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Title: Adaptive Gradient Methods with Dynamic Bound of Learning Rate

Paper content:
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 .
There has been tremendous progress in first-order optimization algorithms for training deep neural networks.
One of the most dominant algorithms is stochastic gradient descent (SGD) BID15 , which performs well across many applications in spite of its simplicity.
However, there is a disadvantage of SGD that it scales the gradient uniformly in all directions.
This may lead to poor performance as well as limited training speed when the training data are sparse.
To address this problem, recent work has proposed a variety of adaptive methods that scale the gradient by square roots of some form of the average of the squared values of past gradients.
Examples of such methods include ADAM BID7 , ADAGRAD BID2 and RMSPROP BID16 .
ADAM in particular has become the default algorithm leveraged across many deep learning frameworks due to its rapid training speed BID17 .Despite
their popularity, the generalization ability and out-of-sample behavior of these adaptive methods are likely worse than their non-adaptive counterparts. Adaptive
methods often display faster progress in the initial portion of the training, but their performance quickly plateaus on the unseen data (development/test set) BID17 . Indeed,
the optimizer is chosen as SGD (or with momentum) in several recent state-of-the-art works in natural language processing and computer vision BID11 BID18 , wherein these instances SGD does perform better than adaptive methods. BID14 have
recently proposed a variant of ADAM called AMSGRAD, hoping to solve this problem. The authors
provide a theoretical guarantee of convergence but only illustrate its better performance on training data. However, the
generalization ability of AMSGRAD on unseen data is found to be similar to that of ADAM while a considerable performance gap still exists between AMSGRAD and SGD BID6 BID1 .In this paper
, we first conduct an empirical study on ADAM and illustrate that both extremely large and small learning rates exist by the end of training. The results
correspond with the perspective pointed out by BID17 that the lack of generalization performance of adaptive methods may stem from unstable and extreme learning rates. In fact, introducing
non-increasing learning rates, the key point in AMSGRAD, may help abate the impact of huge learning rates, while it neglects possible effects of small ones. We further provide an
example of a simple convex optimization problem to elucidate how tiny learning rates of adaptive methods can lead to undesirable non-convergence. In such settings, RMSPROP
and ADAM provably do not converge to an optimal solution, and furthermore, however large the initial step size α is, it is impossible for ADAM to fight against the scale-down term.Based on the above analysis, we propose new variants of ADAM and AMSGRAD, named AD-ABOUND and AMSBOUND, which do not suffer from the negative impact of extreme learning rates. We employ dynamic bounds
on learning rates in these adaptive methods, where the lower and upper bound are initialized as zero and infinity respectively, and they both smoothly converge to a constant final step size. The new variants can be
regarded as adaptive methods at the beginning of training, and they gradually and smoothly transform to SGD (or with momentum) as time step increases. In this framework, we can
enjoy a rapid initial training process as well as good final generalization ability. We provide a convergence
analysis for the new variants in the convex setting.We finally turn to an empirical study of the proposed methods on various popular tasks and models in computer vision and natural language processing. Experimental results demonstrate
that our methods have higher learning speed early in training and in the meantime guarantee strong generalization performance compared to several adaptive and non-adaptive methods. Moreover, they can bring considerable
improvement over their prototypes especially on complex deep networks.
We investigate existing adaptive algorithms and find that extremely large or small learning rates can result in the poor convergence behavior.
A rigorous proof of non-convergence for ADAM is provided to demonstrate the above problem.Motivated by the strong generalization ability of SGD, we design a strategy to constrain the learning rates of ADAM and AMSGRAD to avoid a violent oscillation.
Our proposed algorithms, AD-ABOUND and AMSBOUND, which employ dynamic bounds on their learning rates, achieve a smooth transition to SGD.
They show the great efficacy on several standard benchmarks while maintaining advantageous properties of adaptive methods such as rapid initial progress and hyperparameter insensitivity.

assistant: text: Novel variants of optimization methods that combine the benefits of both adaptive and non-adaptive methods.