The paper analyzes deep ReLU CNNs' approximation properties in 2D space.
problem Establishing L 2 L^2 L 2 approximation properties for deep ReLU CNNs. method Analysis based on decomposition theorem for convolutional kernels, properties of ReLU activation, and connections with one-hidden-layer ReLU NNs.
result Universal approximation theorem for deep ReLU CNNs with classic structure.
This paper examines how noise affects deep neural networks and improves their performance.
problem The impact of noise on the stability of deep ReLU neural networks for nonparametric regression.
method Investigates the optimal rate of convergence for deep ReLU neural networks under Huber loss, considering the p-th moment of noise and the smoothness of the function.
result The optimal rate of convergence cannot be achieved by ordinary least squares but can be by Huber loss with a properly chosen parameter.
Large deviation principle for deep neural networks with ReLU activation.
problem Understanding the behavior of deep neural networks with ReLU activation.
method Proving a large deviation principle for networks with Gaussian weights and ReLU activation functions.
result Simplified expressions and power-series expansions for the ReLU case.
Gradient descent methods for deep ReLU networks achieve optimal generalization rates.
problem Generalization of gradient descent methods for deep neural networks
method Establishing minimax-optimal rates for GD and SGD with deep ReLU networks
result Gradient descent methods for deep ReLU networks achieve optimal generalization rates
Deep ReLU networks can be simplified to a three-layer model.
problem Understanding the behavior of deep neural networks.
method Constructive proof and algorithm to transform deep networks into shallow ones.
result Deep ReLU networks can be represented by a simpler three-layer structure.
In this paper, we theoretically prove that the deep ReLU neural networks do not lie in spurious local minima in the loss landscape under the Neural Tangent Kernel (NTK) regime, that is, in the gradient descent training dynamics of the deep ReLU neural networks whose parameters are initialized by a normal distribution i…
Optimal rates for shallow ReLU networks in nonparametric regression.
problem Approximating smooth and non-smooth functions with shallow ReLU networks.
method Analysis of shallow ReLU k ^k k neural networks, using variation norms and deep learning theory. result Optimal approximation rates for shallow ReLU networks in nonparametric regression.
Bayesian free energy remains bounded for deep ReLU networks in overparametrized cases.
problem Understanding the generalization performance of deep ReLU neural networks.
method Analyzes Bayesian free energy in overparametrized deep ReLU neural networks.
result Bayesian free energy is bounded even in overparametrized deep ReLU networks.
Deep ReLU networks need Ω(N) parameters to interpolate at irregularly spaced points.
problem Interpolating at irregularly spaced data points with deep ReLU networks.
method Analyzing the number of parameters required for interpolation.
result Ω(N) parameters are necessary for interpolation when δ is exponentially small in N.
New theory for local parameterization of deep ReLU networks.
problem Determining local parameters of deep ReLU neural networks.
method Introducing local lifting operators and charts of a manifold, deriving necessary and sufficient conditions for local identifiability.
result Sharp and testable conditions for local identifiability of deep ReLU networks.
Study approximates nonlinear functionals using deep ReLU networks.
problem Approximating nonlinear continuous functionals with neural networks.
method Constructs continuous piecewise linear interpolation under simple triangulation, analyzes rates of approximation.
result Established rates of approximation for functional deep ReLU networks.
Improved bounds on neural network expressivity.
problem Understanding neural network expressivity and approximation capabilities.
method Improved bounds on the maximal number of linear regions of ReLU-networks.
result New insights into the expressivity of neural networks.
Deep neural networks approximate functions in shift-invariant spaces with controlled error.
problem Approximating functions in shift-invariant spaces with neural networks.
method Using deep ReLU neural networks, estimating approximation error bounds based on network width and depth.
result Deep neural networks achieve optimal approximation rates for Sobolev spaces up to a logarithmic factor.
Path regularization reveals convex optimization in deep ReLU networks.
problem Understanding the optimization landscape of deep neural networks.
method Introducing path regularization to make the training problem convex and sparsity-inducing.
result Path regularized parallel ReLU networks are a parsimonious convex model in high dimensions.
Deep neural networks struggle with numerical instability during training.
problem Numerical instability in gradient descent training of deep neural networks.
method Analysis of floating-point arithmetic and gradient descent in ReLU neural networks.
result It is highly unlikely for ReLU networks to maintain a superlinear number of affine pieces during training.
Deep ReLU networks can efficiently approximate Sobolev and Besov functions.
problem Approximating functions in Sobolev and Besov spaces using deep neural networks.
method Used deep ReLU neural networks with varied width and depth to approximate functions in Sobolev and Besov spaces.
result Generalized the approximation rate to hold under the Sobolev embedding condition.
