On the Convergence Proof of AMSGrad and a New Version

The adaptive moment estimation algorithm Adam (Kingma and Ba) is a popular optimizer in the training of deep neural networks. However, Reddi et al. have recently shown that the convergence proof of Adam is problematic and proposed a variant of Adam called AMSGrad as a fix. In this paper, we show that the convergence proof of AMSGrad is also problematic. Concretely, the problem in the convergence proof of AMSGrad is in handling the hyper-parameters, treating them as equal while they are not. This is also the neglected issue in the convergence proof of Adam. We provide an explicit counter-example of a simple convex optimization setting to show this neglected issue. Depending on manipulating the hyper-parameters, we present various fixes for this issue. We provide a new convergence proof for AMSGrad as the first fix. We also propose a new version of AMSGrad called AdamX as another fix. Our experiments on the benchmark dataset also support our theoretical results.Comment: Update publication informatio

On the Convergence of A Class of Adam-Type Algorithms for Non-Convex Optimization

This paper studies a class of adaptive gradient based momentum algorithms that update the search directions and learning rates simultaneously using past gradients. This class, which we refer to as the "Adam-type", includes the popular algorithms such as the Adam, AMSGrad and AdaGrad. Despite their popularity in training deep neural networks, the convergence of these algorithms for solving nonconvex problems remains an open question. This paper provides a set of mild sufficient conditions that guarantee the convergence for the Adam-type methods. We prove that under our derived conditions, these methods can achieve the convergence rate of order $O(\log{T}/\sqrt{T})$ for nonconvex stochastic optimization. We show the conditions are essential in the sense that violating them may make the algorithm diverge. Moreover, we propose and analyze a class of (deterministic) incremental adaptive gradient algorithms, which has the same $O(\log{T}/\sqrt{T})$ convergence rate. Our study could also be extended to a broader class of adaptive gradient methods in machine learning and optimization

Closing the Generalization Gap of Adaptive Gradient Methods in Training Deep Neural Networks

Adaptive gradient methods, which adopt historical gradient information to automatically adjust the learning rate, despite the nice property of fast convergence, have been observed to generalize worse than stochastic gradient descent (SGD) with momentum in training deep neural networks. This leaves how to close the generalization gap of adaptive gradient methods an open problem. In this work, we show that adaptive gradient methods such as Adam, Amsgrad, are sometimes "over adapted". We design a new algorithm, called Partially adaptive momentum estimation method, which unifies the Adam/Amsgrad with SGD by introducing a partial adaptive parameter $p$, to achieve the best from both worlds. We also prove the convergence rate of our proposed algorithm to a stationary point in the stochastic nonconvex optimization setting. Experiments on standard benchmarks show that our proposed algorithm can maintain a fast convergence rate as Adam/Amsgrad while generalizing as well as SGD in training deep neural networks. These results would suggest practitioners pick up adaptive gradient methods once again for faster training of deep neural networks.Comment: 17 pages, 4 figures, 4 tables. In IJCAI 202

Adaptive Gradient Methods with Dynamic Bound of Learning Rate

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 .Comment: Accepted to ICLR 2019. arXiv admin note: text overlap with arXiv:1904.09237 by other author

AdaShift: Decorrelation and Convergence of Adaptive Learning Rate Methods

Adam is shown not being able to converge to the optimal solution in certain cases. Researchers recently propose several algorithms to avoid the issue of non-convergence of Adam, but their efficiency turns out to be unsatisfactory in practice. In this paper, we provide new insight into the non-convergence issue of Adam as well as other adaptive learning rate methods. We argue that there exists an inappropriate correlation between gradient $g_t$ and the second-moment term $v_t$ in Adam ($t$ is the timestep), which results in that a large gradient is likely to have small step size while a small gradient may have a large step size. We demonstrate that such biased step sizes are the fundamental cause of non-convergence of Adam, and we further prove that decorrelating $v_t$ and $g_t$ will lead to unbiased step size for each gradient, thus solving the non-convergence problem of Adam. Finally, we propose AdaShift, a novel adaptive learning rate method that decorrelates $v_t$ and $g_t$ by temporal shifting, i.e., using temporally shifted gradient $g_{t-n}$ to calculate $v_t$. The experiment results demonstrate that AdaShift is able to address the non-convergence issue of Adam, while still maintaining a competitive performance with Adam in terms of both training speed and generalization.Comment: Published as a conference paper at ICLR 201

