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In deep learning tasks, the learning rate determines the update step size in each iteration, which plays a critical role in gradient-based optimization. However, the determination of the appropriate learning rate in practice typically…

Machine Learning · Statistics 2020-04-08 Yingqiu Zhu , Yu Chen , Danyang Huang , Bo Zhang , Hansheng Wang

Training neural networks can be challenging, especially as the complexity of the problem increases. Despite using wider or deeper networks, training them can be a tedious process, especially if a wrong choice of the hyperparameter is made.…

Computational Engineering, Finance, and Science · Computer Science 2025-07-30 D. Veerababu , Ashwin A. Raikar , Prasanta K. Ghosh

The Lipschitz constant of the map between the input and output space represented by a neural network is a natural metric for assessing the robustness of the model. We present a new method to constrain the Lipschitz constant of dense deep…

Machine Learning · Computer Science 2023-08-22 Ouail Kitouni , Niklas Nolte , Mike Williams

Neural networks are typically trained with a single learning rate across all layers. While recent empirical evidence suggests that assigning layer-specific learning rates can accelerate training, a principled understanding of the conditions…

Machine Learning · Computer Science 2026-05-26 Sihan Zeng , Sujay Bhatt , Sumitra Ganesh

Recurrent neural network is a powerful model that learns temporal patterns in sequential data. For a long time, it was believed that recurrent networks are difficult to train using simple optimizers, such as stochastic gradient descent, due…

Neural and Evolutionary Computing · Computer Science 2015-04-20 Tomas Mikolov , Armand Joulin , Sumit Chopra , Michael Mathieu , Marc'Aurelio Ranzato

Back-propagation is a popular machine learning algorithm that uses gradient descent in training neural networks for supervised learning, but can be very slow. A number of algorithms have been developed to speed up convergence and improve…

Neural and Evolutionary Computing · Computer Science 2020-09-29 Ho Ling Li

State-of-the-art approaches for training Differentially Private (DP) Deep Neural Networks (DNN) face difficulties to estimate tight bounds on the sensitivity of the network's layers, and instead rely on a process of per-sample gradient…

We study the convergence properties of gradient descent for training deep linear neural networks, i.e., deep matrix factorizations, by extending a previous analysis for the related gradient flow. We show that under suitable conditions on…

Machine Learning · Computer Science 2021-11-25 Gabin Maxime Nguegnang , Holger Rauhut , Ulrich Terstiege

Learning rate decay (lrDecay) is a \emph{de facto} technique for training modern neural networks. It starts with a large learning rate and then decays it multiple times. It is empirically observed to help both optimization and…

Machine Learning · Computer Science 2019-09-27 Kaichao You , Mingsheng Long , Jianmin Wang , Michael I. Jordan

We propose an algorithm for the adaptation of the learning rate for stochastic gradient descent (SGD) that avoids the need for validation set use. The idea for the adaptiveness comes from the technique of extrapolation: to get an estimate…

Machine Learning · Statistics 2020-08-28 Antti Koskela , Antti Honkela

Although kernel methods are widely used in many learning problems, they have poor scalability to large datasets. To address this problem, sketching and stochastic gradient methods are the most commonly used techniques to derive efficient…

Machine Learning · Statistics 2022-06-03 Shingo Yashima , Atsushi Nitanda , Taiji Suzuki

Improving adversarial robustness of neural networks remains a major challenge. Fundamentally, training a neural network via gradient descent is a parameter estimation problem. In adaptive control, maintaining persistency of excitation (PoE)…

Machine Learning · Statistics 2021-10-18 Kaustubh Sridhar , Oleg Sokolsky , Insup Lee , James Weimer

The performance of gradient-based optimization methods, such as standard gradient descent (GD), greatly depends on the choice of learning rate. However, it can require a non-trivial amount of user tuning effort to select an appropriate…

Machine Learning · Computer Science 2025-10-14 Nikola Surjanovic , Alexandre Bouchard-Côté , Trevor Campbell

This paper tackles the problem of Lipschitz regularization of Convolutional Neural Networks. Lipschitz regularity is now established as a key property of modern deep learning with implications in training stability, generalization,…

Machine Learning · Computer Science 2020-11-10 Alexandre Araujo , Benjamin Negrevergne , Yann Chevaleyre , Jamal Atif

Large-scale optimization problems require algorithms both effective and efficient. One such popular and proven algorithm is Stochastic Gradient Descent which uses first-order gradient information to solve these problems. This paper studies…

Optimization and Control · Mathematics 2021-11-11 Theodoros Mamalis , Dusan Stipanovic , Petros Voulgaris

We develop an approach to efficiently grow neural networks, within which parameterization and optimization strategies are designed by considering their effects on the training dynamics. Unlike existing growing methods, which follow simple…

Machine Learning · Computer Science 2023-06-23 Xin Yuan , Pedro Savarese , Michael Maire

We analyze speed of convergence to global optimum for gradient descent training a deep linear neural network (parameterized as $x \mapsto W_N W_{N-1} \cdots W_1 x$) by minimizing the $\ell_2$ loss over whitened data. Convergence at a linear…

Machine Learning · Computer Science 2019-10-29 Sanjeev Arora , Nadav Cohen , Noah Golowich , Wei Hu

The Lipschitz constant plays a crucial role in certifying the robustness of neural networks to input perturbations. Since calculating the exact Lipschitz constant is NP-hard, efforts have been made to obtain tight upper bounds on the…

Machine Learning · Computer Science 2024-10-30 Yuezhu Xu , S. Sivaranjani

Stochastic gradient descent (SGD), which updates the model parameters by adding a local gradient times a learning rate at each step, is widely used in model training of machine learning algorithms such as neural networks. It is observed…

Machine Learning · Computer Science 2017-06-01 Chang Xu , Tao Qin , Gang Wang , Tie-Yan Liu

Deep neural networks are notorious for being sensitive to small well-chosen perturbations, and estimating the regularity of such architectures is of utmost importance for safe and robust practical applications. In this paper, we investigate…

Machine Learning · Statistics 2019-10-28 Kevin Scaman , Aladin Virmaux