Deep neural networks classify unbounded Gaussian mixture data without dimensionality issues.
problem Binary classification of unbounded Gaussian mixture data.
method Deep ReLU neural networks with non-asymptotic upper bounds and convergence rates.
result Deep ReLU networks can classify unbounded Gaussian mixture data without dimensionality constraints.
Deep ReLU networks can approximate matrix-vector products with error bounds.
problem Can deep ReLU networks accurately approximate matrix-vector products?
method Derived error bounds in Lebesgue and Sobolev norms for deep ReLU FNNs.
result Developed deep approximation theory with successful applications.
This paper proves SGD converges to global minimum for over-parameterized ReLU networks.
problem Theoretical understanding of implicit neural networks is limited.
method Gradient flow analysis of ReLU activated implicit neural networks.
result Randomly initialized gradient descent converges to global minimum at a linear rate for square loss function in over-parameterized ReLU networks.
Deep neural networks can learn smooth functions without parameters.
problem Learning smooth functions from shallow ReLU neural networks.
method Using over-parameterized shallow ReLU neural networks with norm constraints.
result Least squares estimators based on shallow neural networks are minimax optimal.
Rational neural networks approximate functions more efficiently with less depth.
problem Choosing optimal nonlinear activation functions in neural networks.
method Rational activation functions with optimal bounds and efficiency proofs.
result Rational neural networks approximate smooth functions more efficiently than ReLU networks with exponentially smaller depth.
Deep neural networks can approximate complex functions through repeated compositions of a fixed-size ReLU network.
problem Understanding the expressive power of deep neural networks through function compositions.
method Demonstrated the surprising expressive power of repeated compositions of a single fixed-size ReLU network.
result Repeated compositions of a single fixed-size ReLU network can approximate 1-Lipschitz continuous functions on [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d with an error O ( r − 1 / d ) \mathcal{O}(r^{-1/d}) O ( r − 1/ d ) . Activation functions play a key role in providing remarkable performance in deep neural networks, and the rectified linear unit (ReLU) is one of the most widely used activation functions. Various new activation functions and improvements on ReLU have been proposed, but each carry performance drawbacks. In this paper, w…
New maximum score estimators using ReLU functions and deep neural networks.
problem Estimating parameters in models with sign restrictions.
method ReLU-based maximum score criterion and DNN architecture.
result RMS estimator achieves n − s / ( 2 s + 1 ) n^{-s/(2s+1)} n − s / ( 2 s + 1 ) convergence rate and asymptotic normality. The paper explores neural scaling laws for deep operator networks, offering a theoretical foundation.
problem Understanding neural scaling laws in deep operator networks.
method Theoretical analysis of approximation and generalization errors.
result Established a theoretical framework to quantify neural scaling laws for deep operator networks.
We study the problem of training deep neural networks with Rectified Linear Unit (ReLU) activation function using gradient descent and stochastic gradient descent. In particular, we study the binary classification problem and show that for a broad family of loss functions, with proper random weight initialization, both…
Paper studies shallow ReLU networks' approximation rates for Hölder functions.
problem Understanding shallow ReLU networks' efficiency in approximating Hölder functions.
method Analyzes rates of uniform approximation by ReLU shallow neural networks with m m m hidden neurons. result Shows ReLU shallow neural networks can uniformly approximate Hölder functions with rates close to optimal.
SGD fails to converge for deep ReLU networks with limited random initializations.
problem SGD convergence in deep neural networks with limited random initializations.
method Analysis of four discretization parameters: network architecture, training data, gradient steps, and random initializations.
result SGD fails to converge for ReLU networks with depth much larger than width.
New study shows neural networks need many samples for training.
problem How much data is needed to train a ReLU feed-forward neural network?
method Theoretical and empirical analysis of ReLU feed-forward neural networks.
result Generalization error scales at 1 / n 1/\sqrt{n} 1/ n in sample size n n n . Conditions for uniquely identifying parameters of deep ReLU networks.
problem Characterizing networks whose parameters can be uniquely identified.
method Conditions on deep fully-connected feedforward ReLU neural networks.
result Parameters of the network are uniquely identified under certain conditions.
Paper studies the expressivity of Convolutional Neural Networks (CNNs).
problem Understanding the expressivity of CNNs and their superiority in deep learning.
method Mathematical analysis of linear regions in one-layer and multi-layer ReLU CNNs.
result Deeper CNNs and CNNs have more expressivity per parameter than fully-connected NNs.
In this paper, we prove that a shallow neural network with a monotone sigmoid, ReLU, ELU, Softplus, or LeakyReLU activation function can arbitrarily well approximate any L^p(p>=2) integrable functions defined on R*[0,1]^n. We also prove that a shallow neural network with a sigmoid, ReLU, ELU, Softplus, or LeakyReLU act…
This paper provides an overview of activation functions in neural networks.
problem Confusion in activation function selection and properties in deep learning.
method Analytic review of popular activation functions.
result Clarification of activation function properties and selection.
