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Deep neural networks are usually trained with stochastic gradient descent (SGD), which minimizes objective function using very rough approximations of gradient, only averaging to the real gradient. Standard approaches like momentum or ADAM…

Learning-to-optimize is an emerging framework that seeks to speed up the solution of certain optimization problems by leveraging training data. Learned optimization solvers have been shown to outperform classical optimization algorithms in…

Stochastic gradient descent (\textsc{Sgd}) methods are the most powerful optimization tools in training machine learning and deep learning models. Moreover, acceleration (a.k.a. momentum) methods and diagonal scaling (a.k.a. adaptive…

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…

Although gradient descent (GD) almost always escapes saddle points asymptotically [Lee et al., 2016], this paper shows that even with fairly natural random initialization schemes and non-pathological functions, GD can be significantly…

In gradient descent, changing how we parametrize the model can lead to drastically different optimization trajectories, giving rise to a surprising range of meaningful inductive biases: identifying sparse classifiers or reconstructing…

Balancing convergence speed, generalization capability, and computational efficiency remains a core challenge in deep learning optimization. First-order gradient descent methods, epitomized by stochastic gradient descent (SGD) and Adam,…

Adam is widely recognized as one of the most effective optimizers for training deep neural networks (DNNs). Despite its remarkable empirical success, its theoretical convergence analysis remains unsatisfactory. Existing works predominantly…

Mirror descent is an elegant optimization technique that leverages a dual space of parametric models to perform gradient descent. While originally developed for convex optimization, it has increasingly been applied in the field of machine…

We consider distributed optimization under communication constraints for training deep learning models. We propose a new algorithm, whose parameter updates rely on two forces: a regular gradient step, and a corrective direction dictated by…

Learning rules -- prescriptions for updating model parameters to improve performance -- are typically assumed rather than derived. Why do some learning rules work better than others, and under what assumptions can a given rule be considered…

Stochastic mirror descent (SMD) is a fairly new family of algorithms that has recently found a wide range of applications in optimization, machine learning, and control. It can be considered a generalization of the classical stochastic…

Mirror Descent (MD) is a well-known method of solving non-smooth convex optimization problems. This paper analyzes the stochastic variant of MD with adaptive stepsizes. Its convergence on average is shown to be faster than with the fixed…

Satisfaction of the strict saddle property has become a standard assumption in non-convex optimization, and it ensures that many first-order optimization algorithms will almost always escape saddle points. However, functions exist in…

We study the SAM (Sharpness-Aware Minimization) optimizer which has recently attracted a lot of interest due to its increased performance over more classical variants of stochastic gradient descent. Our main contribution is the derivation…

The simplicity of gradient descent (GD) made it the default method for training ever-deeper and complex neural networks. Both loss functions and architectures are often explicitly tuned to be amenable to this basic local optimization. In…

First-order optimization methods tend to inherently favor certain solutions over others when minimizing an underdetermined training objective that has multiple global optima. This phenomenon, known as implicit bias, plays a critical role in…

First-order optimization methods, such as SGD and Adam, are widely used for training large-scale deep neural networks due to their computational efficiency and robust performance. However, relying solely on gradient information, these…

Adaptive gradient methods, e.g. \textsc{Adam}, have achieved tremendous success in machine learning. Scaling the learning rate element-wisely by a certain form of second moment estimate of gradients, such methods are able to attain rapid…

Adaptive Moment Estimation (Adam) is a cornerstone optimization algorithm in deep learning, widely recognized for its flexibility with adaptive learning rates and efficiency in handling large-scale data. However, despite its practical…