Adam Optimizer Convergence Analyzed Under Heavy-Tailed Noise

Yijiang Pang· July 31, 2026 View original

Key takeaways

  • Adam optimizer's convergence is guaranteed even with heavy-tailed noise.
  • Its convergence rate is suboptimal, even in standard bounded-variance cases.
  • Knowing the domain radius can improve Adam's convergence speed.
  • The findings offer new theoretical insights into Adam's behavior in challenging environments.

Who benefits

AI/ML DevelopmentData ScienceAutonomous SystemsFinancial Modeling

Summary

This research provides the first convergence guarantees for the Adam optimizer when dealing with heavy-tailed stochastic noise, a common scenario in deep learning. It reveals that Adam converges to stationary points but with suboptimal iteration complexity, even in bounded-variance cases.

The Adam optimizer, widely used in deep learning, has been theoretically analyzed for its convergence properties under conditions of heavy-tailed stochastic noise. This type of noise, where extreme values are more probable, is increasingly observed in modern deep learning environments. The study establishes the first formal convergence guarantees for the plain vector-form Adam. The findings indicate that Adam does converge to a stable state, specifically to (rho,epsilon)-stationary points, even with heavy-tailed noise. However, this convergence comes with a suboptimal iteration complexity, meaning it takes more steps than theoretically ideal. This suboptimality was found to persist even in standard bounded-variance scenarios. The research suggests that if the domain radius is known and used to control the online-learner output, the convergence rate can improve to match optimal complexity, offering new insights into Adam's robustness and limitations.

Why it matters

Professionals using Adam for deep learning models, especially in noisy or complex data environments, gain a deeper understanding of its theoretical performance limits and potential inefficiencies. This insight can guide hyperparameter tuning and algorithm selection for more robust model training.

How to implement this in your domain

  1. 1Review current Adam implementations for potential heavy-tailed noise issues in data.
  2. 2Consider alternative optimizers or Adam variants if convergence speed is critical in noisy settings.
  3. 3Experiment with explicit domain radius control if applicable to improve Adam's convergence rate.
  4. 4Monitor training stability and convergence behavior more closely in applications prone to heavy-tailed noise.

Original post by Yijiang Pang

"arXiv:2607.27383v1 Announce Type: new Abstract: We establish the first convergence guarantees for the plain vector-form \emph{Adam} optimizer under heavy-tailed stochastic noise. While several Adam variants are known to achieve optimal iteration complexity in bounded-variance non…"

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