-dependent constants as the dataset size increases. Furthermore, under Lipschitz smoothness, we formulate multistage Adam with practical learning rate schedules and demonstrate that increasing batch sizes can reduce index-sampling noise and improve convergence guarantees and SFO complexity. Numerical experiments support our theoretical findings on the effects of heavy-tailed noise and the computational benefits of increasing batch sizes.