Related papers: Finite Depth and Width Corrections to the Neural T…
We study depth separation in infinite-width neural networks, where complexity is controlled by the overall squared $\ell_2$-norm of the weights (sum of squares of all weights in the network). Whereas previous depth separation results…
In this paper, we advance the understanding of neural network training dynamics by examining the intricate interplay of various factors introduced by weight parameters in the initialization process. Motivated by the foundational work of Luo…
We consider the dynamic of gradient descent for learning a two-layer neural network. We assume the input $x\in\mathbb{R}^d$ is drawn from a Gaussian distribution and the label of $x$ satisfies $f^{\star}(x) = a^{\top}|W^{\star}x|$, where…
In multi-objective optimization, multiple loss terms are weighted and added together to form a single objective. These weights are chosen to properly balance the competing losses according to some meta-goal. For example, in physics-informed…
The Neural Tangent Kernel (NTK) framework has provided deep insights into the training dynamics of neural networks under gradient flow. However, it relies on the assumption that the network is differentiable with respect to its parameters,…
Neural networks with REctified Linear Unit (ReLU) activation functions (a.k.a. ReLU networks) have achieved great empirical success in various domains. Nonetheless, existing results for learning ReLU networks either pose assumptions on the…
Spectral bias is a significant phenomenon in neural network training and can be explained by neural tangent kernel (NTK) theory. In this work, we develop the NTK theory for deep neural networks with physics-informed loss, providing insights…
Empirical studies show that gradient-based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. Very…
A primary advantage of neural networks lies in their feature learning characteristics, which is challenging to theoretically analyze due to the complexity of their training dynamics. We propose a new paradigm for studying feature learning…
We present a novel neural network Maximum Mean Discrepancy (MMD) statistic by identifying a new connection between neural tangent kernel (NTK) and MMD. This connection enables us to develop a computationally efficient and memory-efficient…
Assessing the complexity of functions computed by a neural network helps us understand how the network will learn and generalize. One natural measure of complexity is how the network distorts length - if the network takes a unit-length…
It has been recognized that a heavily overparameterized artificial neural network exhibits surprisingly good generalization performance in various machine-learning tasks. Recent theoretical studies have made attempts to unveil the mystery…
Wide neural networks with linear output layer have been shown to be near-linear, and to have near-constant neural tangent kernel (NTK), in a region containing the optimization path of gradient descent. These findings seem counter-intuitive…
Several recent trends in machine learning theory and practice, from the design of state-of-the-art Gaussian Process to the convergence analysis of deep neural nets (DNNs) under stochastic gradient descent (SGD), have found it fruitful to…
We study the properties of the Neural Tangent Kernel (NTK) $\overset{\scriptscriptstyle\infty}{K} : \mathbb{R}^{m_0} \times \mathbb{R}^{m_0} \to \mathbb{R}^{m_l \times m_l}$ corresponding to infinitely wide $l$-layer Multilayer Perceptrons…
A recurrent neural network (RNN) is a widely used deep-learning network for dealing with sequential data. Imitating a dynamical system, an infinite-width RNN can approximate any open dynamical system in a compact domain. In general, deep…
Recent advances have significantly improved our understanding of the generalization performance of gradient descent (GD) methods in deep neural networks. A natural and fundamental question is whether GD can achieve generalization rates…
In this work, beyond width and depth, we augment a neural network with a new dimension called height by intra-linking neurons in the same layer to create an intra-layer hierarchy, which gives rise to the notion of height. We call a neural…
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 study multigrade deep learning (MGDL) as a principled framework for structured error refinement in deep neural networks. While the approximation power of neural networks is now relatively well understood, training very deep architectures…