Why Adam Can Beat SGD: Second-Moment Normalization Yields Sharper Tails

We provide a theoretical analysis of Adam under non-stationary stochastic objectives, separating two regimes: Euclidean tracking under adaptive strong monotonicity of the Adam-preconditioned mean-gradient operator, and high-probability projected stationarity guarantees under general $L$-smooth objectives. In the tracking regime, we derive finite-time expected and high-probability bounds that decompose sharply into four components: initialization, objective drift, a first-moment tracking error governed by $\beta_1$, and a preconditioner perturbation governed by $\beta_2$. We characterize the burn-in time required for the transient terms to decay to the asymptotic tracking bound under constant and step-decay schedules. We also prove a high-probability bound on the average projected stationarity gap for Adam under distribution shift. Across both analyses, our bounds reveal a noise--drift tradeoff: in noise-dominated regimes, first-moment averaging and adaptive preconditioning can yield favorable upper guarantees, whereas in drift-dominated regimes, stale first-moment information and preconditioner perturbations can enlarge Adam's tracking guarantee, potentially allowing vanilla SGD to attain a smaller tracking error. Our explicit $(\beta_1,\beta_2,\epsilon)$-dependent bounds identify mechanisms through which adaptive step-sizing can help or hurt under nonstationarity and provide theoretical explanations consistent with Adam's empirical instability and stabilization under distribution shift.

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 $δ^{-1/2}$, while the stepsize prefactor depends on $δ$ only through a single logarithmic factor. We further construct a hard instance showing that, under only second-moment information, this $δ^{-1/2}$-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.

We establish the first convergence guarantees for the plain vector-form Adam optimizer under heavy-tailed stochastic noise. While several Adam variants are known to achieve optimal iteration complexity in bounded-variance nonsmooth nonconvex optimization, little is understood about their behavior when stochastic gradients admit only a bounded $p$-th central moment for some $p \in (1,2]$, a setting increasingly observed in modern deep learning. To address this gap, we generalize the recent online-to-nonconvex conversion framework to accommodate heavy-tailed martingale-difference noise. Building on this generalized framework, we develop a discounted regret analysis for Adam, without restrictive parameter coupling. Our results show that Adam converges to $(ρ,ε)$-stationary points under heavy-tailed noise. However, it exhibits a suboptimal iteration complexity and $p$-dependent convergence, a suboptimality that persists even in the bounded-variance case ($p=2$). Specifically, the $ε$-dominant term in the iteration complexity for reaching in-expectation stationarity is $T=\mathrm{O}\left(Δρ^{1/2}(G+σ)^{\frac{5p}{3p-4}}ε^{-\left(\frac{5p}{3p-4}+\frac{3}{2}\right)}\right)$ for $p\in(\frac{4}{3},2]$, which simplifies to $T=\mathrm{O}(ε^{-13/2})$ when $p=2$. When the domain radius is known and used to control the online-learner output, a standard setup in related literature, the convergence rate improves to match the optimal complexity. In this case, the $ε$-dominant iteration complexity is $T=\mathrm{O}\left(Δρ^{1/2}(G+σ)^{\frac{p}{p-1}}ε^{-\left(\frac{p}{p-1}+\frac{3}{2}\right)}\right)$ for $p\in(1,2]$, which simplifies to $T=\mathrm{O}(ε^{-7/2})$ when $p=2$. These findings provide new theoretical insight into the robustness and limitations of Adam in heavy-tailed regimes.