Adam under Generalized Smoothness with Second-Moment-Type Stochastic Gradients

arXiv:2609.37787v1 Announce Type: new Abstract: Adam is widely observed to remain stable even when the objective deviates significantly from global smoothness. Under the generalized smoothness framework, however, existing analyses rely on strong tail assumptions on the stochastic gradients, such as almost-sure boundedness or sub-Gaussianity. Whether Adam converges on generalized smooth objectives under only second moment information on the stochastic gradients, without such concentration assumptions, was identified as an important open direction by Li et al. (2023). This paper gives an affirmative answer under fairly general conditions: such tail assumptions are not necessary. Building on the Adam self-normalization framework of Jin et al. (2026), developed for classical smoothness and bounded variance, we extend the stopping-time and de-preconditioning strategy to the $L_0$-$L_p$ generalized smoothness condition and a generalized second moment ABC condition. Even when the stochastic-gradient condition provides only second moment information that may grow along the trajectory, the stochastic trajectory of Adam remains in a locally well-behaved smoothness region, with stretched-exponential tail decay under bounded variance and global smoothness. Consequently, we establish high-probability convergence rate guarantees over the full range $p<2$, with confidence dependence of order $\delta^{-1/2}$, while the stepsize prefactor depends on $\delta$ only through a single logarithmic factor. We further construct a hard instance showing that, under only second-moment information, this $\delta^{-1/2}$-type confidence dependence is sharp. Finally, in the regime $p<1$, we combine the trajectory control with polynomial-growth estimates on rare events to obtain convergence rate guarantees in expectation.

Abstract:Adam is widely observed to remain stable even when the objective deviates significantly from global smoothness. Under the generalized smoothness framework, however, existing analyses rely on strong tail assumptions on the stochastic gradients, such as almost-sure boundedness or sub-Gaussianity. Whether Adam converges on generalized smooth objectives under only second moment information on the stochastic gradients, without such concentration assumptions, was identified as an important open direction by Li et al. (2023). This paper gives an affirmative answer under fairly general conditions: such tail assumptions are not necessary. Building on the Adam self-normalization framework of Jin et al. (2026), developed for classical smoothness and bounded variance, we extend the stopping-time and de-preconditioning strategy to the $L_0$-$L_p$ generalized smoothness condition and a generalized second moment ABC condition. Even when the stochastic-gradient condition provides only second moment information that may grow along the trajectory, the stochastic trajectory of Adam remains in a locally well-behaved smoothness region, with stretched-exponential tail decay under bounded variance and global smoothness. Consequently, we establish high-probability convergence rate guarantees over the full range $p<2$, with confidence dependence of order $\delta^{-1/2}$, while the stepsize prefactor depends on $\delta$ only through a single logarithmic factor. We further construct a hard instance showing that, under only second-moment information, this $\delta^{-1/2}$-type confidence dependence is sharp. Finally, in the regime $p<1$, we combine the trajectory control with polynomial-growth estimates on rare events to obtain convergence rate guarantees in expectation.

37 pages, 4 figures. Accepted at NeurIPS 2026

Artificial Intelligence (cs.AI)

arXiv:2609.37787 [cs.AI]

(or arXiv:2609.37787v1 [cs.AI] for this version)

https://doi.org/10.48550/arXiv.2609.37787 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

aiqbench analysis

Adam under Generalized Smoothness with Second-Moment-Type Stochastic Gradients

Related coverage