Riemannian Adaptive Optimization Methods

Several first order stochastic optimization methods commonly used in the Euclidean domain such as stochastic gradient descent (SGD), accelerated gradient descent or variance reduced methods have already been adapted to certain Riemannian settings. However, some of the most popular of these optimization tools - namely Adam , Adagrad and the more recent Amsgrad - remain to be generalized to Riemannian manifolds. We discuss the difficulty of generalizing such adaptive schemes to the most agnostic Riemannian setting, and then provide algorithms and convergence proofs for geodesically convex objectives in the particular case of a product of Riemannian manifolds, in which adaptivity is implemented across manifolds in the cartesian product. Our generalization is tight in the sense that choosing the Euclidean space as Riemannian manifold yields the same algorithms and regret bounds as those that were already known for the standard algorithms. Experimentally, we show faster convergence and to a lower train loss value for Riemannian adaptive methods over their corresponding baselines on the realistic task of embedding the WordNet taxonomy in the Poincare ball.Comment: Accepted at International Conference on Learning Representations (ICLR), 201

On the Convergence of Adam and Beyond

Several recently proposed stochastic optimization methods that have been successfully used in training deep networks such as RMSProp, Adam, Adadelta, Nadam are based on using gradient updates scaled by square roots of exponential moving averages of squared past gradients. In many applications, e.g. learning with large output spaces, it has been empirically observed that these algorithms fail to converge to an optimal solution (or a critical point in nonconvex settings). We show that one cause for such failures is the exponential moving average used in the algorithms. We provide an explicit example of a simple convex optimization setting where Adam does not converge to the optimal solution, and describe the precise problems with the previous analysis of Adam algorithm. Our analysis suggests that the convergence issues can be fixed by endowing such algorithms with `long-term memory' of past gradients, and propose new variants of the Adam algorithm which not only fix the convergence issues but often also lead to improved empirical performance.Comment: Appeared in ICLR 201

Deep Frank-Wolfe For Neural Network Optimization

Learning a deep neural network requires solving a challenging optimization problem: it is a high-dimensional, non-convex and non-smooth minimization problem with a large number of terms. The current practice in neural network optimization is to rely on the stochastic gradient descent (SGD) algorithm or its adaptive variants. However, SGD requires a hand-designed schedule for the learning rate. In addition, its adaptive variants tend to produce solutions that generalize less well on unseen data than SGD with a hand-designed schedule. We present an optimization method that offers empirically the best of both worlds: our algorithm yields good generalization performance while requiring only one hyper-parameter. Our approach is based on a composite proximal framework, which exploits the compositional nature of deep neural networks and can leverage powerful convex optimization algorithms by design. Specifically, we employ the Frank-Wolfe (FW) algorithm for SVM, which computes an optimal step-size in closed-form at each time-step. We further show that the descent direction is given by a simple backward pass in the network, yielding the same computational cost per iteration as SGD. We present experiments on the CIFAR and SNLI data sets, where we demonstrate the significant superiority of our method over Adam, Adagrad, as well as the recently proposed BPGrad and AMSGrad. Furthermore, we compare our algorithm to SGD with a hand-designed learning rate schedule, and show that it provides similar generalization while converging faster. The code is publicly available at https://github.com/oval-group/dfw.Comment: Published as a conference paper at ICLR 201

Convergence Analyses of Online ADAM Algorithm in Convex Setting and Two-Layer ReLU Neural Network

Nowadays, online learning is an appealing learning paradigm, which is of great interest in practice due to the recent emergence of large scale applications such as online advertising placement and online web ranking. Standard online learning assumes a finite number of samples while in practice data is streamed infinitely. In such a setting gradient descent with a diminishing learning rate does not work. We first introduce regret with rolling window, a new performance metric for online streaming learning, which measures the performance of an algorithm on every fixed number of contiguous samples. At the same time, we propose a family of algorithms based on gradient descent with a constant or adaptive learning rate and provide very technical analyses establishing regret bound properties of the algorithms. We cover the convex setting showing the regret of the order of the square root of the size of the window in the constant and dynamic learning rate scenarios. Our proof is applicable also to the standard online setting where we provide the first analysis of the same regret order (the previous proofs have flaws). We also study a two layer neural network setting with ReLU activation. In this case we establish that if initial weights are close to a stationary point, the same square root regret bound is attainable. We conduct computational experiments demonstrating a superior performance of the proposed algorithms

SAdam: A Variant of Adam for Strongly Convex Functions

The Adam algorithm has become extremely popular for large-scale machine learning. Under convexity condition, it has been proved to enjoy a data-dependant $O(\sqrt{T})$ regret bound where $T$ is the time horizon. However, whether strong convexity can be utilized to further improve the performance remains an open problem. In this paper, we give an affirmative answer by developing a variant of Adam (referred to as SAdam) which achieves a data-dependant $O(\log T)$ regret bound for strongly convex functions. The essential idea is to maintain a faster decaying yet under controlled step size for exploiting strong convexity. In addition, under a special configuration of hyperparameters, our SAdam reduces to SC-RMSprop, a recently proposed variant of RMSprop for strongly convex functions, for which we provide the first data-dependent logarithmic regret bound. Empirical results on optimizing strongly convex functions and training deep networks demonstrate the effectiveness of our method.Comment: 19 pages, 9 figure