A recent line of research on deep learning focuses on the extremely over-parameterized setting, and shows that when the network width is larger than a high degree polynomial of the training sample size n n n and the inverse of the target error ε − 1 ε^{-1} ε − 1 , deep neural networks learned by (stochastic) gradient descent enjoy …
This paper analyzes how normalization layers improve neural network training.
problem Improving generalization performance and training speed of neural networks.
method Global convergence analysis of two-layer neural networks with ReLU activations and Weight Normalization.
result Introduction of normalization layers changes the optimization landscape, enabling faster convergence.
Deep networks can approximate various activation functions with modest adjustments.
problem Expressive power of deep neural networks with diverse activation functions.
method Approximation of any activation function in set A by ReLU networks with specific scaling factors.
result Approximation of any activation function in a specific subset of A by ReLU networks with (1,1) scaling factors.
Paper analyzes sample complexity for offline RL with deep ReLU networks.
problem Theoretical analysis of sample complexity for offline RL with deep ReLU networks.
method Establishes sample complexity for offline RL with deep ReLU networks, considering Besov dynamic closure and correlated structure.
result First theoretical characterization of sample complexity for offline RL with deep neural network function approximation.
MGDL refines deep neural networks by training grades sequentially, improving stability.
problem Training deep neural networks is challenging due to nonconvex optimization landscapes.
method MGDL trains deep networks grade by grade, freezing previously learned grades and training new ones to fit residuals.
result MGDL guarantees vanishing error in a fixed-width multigrade ReLU architecture.
Adam achieves optimal convergence in deep ReLU networks via novel Kakeya bounds.
problem Training deep ReLU networks using Adam in non-smooth settings.
method Stratified Morse theory and Kakeya bounds to analyze region crossings and convergence.
result First global-optimal convergence for Adam in non-smooth, non-convex ReLU landscapes.
This paper tightens bounds on the smallest eigenvalue of NTK for deep ReLU networks.
problem Analyzing the smallest eigenvalue of Neural Tangent Kernel for deep ReLU networks.
method Analyzing various quantities of independent interest, including lower bounds on the smallest singular value of hidden feature matrices and upper bounds on the Lipschitz constant of input-output feature maps.
result Tight bounds on the smallest eigenvalue of NTK matrices for deep ReLU nets, both in the limiting case of infinite widths and for finite widths.
Early training of deep neural networks leads to small, directionally converging weights.
problem Training dynamics of deep homogeneous neural networks with small initializations.
method Gradient flow analysis and study of KKT points for neural correlation function.
result Weights converge in direction to KKT points during early training stages.
The paper calculates bounds on the local Lipschitz constants of neural network layers.
problem Understanding the Lipschitz constants of neural network layers for robustness analysis.
method Analytical approach to determine upper bounds on local Lipschitz constants of affine-ReLU functions.
result The method produces tighter bounds than the standard conservative bound, especially for small perturbations.
Robust deep neural networks estimate multi-dimensional functional data robustly.
problem Estimating location function from multi-dimensional functional data robustly.
method Deep neural networks with ReLU activation, robust to outliers and model misspecification.
result Uniform convergence rates for robust deep neural network estimators.
Study on the complexity of 1D ReLU neural networks, proving growth in linear regions.
problem Understanding the complexity and expressivity of 1D ReLU neural networks.
method Analyzing the number of linear regions in randomly initialized, fully connected 1D ReLU networks in the infinite-width limit.
result The expected number of linear regions grows as a function of the number of neurons in each layer.
This paper simplifies deep ReLU networks into local linear models for better interpretability.
problem Limited transparency and interpretability of deep neural networks, especially ReLU networks.
method Local linear representation and equivalent set of local linear models (LLMs).
result Simplified deep ReLU networks for better interpretability and diagnostics.
The paper improves theoretical bounds on deep neural networks' convergence.
problem Understanding convergence of over-parameterized deep neural networks.
method Surrogate network construction with fixed activation patterns.
result Convergence to a global minimum guaranteed for networks with quadratic width and linear depth.
The paper analyzes the role of ReLU gates in deep learning networks.
problem Understanding the role of gates in deep learning networks.
method Developed neural path features (NPF) and neural path values (NPV) to characterize the active sub-networks during training.
result The neural path kernel associated with NPFs is a fundamental quantity that characterizes the information stored in the gates of a DNN.
Two-layer ReLU networks can overfit without harm, study finds.
problem Understanding when and how two-layer ReLU networks can overfit without harming generalization.
method Established algorithm-dependent risk bounds for two-layer ReLU convolutional neural networks with label-flipping noise.
result Gradient descent-trained ReLU networks can achieve near-zero training loss and Bayes optimal test risk.