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In this paper we study the stability and its trade-off with optimization error for stochastic gradient descent (SGD) algorithms in the pairwise learning setting. Pairwise learning refers to a learning task which involves a loss function…

Machine Learning · Computer Science 2019-04-29 Wei Shen , Zhenhuan Yang , Yiming Ying , Xiaoming Yuan

Deep neural network training often exhibits highly anisotropic loss geometry, where a few sharp dominant Hessian directions coexist with a large flatter bulk. Gradients tend to align disproportionately with these dominant directions,…

Machine Learning · Computer Science 2026-05-28 Tolga Dimlioglu , Kristi Topollai , Anna Choromanska

Variational Learning (VL) has recently gained popularity for training deep neural networks. Part of its empirical success can be explained by theories such as PAC-Bayes bounds, minimum description length and marginal likelihood, but little…

A leading hypothesis for the surprising generalization of neural networks is that the dynamics of gradient descent bias the model towards simple solutions, by searching through the solution space in an incremental order of complexity. We…

Machine Learning · Computer Science 2020-01-01 Daniel Gissin , Shai Shalev-Shwartz , Amit Daniely

Despite the widespread adoption of neural networks, their training dynamics remain poorly understood. We show experimentally that as the size of the dataset increases, a point forms where the magnitude of the gradient of the loss becomes…

Machine Learning · Computer Science 2024-07-23 Mark Lowell

Gradient dynamics play a central role in determining the stability and generalization of deep neural networks. In this work, we provide an empirical analysis of how variance and standard deviation of gradients evolve during training,…

Machine Learning · Computer Science 2025-09-09 Vincent-Daniel Yun

While stochastic gradient descent (SGD) and variants have been surprisingly successful for training deep nets, several aspects of the optimization dynamics and generalization are still not well understood. In this paper, we present new…

Machine Learning · Computer Science 2019-07-26 Xinyan Li , Qilong Gu , Yingxue Zhou , Tiancong Chen , Arindam Banerjee

In the era of large-scale neural network models, optimization algorithms often struggle with generalization due to an overreliance on training loss. One key insight widely accepted in the machine learning community is the idea that wide…

Machine Learning · Computer Science 2025-09-01 Bodu Gong , Gustavo Enrique Batista , Pierre Lafaye de Micheaux

Private machine learning involves addition of noise while training, resulting in lower accuracy. Intuitively, greater stability can imply greater privacy and improve this privacy-utility tradeoff. We study this role of stability in private…

Machine Learning · Computer Science 2020-06-26 Lauren Watson , Benedek Rozemberczki , Rik Sarkar

Stochastic gradient descent (SGD) and adaptive gradient methods, such as Adam and RMSProp, have been widely used in training deep neural networks. We empirically show that while the difference between the standard generalization performance…

Machine Learning · Computer Science 2023-11-30 Avery Ma , Yangchen Pan , Amir-massoud Farahmand

The key to generalization is controlling the complexity of the network. However, there is no obvious control of complexity -- such as an explicit regularization term -- in the training of deep networks for classification. We will show that…

Machine Learning · Computer Science 2020-04-14 Andrzej Banburski , Qianli Liao , Brando Miranda , Lorenzo Rosasco , Fernanda De La Torre , Jack Hidary , Tomaso Poggio

Despite their widespread use, training deep Transformers can be unstable. Layer normalization, a standard component, improves training stability, but its placement has often been ad-hoc. In this paper, we conduct a principled study on the…

Machine Learning · Computer Science 2025-10-14 Kelvin Kan , Xingjian Li , Benjamin J. Zhang , Tuhin Sahai , Stanley Osher , Krishna Kumar , Markos A. Katsoulakis

Models are expected to engage in invariance learning, which involves distinguishing the core relations that remain consistent across varying environments to ensure the predictions are safe, robust and fair. While existing works consider…

Machine Learning · Computer Science 2025-03-11 Yang Xu , Yihong Gu , Cong Fang

Shortcut learning causes deep learning models to rely on non-essential features within the data. However, its formation in deep neural network training still lacks theoretical understanding. In this paper, we provide a formal definition of…

Artificial Intelligence · Computer Science 2026-05-06 Xiayang Li , Kuo Gai , Shihua Zhang

This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of…

Adaptive gradient methods such as Adam have gained increasing popularity in deep learning optimization. However, it has been observed that compared with (stochastic) gradient descent, Adam can converge to a different solution with a…

Machine Learning · Computer Science 2021-08-26 Difan Zou , Yuan Cao , Yuanzhi Li , Quanquan Gu

We examine the stability of loss-minimizing training processes that are used for deep neural networks (DNN) and other classifiers. While a classifier is optimized during training through a so-called loss function, the performance of…

Analysis of PDEs · Mathematics 2020-10-05 Leonid Berlyand , Pierre-Emmanuel Jabin , C. Alex Safsten

It has been empirically observed that the flatness of minima obtained from training deep networks seems to correlate with better generalization. However, for deep networks with positively homogeneous activations, most measures of…

Machine Learning · Statistics 2019-02-08 Akshay Rangamani , Nam H. Nguyen , Abhishek Kumar , Dzung Phan , Sang H. Chin , Trac D. Tran

Deep neural networks have become invaluable tools for supervised machine learning, e.g., classification of text or images. While often offering superior results over traditional techniques and successfully expressing complicated patterns in…

Machine Learning · Computer Science 2019-02-19 Eldad Haber , Lars Ruthotto

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…

Machine Learning · Computer Science 2017-11-23 Haiping Huang , Taro Toyoizumi