Related papers: Analysis of a Two-Layer Neural Network via Displac…
The paper contains approximation guarantees for neural networks that are trained with gradient flow, with error measured in the continuous $L_2(\mathbb{S}^{d-1})$-norm on the $d$-dimensional unit sphere and targets that are Sobolev smooth.…
In this paper we develop random block coordinate gradient descent methods for minimizing large scale linearly constrained separable convex problems over networks. Since we have coupled constraints in the problem, we devise an algorithm that…
A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such…
The proximal gradient algorithm for minimizing the sum of a smooth and a nonsmooth convex function often converges linearly even without strong convexity. One common reason is that a multiple of the step length at each iteration may…
We develop a framework for convexifying a fairly general class of optimization problems. Under additional assumptions, we analyze the suboptimality of the solution to the convexified problem relative to the original nonconvex problem and…
We consider the stochastic approximation problem where a convex function has to be minimized, given only the knowledge of unbiased estimates of its gradients at certain points, a framework which includes machine learning methods based on…
We propose a distributionally robust approach to learning hyperparameters for first-order methods in convex optimization. Given a dataset of problem instances, we minimize a Wasserstein distributionally robust version of the performance…
The problem of minimizing convex functionals of probability distributions is solved under the assumption that the density of every distribution is bounded from above and below. A system of sufficient and necessary first-order optimality…
Soft-thresholding has been widely used in neural networks. Its basic network structure is a two-layer convolution neural network with soft-thresholding. Due to the network's nature of nonlinearity and nonconvexity, the training process…
Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave…
Transformer models have emerged as fundamental tools across various scientific and engineering disciplines, owing to their outstanding performance in diverse applications. Despite this empirical success, the theoretical foundations of…
We present a framework to define a large class of neural networks for which, by construction, training by gradient flow provably reaches arbitrarily low loss when the number of parameters grows. Distinct from the fixed-space global…
We study the generalization of two-layer ReLU neural networks in a univariate nonparametric regression problem with noisy labels. This is a problem where kernels (\emph{e.g.} NTK) are provably sub-optimal and benign overfitting does not…
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…
This paper argues that DNNs implement a computational Occam's razor -- finding the `simplest' algorithm that fits the data -- and that this could explain their incredible and wide-ranging success over more traditional statistical methods.…
We study the generalization performance of gradient methods in the fundamental stochastic convex optimization setting, focusing on its dimension dependence. First, for full-batch gradient descent (GD) we give a construction of a learning…
Neural networks are a powerful class of functions that can be trained with simple gradient descent to achieve state-of-the-art performance on a variety of applications. Despite their practical success, there is a paucity of results that…
We prove that finding all globally optimal two-layer ReLU neural networks can be performed by solving a convex optimization program with cone constraints. Our analysis is novel, characterizes all optimal solutions, and does not leverage…
We study the implicit regularization imposed by gradient descent for learning multi-layer homogeneous functions including feed-forward fully connected and convolutional deep neural networks with linear, ReLU or Leaky ReLU activation. We…
We study distributed optimization problems over a network when the communication between the nodes is constrained, and so information that is exchanged between the nodes must be quantized. This imperfect communication poses a fundamental…