The Convergence Behavior of Adam under Heavy-Tailed Noise
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 $(\rho,\epsilon)$-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 $\epsilon$-dominant term in the iteration complexity for reaching in-expectation stationarity is $T=\mathrm{O}\left(\Delta \rho^{1/2}(G+\sigma)^{\frac{5p}{3p-4}}\epsilon^{-\left(\frac{5p}{3p-4}+\frac{3}{2}\right)}\right)$ for $p\in(\frac{4}{3},2]$, which simplifies to $T=\mathrm{O}(\epsilon^{-13/2})$ 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 $\epsilon$-dominant iteration complexity is $T=\mathrm{O}\left(\Delta \rho^{1/2}(G+\sigma)^{\frac{p}{p-1}}\epsilon^{-\left(\frac{p}{p-1}+\frac{3}{2}\right)}\right)$ for $p\in(1,2]$, which simplifies to $T=\mathrm{O}(\epsilon^{-7/2})$ when $p=2$. These findings provide new theoretical insight into the robustness and limitations of Adam in heavy-tailed regimes.