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Convergence detection of iterative stochastic optimization methods is of great practical interest. This paper considers stochastic gradient descent (SGD) with a constant learning rate and momentum. We show that there exists a transient…

Modern adaptive optimization methods, such as Adam and its variants, have emerged as the most widely used tools in deep learning over recent years. These algorithms offer automatic mechanisms for dynamically adjusting the update step based…

Gradient descent (GD) and stochastic gradient descent (SGD) have been widely used in a large number of application domains. Therefore, understanding the dynamics of GD and improving its convergence speed is still of great importance. This…

Optimization algorithms are core methods by which machine learning models iteratively minimize loss functions, update parameters, learn from data, and improve performance. Momentum SGD and AdamW represent two important optimization…

This paper establishes a mathematical foundation for the Adam optimizer, elucidating its connection to natural gradient descent through Riemannian and information geometry. We provide an accessible and detailed analysis of the diagonal…

Currently, widely used first-order deep learning optimizers include non-adaptive learning rate optimizers and adaptive learning rate optimizers. The former is represented by SGDM (Stochastic Gradient Descent with Momentum), while the latter…

The success of the Adam optimizer on a wide array of architectures has made it the default in settings where stochastic gradient descent (SGD) performs poorly. However, our theoretical understanding of this discrepancy is lagging,…

In this paper, we investigate the popular deep learning optimization routine, Adam, from the perspective of statistical moments. While Adam is an adaptive lower-order moment based (of the stochastic gradient) method, we propose an extension…

The impressive performance of large language models (LLMs) arises from their massive scale and heterogeneous module composition. However, this structural heterogeneity introduces additional optimization challenges. While adaptive optimizers…

Stochastic gradient descent (SGD) is a standard optimization method to minimize a training error with respect to network parameters in modern neural network learning. However, it typically suffers from proliferation of saddle points in the…

Douglas-Rachford splitting and its equivalent dual formulation ADMM are widely used iterative methods in composite optimization problems arising in control and machine learning applications. The performance of these algorithms depends on…

Convolutional neural networks (CNNs) are trained using stochastic gradient descent (SGD)-based optimizers. Recently, the adaptive moment estimation (Adam) optimizer has become very popular due to its adaptive momentum, which tackles the…

Momentum methods have been shown to accelerate the convergence of the standard gradient descent algorithm in practice and theory. In particular, the minibatch-based gradient descent methods with momentum (MGDM) are widely used to solve…

Adaptive gradient algorithms such as ADAGRAD and its variants have gained popularity in the training of deep neural networks. While many works as for adaptive methods have focused on the static regret as a performance metric to achieve a…

Stochastic optimization plays a crucial role in the advancement of deep learning technologies. Over the decades, significant effort has been dedicated to improving the training efficiency and robustness of deep neural networks, via various…

It is well known that we need to choose the hyper-parameters in Momentum, AdaGrad, AdaDelta, and other alternative stochastic optimizers. While in many cases, the hyper-parameters are tuned tediously based on experience becoming more of an…

Adam is known to perform significantly better than Stochastic Gradient Descent (SGD) in language models, a phenomenon for which a number of explanations have been proposed. In this work, we revisit this "optimizer gap" through a series of…

Understanding the training dynamics of transformers is important to explain the impressive capabilities behind large language models. In this work, we study the dynamics of training a shallow transformer on a task of recognizing…

Optimization is essential in deep learning. The foundational method upon which most optimizers are built is momentum-based stochastic gradient descent. However, it suffers from two key drawbacks. First, it has noisy and varying gradients,…

We present Amos, a stochastic gradient-based optimizer designed for training deep neural networks. It can be viewed as an Adam optimizer with theoretically supported, adaptive learning-rate decay and weight decay. A key insight behind Amos…