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arXiv · 2605.04269

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

Sharan Sahu·Abir Sarkar·Cameron J. Hogan·Martin T. Wells
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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 $β_1$, and a preconditioner perturbation governed by $β_2$. 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 $(β_1,β_2,ε)$-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.

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Sharan Sahu, Abir Sarkar, Cameron J. Hogan, Martin T. Wells. 2026-09-11. Adapt or Forget: Provable Tradeoffs Between Adam and SGD in Nonstationary Optimization. https://arxiv.org/abs/2605.04269

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