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Adaptive Gradient Methods with Dynamic Bound of Learning Rate

Liangchen Luo, Yuanhao Xiong, Yan Liu, Xu Sun

TL;DR

Adaptive optimizers can suffer from unstable learning rates that hurt generalization. AdaBound and AMSBound impose dynamic bounds on per-parameter learning rates, enabling a smooth transition from Adam-like updates to SGD(M) and achieving convergence guarantees in the convex setting. The methods deliver fast early training and improved generalization across vision and NLP tasks, with larger gains on deeper networks. This approach offers a practical, theoretically motivated way to combine the strengths of adaptive methods with SGD's generalization benefits.

Abstract

Adaptive optimization methods such as AdaGrad, RMSprop and Adam have been proposed to achieve a rapid training process with an element-wise scaling term on learning rates. Though prevailing, they are observed to generalize poorly compared with SGD or even fail to converge due to unstable and extreme learning rates. Recent work has put forward some algorithms such as AMSGrad to tackle this issue but they failed to achieve considerable improvement over existing methods. In our paper, we demonstrate that extreme learning rates can lead to poor performance. We provide new variants of Adam and AMSGrad, called AdaBound and AMSBound respectively, which employ dynamic bounds on learning rates to achieve a gradual and smooth transition from adaptive methods to SGD and give a theoretical proof of convergence. We further conduct experiments on various popular tasks and models, which is often insufficient in previous work. Experimental results show that new variants can eliminate the generalization gap between adaptive methods and SGD and maintain higher learning speed early in training at the same time. Moreover, they can bring significant improvement over their prototypes, especially on complex deep networks. The implementation of the algorithm can be found at https://github.com/Luolc/AdaBound .

Adaptive Gradient Methods with Dynamic Bound of Learning Rate

TL;DR

Adaptive optimizers can suffer from unstable learning rates that hurt generalization. AdaBound and AMSBound impose dynamic bounds on per-parameter learning rates, enabling a smooth transition from Adam-like updates to SGD(M) and achieving convergence guarantees in the convex setting. The methods deliver fast early training and improved generalization across vision and NLP tasks, with larger gains on deeper networks. This approach offers a practical, theoretically motivated way to combine the strengths of adaptive methods with SGD's generalization benefits.

Abstract

Adaptive optimization methods such as AdaGrad, RMSprop and Adam have been proposed to achieve a rapid training process with an element-wise scaling term on learning rates. Though prevailing, they are observed to generalize poorly compared with SGD or even fail to converge due to unstable and extreme learning rates. Recent work has put forward some algorithms such as AMSGrad to tackle this issue but they failed to achieve considerable improvement over existing methods. In our paper, we demonstrate that extreme learning rates can lead to poor performance. We provide new variants of Adam and AMSGrad, called AdaBound and AMSBound respectively, which employ dynamic bounds on learning rates to achieve a gradual and smooth transition from adaptive methods to SGD and give a theoretical proof of convergence. We further conduct experiments on various popular tasks and models, which is often insufficient in previous work. Experimental results show that new variants can eliminate the generalization gap between adaptive methods and SGD and maintain higher learning speed early in training at the same time. Moreover, they can bring significant improvement over their prototypes, especially on complex deep networks. The implementation of the algorithm can be found at https://github.com/Luolc/AdaBound .

Paper Structure

This paper contains 20 sections, 9 theorems, 39 equations, 8 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. The Non-Convergence Caused by Extreme Learning Rate
  4. Adaptive Moment Estimation with Dynamic Bound
  5. Experiments
  6. Hyperparameter Tuning
  7. Feedforward Neural Network
  8. Convolutional Neural Network
  9. Recurrent Neural Network
  10. Analysis
  11. Future Work
  12. Conclusion
  13. Auxiliary Lemmas
  14. Proof of Theorem \ref{['the:spec-example']}
  15. Proof of Theorem \ref{['the:gen-example']}
  16. ...and 5 more sections

Key Result

Theorem 1

There is an online convex optimization problem where for any initial step size $\alpha$, Adam has non-zero average regret i.e., $R_T/T \nrightarrow 0$ as $T \rightarrow \infty$.

Figures (8)

  • Figure 1: Learning rates of sampled parameters. Each cell contains a value obtained by conducting a logarithmic operation on the learning rate. The lighter cell stands for the smaller learning rate.
  • Figure 2: Training (left) and test accuracy (right) for feedforward neural network on MNIST.
  • Figure 3: Training and test accuracy for DenseNet-121 and ResNet-34 on CIFAR-10.
  • Figure 4: Perplexity curves on the test set comparing Sgd, Adam, AdaBound and AMSBound for the LSTM with different layers on Penn Treebank.
  • Figure 5: Test accuracy of AdaBound with different $\beta$ using ResNet-34 on CIFAR-10.
  • ...and 3 more figures

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark
  • Theorem 4
  • Corollary 4.1
  • Lemma 1: Mcmahan2010AdaptiveOptimization
  • proof
  • Lemma 2
  • proof
  • ...and 7 more