A Full Adam Theorem for Spectral Heavy-Tail Onset

Heavy-tailed empirical spectral densities of neural-network weight matrices are widely used as diagnostics of implicit self-regularization, but the step complexity of heavy-tail emergence remains poorly understood. We formulate spectral heavy-tail formation as a right-censored hitting-time problem: a run that does not reach a heavy-tail diagnostic within the observation horizon is treated as censored rather than discarded. In controlled full-batch teacher--student dynamics, we find that the first-step spike--bulk gap alone does not explain onset time. Instead, finite-onset regression supports a dimension-corrected spectral-gap law, (\tau_{\mathrm{HT}}\approx C\Delta_1^{-\gamma}d^\rho), with (R^2=0.683), (\gamma=0.626), and (\rho=0.772) across 330 completed runs. Right-censored lognormal accelerated-failure-time models further favor the dimension-corrected model over a gap-only model, improving AIC from 706.62 to 628.70. Theoretically, we prove that exact early loss dynamics in linear networks do not determine factor spectral tails, that Adam recurrences alone do not imply spectral redistribution, and that projected singular-basis spreading implies contraction of a spectral-tail potential and hence a dimension-corrected hitting-time bound. Empirically, projected-kernel profiles support the sufficient spreading mechanism, Adam and AdamW agree under tested grids, GD and signGD do not reach onset in the same regimes, and real pretrained Qwen2.5-0.5B and Pythia-70M transformer weights show non-Gaussian spectral-tail structure relative to matched Gaussian nulls. The result is a reproducible spectral hitting-time law with rigorous conditional theory, not a claim that Adam necessarily generates heavy tails from first principles.

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.

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.