The Limited Benefit of Momentum in Stochastic Gradient Descent for Small Learning Rates

Kernekoncepter

Momentum does not provide significant optimization or generalization benefits in stochastic gradient descent when the learning rate is small, as SGD and SGD with Momentum (SGDM) exhibit similar behaviors in both short-term and long-term training.

Resumé

The paper investigates the role of momentum in stochastic gradient descent (SGD) optimization, particularly in settings where the learning rate is small. The key findings are:

Theoretical Analysis:

In the short-term (O(1/η) steps), SGD and SGDM are shown to be weakly approximated, meaning their distributions are similar. This holds even when the gradient noise scales inversely with the learning rate.
In the long-term (O(1/η^2) steps), SGD and SGDM are proven to have the same limiting dynamics when the iterates are close to a manifold of local minimizers. This suggests momentum does not provide extra generalization benefits over SGD in this regime.

Empirical Verification:

Experiments on ImageNet training of ResNet-50 and fine-tuning of RoBERTa-large confirm that SGD and SGDM perform comparably when the optimal learning rate is small.
For large-batch training on CIFAR-10, the authors show that the benefit of momentum can be attributed to its ability to mitigate curvature-induced impacts, rather than noise reduction. When the curvature-induced impact is reduced, the performance gap between SGD and SGDM diminishes.

The paper concludes that momentum provides limited benefits in practical training regimes where the optimal learning rate is not very large, and that model performance is generally insensitive to the choice of momentum hyperparameters in such settings.

Statistik

The paper does not provide any specific numerical data or metrics to support the claims. The analysis is primarily theoretical, with empirical results presented in a qualitative manner.

Citater

None.

Vigtigste indsigter udtrukket fra

The Marginal Value of Momentum for Small Learning Rate SGD

by Runzhe Wang,... kl. arxiv.org 04-17-2024

https://arxiv.org/pdf/2307.15196.pdf

Dybere Forespørgsler

What are the implications of this work for the design of optimization algorithms beyond SGD and SGDM, such as adaptive methods or second-order methods

The implications of this work for the design of optimization algorithms beyond SGD and SGDM are significant. The findings suggest that in scenarios where the optimal learning rate is not very large, the benefits of momentum are limited. This insight can guide the design of adaptive methods by highlighting the importance of considering the interplay between momentum and gradient noise.
For adaptive methods, understanding the marginal value of momentum in small learning rate regimes can inform the development of algorithms that dynamically adjust momentum based on the noise level in the gradients. By incorporating this knowledge, adaptive methods can potentially achieve better convergence and generalization performance, especially in scenarios where the noise in gradient estimates plays a significant role in optimization stability.
Moreover, for second-order methods, the insights from this work can guide the integration of momentum to enhance convergence properties. By considering the trade-off between momentum and gradient noise, second-order methods can be designed to leverage momentum effectively in scenarios where it provides tangible benefits, while mitigating its impact in situations where the noise dominates the optimization process.

Can the theoretical insights be extended to understand the role of momentum in other stochastic optimization settings, such as non-convex problems or distributed training

The theoretical insights from this work can be extended to understand the role of momentum in various other stochastic optimization settings, including non-convex problems and distributed training. In non-convex optimization, momentum is often used to accelerate convergence and escape local minima. By applying the framework developed in this work, researchers can analyze how momentum interacts with gradient noise in non-convex settings and determine its impact on optimization performance.
In distributed training, where multiple devices collaborate to train a model, the role of momentum becomes crucial in ensuring convergence across different nodes. By extending the theoretical analyses to distributed settings, researchers can investigate how momentum affects the stability and convergence of distributed optimization algorithms. Understanding the interplay between momentum, learning rates, and noise in distributed training can lead to the development of more efficient and robust distributed optimization strategies.

Are there specific problem domains or model architectures where momentum may still provide significant benefits, even in the small learning rate regime

While the study suggests that momentum has limited benefits in practical training regimes with small learning rates, there are specific problem domains and model architectures where momentum may still provide significant advantages. In scenarios where the optimization landscape is highly non-convex and contains sharp minima, momentum can help navigate the complex surface and escape saddle points more effectively.
Additionally, for models with intricate architectures such as recurrent neural networks or transformers, momentum can aid in smoothing out the optimization trajectory and accelerating convergence. In these cases, momentum may play a crucial role in improving optimization efficiency and generalization performance, even in scenarios with small learning rates.
Overall, while the study highlights the marginal value of momentum in certain settings, there are still domains and architectures where momentum remains a valuable tool for enhancing optimization outcomes.