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The mini-batch stochastic gradient descent (SGD) algorithm is widely used in training machine learning models, in particular deep learning models. We study SGD dynamics under linear regression and two-layer linear networks, with an easy…
In deep learning, it is common to use more network parameters than training points. In such scenarioof over-parameterization, there are usually multiple networks that achieve zero training error so that thetraining algorithm induces an…
We analyze the variance of stochastic gradients along negative curvature directions in certain non-convex machine learning models and show that stochastic gradients exhibit a strong component along these directions. Furthermore, we show…
Stochastic gradient descent (SGD), a widely used algorithm in deep-learning neural networks has attracted continuing studies for the theoretical principles behind its success. A recent work reports an anomaly (inverse) relation between the…
Stochastic gradient descent (SGD) with momentum is widely used for training modern deep learning architectures. While it is well-understood that using momentum can lead to faster convergence rate in various settings, it has also been…
In this paper, we consider a general stochastic optimization problem which is often at the core of supervised learning, such as deep learning and linear classification. We consider a standard stochastic gradient descent (SGD) method with a…
Several works have aimed to explain why overparameterized neural networks generalize well when trained by Stochastic Gradient Descent (SGD). The consensus explanation that has emerged credits the randomized nature of SGD for the bias of the…
In the vanishing learning rate regime, stochastic gradient descent (SGD) is now relatively well understood. In this work, we propose to study the basic properties of SGD and its variants in the non-vanishing learning rate regime. The focus…
In this paper, we study the implicit regularization of stochastic gradient descent (SGD) through the lens of {\em dynamical stability} (Wu et al., 2018). We start by revising existing stability analyses of SGD, showing how the Frobenius…
Neural networks are usually trained by some form of stochastic gradient descent (SGD)). A number of strategies are in common use intended to improve SGD optimization, such as learning rate schedules, momentum, and batching. These are…
Stochastic gradient descent (SGD) and its variants enable modern artificial intelligence. However, theoretical understanding lags far behind their empirical success. It is widely believed that SGD has a curious ability to avoid sharp local…
We study the limiting dynamics of a large class of noisy gradient descent systems in the overparameterized regime. In this regime the set of global minimizers of the loss is large, and when initialized in a neighbourhood of this zero-loss…
Stochastic gradient descent (SGD) is of fundamental importance in deep learning. Despite its simplicity, elucidating its efficacy remains challenging. Conventionally, the success of SGD is ascribed to the stochastic gradient noise (SGN)…
Previous work has examined the ability of larger capacity neural networks to generalize better than smaller ones, even without explicit regularizers, by analyzing gradient based algorithms such as GD and SGD. The presence of noise and its…
We present novel empirical observations regarding how stochastic gradient descent (SGD) navigates the loss landscape of over-parametrized deep neural networks (DNNs). These observations expose the qualitatively different roles of learning…
We study to what extent may stochastic gradient descent (SGD) be understood as a "conventional" learning rule that achieves generalization performance by obtaining a good fit to training data. We consider the fundamental stochastic convex…
Recent studies have demonstrated that noise in stochastic gradient descent (SGD) is closely related to generalization: A larger SGD noise, if not too large, results in better generalization. Since the covariance of the SGD noise is…
In overparametrized models, the noise in stochastic gradient descent (SGD) implicitly regularizes the optimization trajectory and determines which local minimum SGD converges to. Motivated by empirical studies that demonstrate that training…
Stochastic Gradient Descent (SGD) has proven to be remarkably effective in optimizing deep neural networks that employ ever-larger numbers of parameters. Yet, improving the efficiency of large-scale optimization remains a vital and highly…
The success of SGD in deep learning has been ascribed by prior works to the implicit bias induced by finite batch sizes ("SGD noise"). While prior works focused on offline learning (i.e., multiple-epoch training), we study the impact of SGD…