Theoretical and Empirical Analysis of Adam Optimizer with Constant Step Size in Non-Convex Settings

Conceitos Básicos

This work provides theoretical guarantees for the convergence of the Adam optimizer with a constant step size in non-convex settings. It also proposes a method to estimate the Lipschitz constant of the loss function, which is crucial for determining the optimal constant step size.

Resumo

The paper presents a theoretical and empirical study on the convergence of the Adam optimizer with a constant step size in non-convex settings.
Key highlights:

The authors derive an exact constant step size that guarantees the convergence of deterministic and stochastic Adam algorithms in non-convex settings.
They provide runtime bounds for deterministic and stochastic Adam to achieve approximate criticality with smooth non-convex functions.
They introduce a method to estimate the Lipschitz constant of the loss function with respect to the network parameters, which is crucial for determining the optimal constant step size.
The empirical analysis suggests that even with the accumulation of past gradients, the key driver for convergence in Adam is the non-increasing nature of the step sizes.
The experiments validate the effectiveness of the proposed constant step size, which drives the gradient norm towards zero more aggressively than commonly used schedulers and a range of constant step sizes.

The authors conclude that their derived step size is easy to use and estimate, and can be applied to a wide range of tasks.

Estatísticas

The paper does not provide any specific numerical data or metrics to support the key claims. The analysis is primarily based on theoretical derivations and empirical observations.

Citações

"This work demonstrates the derivation and effective implementation of a constant step size for Adam, offering insights into its performance and efficiency in non-convex optimisation scenarios."
"Our empirical findings suggest that even with the accumulation of the past few gradients, the key driver for convergence in Adam is the non-increasing nature of step sizes."

Principais Insights Extraídos De

A Theoretical and Empirical Study on the Convergence of Adam with an "Exact" Constant Step Size in Non-Convex Settings

by Alokendu Maz... às arxiv.org 04-05-2024

https://arxiv.org/pdf/2309.08339.pdf

Perguntas Mais Profundas

How can the convergence rate of Adam be further tightened beyond the current analysis

To further tighten the convergence rate of Adam beyond the current analysis, several approaches can be considered. One potential avenue is to explore adaptive step size strategies that dynamically adjust the step size based on the local curvature of the loss landscape. By incorporating more sophisticated techniques for adapting the step size, such as adaptive learning rate schedules that respond to the geometry of the optimization problem, it may be possible to achieve faster convergence rates for Adam. Additionally, investigating the impact of momentum parameters and their interplay with the step size could provide insights into optimizing the convergence rate of Adam further.

What are the potential limitations or drawbacks of the proposed constant step size approach in practical scenarios

While the proposed constant step size approach offers theoretical guarantees for the convergence of Adam in non-convex settings, there are potential limitations and drawbacks to consider in practical scenarios. One limitation is the sensitivity of the constant step size to the Lipschitz constant estimation of the loss function. In real-world applications, accurately estimating the Lipschitz constant may be challenging, leading to suboptimal performance of the constant step size approach. Additionally, the fixed step size may not adapt well to varying gradients and curvature in the loss landscape, potentially hindering convergence in complex optimization problems. Furthermore, the proposed approach may require fine-tuning of hyperparameters to achieve optimal performance, which can be time-consuming and computationally intensive.

How can the theoretical analysis be extended to account for the effect of batch size on the convergence of stochastic Adam

To extend the theoretical analysis to account for the effect of batch size on the convergence of stochastic Adam, one approach is to incorporate the batch size parameter into the convergence analysis framework. By considering the impact of batch size on the variance of stochastic gradients and the estimation of the Lipschitz constant, the theoretical analysis can be extended to provide insights into how different batch sizes influence the convergence properties of Adam. Additionally, exploring the relationship between batch size, learning rate, and convergence guarantees in stochastic optimization algorithms can offer a more comprehensive understanding of the interplay between these factors in practical optimization scenarios.