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One of the central problems in the study of deep learning theory is to understand how the structure properties, such as depth, width and the number of nodes, affect the expressivity of deep neural networks. In this work, we show a new…
This paper establishes a comprehensive approximation result for deep fully-connected neural networks with commonly-used and general activation functions in Sobolev spaces $W^{n,\infty}$, with errors measured in the $W^{m,p}$-norm for $m <…
This paper studies the problem of how efficiently functions in the Sobolev spaces $\mathcal{W}^{s,q}([0,1]^d)$ and Besov spaces $\mathcal{B}^s_{q,r}([0,1]^d)$ can be approximated by deep ReLU neural networks with width $W$ and depth $L$,…
Despite remarkable performance on a variety of tasks, many properties of deep neural networks are not yet theoretically understood. One such mystery is the depth degeneracy phenomenon: the deeper you make your network, the closer your…
We present SiLU network constructions whose approximation efficiency depends critically on proper hyperparameter tuning. For the square function $x^2$, with optimally chosen shift $a$ and scale $\beta$, we achieve approximation error…
Training neural networks to be certifiably robust is critical to ensure their safety against adversarial attacks. However, it is currently very difficult to train a neural network that is both accurate and certifiably robust. In this work…
This paper presents an algorithm for searching for the minimum number of neurons in fully connected layers of an arbitrary network solving given problem, which does not require multiple training of the network with different number of…
In this paper, we theoretically prove that gradient descent can find a global minimum of non-convex optimization of all layers for nonlinear deep neural networks of sizes commonly encountered in practice. The theory developed in this paper…
In this article, we study approximation properties of the variation spaces corresponding to shallow neural networks with a variety of activation functions. We introduce two main tools for estimating the metric entropy, approximation rates,…
We explore convergence of deep neural networks with the popular ReLU activation function, as the depth of the networks tends to infinity. To this end, we introduce the notion of activation domains and activation matrices of a ReLU network.…
The primary neural networks decision-making units are activation functions. Moreover, they evaluate the output of networks neural node; thus, they are essential for the performance of the whole network. Hence, it is critical to choose the…
In this paper, we investigate the relationship between deep neural networks (DNN) with rectified linear unit (ReLU) function as the activation function and continuous piecewise linear (CPWL) functions, especially CPWL functions from the…
It is shown that over-parameterized neural networks can achieve minimax optimal rates of convergence (up to logarithmic factors) for learning functions from certain smooth function classes, if the weights are suitably constrained or…
We consider in this paper the optimal approximations of convex univariate functions with feed-forward Relu neural networks. We are interested in the following question: what is the minimal approximation error given the number of…
A recent line of work has analyzed the theoretical properties of deep neural networks via the Neural Tangent Kernel (NTK). In particular, the smallest eigenvalue of the NTK has been related to the memorization capacity, the global…
Neural networks have shown high successful performance in a wide range of tasks, but further studies are needed to improve its performance. We analyze the approximation error of the specific neural network architecture with a local…
In recent years, deep neural networks (DNNs) achieved unprecedented performance in many low-level vision tasks. However, state-of-the-art results are typically achieved by very deep networks, which can reach tens of layers with tens of…
We establish the fundamental limits in the approximation of Lipschitz functions by deep ReLU neural networks with finite-precision weights. Specifically, three regimes, namely under-, over-, and proper quantization, in terms of minimax…
We consider a deep ReLU / Leaky ReLU student network trained from the output of a fixed teacher network of the same depth, with Stochastic Gradient Descent (SGD). The student network is \emph{over-realized}: at each layer $l$, the number…
The massive computational costs of scaling modern deep learning architectures have driven the widespread use of parameter-efficient low-rank structures, such as LoRA and low-rank factorization. However, theoretical guarantees for their…