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Model depth is a double-edged sword in deep learning: deeper models achieve higher accuracy but require higher computational cost. To efficiently train models at scale, an effective strategy is the progressive training, which scales up…

Understanding the asymptotic behavior of gradient-descent training of deep neural networks is essential for revealing inductive biases and improving network performance. We derive the infinite-time training limit of a mathematically…

Differentially private gradient descent (DP-GD) is a popular algorithm to train deep learning models with provable guarantees on the privacy of the training data. In the last decade, the problem of understanding its performance cost with…

Two aspects of neural networks that have been extensively studied in the recent literature are their function approximation properties and their training by gradient descent methods. The approximation problem seeks accurate approximations…

Energy and resource efficient training of DNNs will greatly extend the applications of deep learning. However, there are three major obstacles which mandate accurate calculation in high precision. In this paper, we tackle two of them…

We establish conditions under which gradient descent applied to fixed-width deep networks drives the logistic loss to zero, and prove bounds on the rate of convergence. Our analysis applies for smoothed approximations to the ReLU, such as…

This article provides a comprehensive understanding of optimization in deep learning, with a primary focus on the challenges of gradient vanishing and gradient exploding, which normally lead to diminished model representational ability and…

We prove limitations on what neural networks trained by noisy gradient descent (GD) can efficiently learn. Our results apply whenever GD training is equivariant, which holds for many standard architectures and initializations. As…

Normalized gradient descent has shown substantial success in speeding up the convergence of exponentially-tailed loss functions (which includes exponential and logistic losses) on linear classifiers with separable data. In this paper, we go…

We prove linear convergence of gradient descent to a global optimum for the training of deep residual networks with constant layer width and smooth activation function. We show that if the trained weights, as a function of the layer index,…

In the context of classification problems, Deep Learning (DL) approaches represent state of art. Many DL approaches are based on variations of standard multi-layer feed-forward neural networks. These are also referred to as deep networks.…

We provide a detailed asymptotic study of gradient flow trajectories and their implicit optimization bias when minimizing the exponential loss over "diagonal linear networks". This is the simplest model displaying a transition between…

Neural Operators that directly learn mappings between function spaces, such as Deep Operator Networks (DONs) and Fourier Neural Operators (FNOs), have received considerable attention. Despite the universal approximation guarantees for DONs…

Linear programming has played a crucial role in shaping decision-making, resource allocation, and cost reduction in various domains. In this paper, we investigate the application of overparametrized neural networks and their implicit bias…

Neural networks are predominantly trained using gradient-based methods, yet in many applications their final predictions remain far from the accuracy attainable within the model's expressive capacity. We introduce Linearized Subspace…

Deep learning has been applied to various tasks in the field of machine learning and has shown superiority to other common procedures such as kernel methods. To provide a better theoretical understanding of the reasons for its success, we…

We perform an average case analysis of the generalization dynamics of large neural networks trained using gradient descent. We study the practically-relevant "high-dimensional" regime where the number of free parameters in the network is on…

This paper investigates the deep learning optimization problem with softmax cross-entropy loss. We propose a layer separation strategy to alleviate the strong nonconvexity encountered during training deep networks. For cross-entropy models…

Incorporating scientific knowledge into deep learning (DL) models for materials-based simulations can constrain the network's predictions to be within the boundaries of the material system. Altering loss functions or adding physics-based…

Traditional landscape analysis of deep neural networks aims to show that no sub-optimal local minima exist in some appropriate sense. From this, one may be tempted to conclude that descent algorithms which escape saddle points will reach a…