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Hessian-free (HF) optimization has been successfully used for training deep autoencoders and recurrent networks. HF uses the conjugate gradient algorithm to construct update directions through curvature-vector products that can be computed…

Machine Learning · Computer Science 2013-05-02 Ryan Kiros

Optimizing neural networks with loss that contain high-dimensional and high-order differential operators is expensive to evaluate with back-propagation due to $\mathcal{O}(d^{k})$ scaling of the derivative tensor size and the…

Machine Learning · Computer Science 2025-01-14 Zekun Shi , Zheyuan Hu , Min Lin , Kenji Kawaguchi

Gradient Descent (GD) and Conjugate Gradient (CG) methods are among the most effective iterative algorithms for solving unconstrained optimization problems, particularly in machine learning and statistical modeling, where they are employed…

Optimization and Control · Mathematics 2024-12-19 Xianqi Jiao , Jia Liu , Zhiping Chen

Gradient descent based optimization methods are the methods of choice to train deep neural networks in machine learning. Beyond the standard gradient descent method, also suitable modified variants of standard gradient descent involving…

Optimization and Control · Mathematics 2025-04-29 Steffen Dereich , Arnulf Jentzen , Adrian Riekert

We consider the problem of Cost-Aware Learning, where sampling different component functions of a finite-sum objective incurs different costs. The objective is to reach a target error while minimizing the total cost. First, we propose the…

Machine Learning · Computer Science 2026-05-01 Clara Mohri , Amir Globerson , Haim Kaplan , Tomer Koren , Yishay Mansour

Research into optimisation for deep learning is characterised by a tension between the computational efficiency of first-order, gradient-based methods (such as SGD and Adam) and the theoretical efficiency of second-order, curvature-based…

Machine Learning · Computer Science 2024-06-17 Ross M. Clarke , José Miguel Hernández-Lobato

Hyperparameter tuning is a bothersome step in the training of deep learning models. One of the most sensitive hyperparameters is the learning rate of the gradient descent. We present the 'All Learning Rates At Once' (Alrao) optimization…

Machine Learning · Computer Science 2019-01-30 Léonard Blier , Pierre Wolinski , Yann Ollivier

Most methods for decision-theoretic online learning are based on the Hedge algorithm, which takes a parameter called the learning rate. In most previous analyses the learning rate was carefully tuned to obtain optimal worst-case…

Machine Learning · Statistics 2015-03-04 Tim van Erven , Peter Grünwald , Wouter M. Koolen , Steven de Rooij

Recent work has established an empirically successful framework for adapting learning rates for stochastic gradient descent (SGD). This effectively removes all needs for tuning, while automatically reducing learning rates over time on…

Machine Learning · Computer Science 2013-03-28 Tom Schaul , Yann LeCun

We aim to design strategies for sequential decision making that adjust to the difficulty of the learning problem. We study this question both in the setting of prediction with expert advice, and for more general combinatorial decision…

Machine Learning · Computer Science 2015-03-02 Wouter M. Koolen , Tim van Erven

We consider Sharpness-Aware Minimization (SAM), a gradient-based optimization method for deep networks that has exhibited performance improvements on image and language prediction problems. We show that when SAM is applied with a convex…

Machine Learning · Computer Science 2023-04-12 Peter L. Bartlett , Philip M. Long , Olivier Bousquet

Learning discrete representations of data is a central machine learning task because of the compactness of the representations and ease of interpretation. The task includes clustering and hash learning as special cases. Deep neural networks…

Machine Learning · Statistics 2017-06-15 Weihua Hu , Takeru Miyato , Seiya Tokui , Eiichi Matsumoto , Masashi Sugiyama

We address the challenge of estimating the learning rate for adaptive gradient methods used in training deep neural networks. While several learning-rate-free approaches have been proposed, they are typically tailored for steepest descent.…

Machine Learning · Computer Science 2024-01-09 Min-Kook Suh , Seung-Woo Seo

We here adapt an extended version of the adaptive cubic regularisation method with dynamic inexact Hessian information for nonconvex optimisation in [3] to the stochastic optimisation setting. While exact function evaluations are still…

Numerical Analysis · Mathematics 2020-09-15 Stefania Bellavia , Gianmarco Gurioli

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

Progress in deep learning is slowed by the days or weeks it takes to train large models. The natural solution of using more hardware is limited by diminishing returns, and leads to inefficient use of additional resources. In this paper, we…

Machine Learning · Computer Science 2017-12-12 Shankar Krishnan , Ying Xiao , Rif A. Saurous

Differentiable optimization layers enable learning systems to make decisions by solving embedded optimization problems. However, computing gradients via implicit differentiation requires solving a linear system with Hessian terms, which is…

Machine Learning · Computer Science 2025-12-03 Zihao Zhao , Kai-Chia Mo , Shing-Hei Ho , Brandon Amos , Kai Wang

A multiplicative constant scaling factor is often applied to the model output to adjust the dynamics of neural network parameters. This has been used as one of the key interventions in an empirical study of lazy and active behavior.…

Machine Learning · Computer Science 2021-07-05 Ryuichi Kanoh , Mahito Sugiyama

One of the most important parts of Artificial Neural Networks is minimizing the loss functions which tells us how good or bad our model is. To minimize these losses we need to tune the weights and biases. Also to calculate the minimum value…

Machine Learning · Computer Science 2021-01-08 Kaustubh Yadav

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…

Machine Learning · Computer Science 2022-11-22 Ran Tian , Ankur P. Parikh