Adapt or Forget: Provable Tradeoffs Between Adam and SGD in Nonstationary Optimization

Abstract: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.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2605.04269 [stat.ML]

(or arXiv:2605.04269v2 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2605.04269 Focus to learn more arXiv-issued DOI via DataCite

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