Analyzing Convergence of Adam vs. SGDM under Non-uniform Smoothness

Concepts de base

Adam converges faster than SGDM under non-uniform smoothness conditions.

Résumé

This paper compares the convergence rates of Adam and Stochastic Gradient Descent with Momentum (SGDM) under non-uniform smoothness conditions. It highlights that Adam achieves faster convergence rates compared to SGDM, providing theoretical insights into their performance. The analysis is conducted in both deterministic and stochastic settings, showcasing the superiority of Adam over SGDM in complex optimization landscapes.

Directory:

1. Abstract:
   • Adam converges faster than SGDM under non-uniform smoothness.
2. Introduction:
   • Adam's success in deep learning applications.
3. Data Extraction:
   • "Adam can attain the known lower bound for the convergence rate of deterministic first-order optimizers."
   • "In stochastic setting, Adam’s convergence rate upper bound matches the lower bounds of stochastic first-order optimizers."
4. Related Works:
   • Various optimization techniques compared to Adam.
5. Preliminary Notations:
   • Asymptotic notations used in the paper.
6. Separating Convergence Rates:
   • Theoretical analysis for deterministic and stochastic settings.
7. Conclusion:
   • Summary of findings and impact statement.

Stats

"Adam can attain the known lower bound for the convergence rate of deterministic first-order optimizers."
"In stochastic setting, Adam’s convergence rate upper bound matches the lower bounds of stochastic first-order optimizers."

Citations

"Adam can attain the known lower bound for the convergence rate of deterministic first-order optimizers."
"In stochastic setting, Adam’s convergence rate upper bound matches the lower bounds of stochastic first-order optimizers."

Idées clés tirées de

On the Convergence of Adam under Non-uniform Smoothness

by Bohan Wang,H... à arxiv.org 03-25-2024

https://arxiv.org/pdf/2403.15146.pdf

Questions plus approfondies

Can SGD achieve similar convergence rates as Adam

In the context of non-uniform smoothness, SGD typically struggles to achieve similar convergence rates as Adam. The research outlined in the provided text highlights that under certain conditions and assumptions, Adam outperforms SGD in terms of convergence speed. Specifically, Adam has been shown to converge faster than SGD in both deterministic and stochastic settings when considering factors such as initial function value gaps, final errors, gradient norms, and adaptive learning rates. The theoretical analysis presented indicates that Adam can match or even surpass lower bounds for convergence rates compared to SGD under non-uniform smoothness conditions.

Does practical implementation align with theoretical findings

Practical implementation often aligns with theoretical findings but may require some adjustments based on real-world constraints and considerations. While the research demonstrates the superior performance of Adam over SGD in terms of convergence rates under specific conditions, practical implementations may need to consider additional factors such as computational resources, dataset characteristics, hyperparameter tuning strategies, and model complexity. It is essential for practitioners to validate these theoretical findings through empirical studies across various datasets and models to ensure applicability in real-world scenarios.

How does this research impact future optimization algorithms

The research on the convergence behavior of optimization algorithms like Adam and SGDM under non-uniform smoothness has significant implications for future optimization algorithms. By providing insights into how different algorithms perform under varying levels of smoothness and noise variance, this research can guide the development of more efficient optimization techniques tailored to specific problem domains. The identification of optimal convergence rates for different algorithms can inform algorithm selection decisions in deep learning applications where fast training times are crucial. Additionally, by introducing novel stopping-time techniques like those discussed in the paper, researchers can further refine optimization algorithms' performance across diverse problem hyperparameters leading to advancements in machine learning efficiency and effectiveness.