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While Adam is one of the most effective optimizer for training large-scale machine learning models, a theoretical understanding of how to optimally set its momentum factors, $\beta_1$ and $\beta_2$, remains largely incomplete. Prior works…

We provide a simple proof of convergence covering both the Adam and Adagrad adaptive optimization algorithms when applied to smooth (possibly non-convex) objective functions with bounded gradients. We show that in expectation, the squared…

Adam is a popular variant of stochastic gradient descent for finding a local minimizer of a function. In the constant stepsize regime, assuming that the objective function is differentiable and non-convex, we establish the convergence in…

Adaptive gradient methods have achieved remarkable success in training deep neural networks on a wide variety of tasks. However, not much is known about the mathematical and statistical properties of this family of methods. This work aims…

Adam is a widely used stochastic optimization method for deep learning applications. While practitioners prefer Adam because it requires less parameter tuning, its use is problematic from a theoretical point of view since it may not…

The adaptive optimizer for training neural networks has continually evolved to overcome the limitations of the previously proposed adaptive methods. Recent studies have found the rare counterexamples that Adam cannot converge to the optimal…

Recently, Arjevani et al. [1] established a lower bound of iteration complexity for the first-order optimization under an $L$-smooth condition and a bounded noise variance assumption. However, a thorough review of existing literature on…

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…

Convergence and convergence rate analyses of adaptive methods, such as Adaptive Moment Estimation (Adam) and its variants, have been widely studied for nonconvex optimization. The analyses are based on assumptions that the expected or…

AdaBelief, one of the current best optimizers, demonstrates superior generalization ability compared to the popular Adam algorithm by viewing the exponential moving average of observed gradients. AdaBelief is theoretically appealing in that…

In this paper, we provide a rigorous proof of convergence of the Adaptive Moment Estimate (Adam) algorithm for a wide class of optimization objectives. Despite the popularity and efficiency of the Adam algorithm in training deep neural…

Adaptive optimization algorithms such as Adam are widely used in deep learning. The stability of such algorithms is often improved with a warmup schedule for the learning rate. Motivated by the difficulty of choosing and tuning warmup…

Recent evidence suggests that Adam performs robustly when its momentum parameters are tied, $\beta_1=\beta_2$, reducing the optimizer to a single remaining parameter. However, how this parameter should be set remains poorly understood. We…

Beside the standard stochastic gradient descent (SGD) method, the Adam optimizer due to Kingma & Ba (2014) is currently probably the best-known optimization method for the training of deep neural networks in artificial intelligence (AI)…

Despite Adam demonstrating faster empirical convergence than SGD in many applications, much of the existing theory yields guarantees essentially comparable to those of SGD, leaving the empirical performance gap insufficiently explained. In…

This paper aims to clearly distinguish between Stochastic Gradient Descent with Momentum (SGDM) and Adam in terms of their convergence rates. We demonstrate that Adam achieves a faster convergence compared to SGDM under the condition of…

In the training of neural networks, adaptive moment estimation (Adam) typically converges fast but exhibits suboptimal generalization performance. A widely accepted explanation for its defect in generalization is that it often tends to…

In this paper, we focus on a theory-practice gap for Adam and its variants (AMSgrad, AdamNC, etc.). In practice, these algorithms are used with a constant first-order moment parameter $\beta_{1}$ (typically between $0.9$ and $0.99$). In…

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…

With limited high-quality data and growing compute, multi-epoch training is gaining back its importance across sub-areas of deep learning. Adam(W), versions of which are go-to optimizers for many tasks such as next token prediction, has two…