This study connects ReLU neural networks to toric geometry to analyze function realization.
problem Determining which continuous piecewise linear functions can be realized by ReLU neural networks.
method Established a connection between toric geometry and ReLU neural networks, defining key structures like the ReLU fan, toric variety, and Cartier divisor.
result Proved a criterion for functions realizable by unbiased shallow ReLU networks using intersection numbers.
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.
Single ReLU neuron's gradient dynamics reveal support vectors as key to generalization.
problem Understanding the generalization capability of ReLU networks.
method Examined gradient flow dynamics and support vectors in single ReLU neuron training.
result Support vectors play a crucial role in the generalization of ReLU networks.
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.
Proves existence of optimal shallow neural networks with ReLU activation.
problem Proving the existence of optimal shallow feedforward networks with ReLU activation.
method Proves existence of global minima in the loss landscape for continuous target functions using shallow feedforward neural networks with ReLU activation.
result Existence of global minima in the loss landscape for shallow feedforward networks with ReLU activation.
Modified ReLU networks improve regression estimation rates.
problem Regression estimation with smooth functions.
method Using modified ReLU neural networks with specific weight modifications.
result Empirical risk minimizers achieve minimax rate of prediction.
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.
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.
Study shows efficient neural network approach for stochastic bandits.
problem Optimizing decisions in uncertain environments with neural network models.
method OFU-ReLU algorithm that balances exploration and exploitation, using a transformed feature space.
result Achieves i l d e O ( T ) ilde{O}(\sqrt{T}) i l d e O ( T ) regret guarantee for stochastic bandits with ReLU neural networks. PHP connects to ReLU neural networks for scalable Bayesian inference.
problem Scalability and Bayesian inference in two-layer ReLU neural networks.
method PHP with Gaussian prior, decomposition propositions, annealed sequential Monte Carlo.
result PHP provides an alternative scalable representation for two-layer ReLU neural networks.
The paper investigates how target normalization and momentum affect dying ReLUs in neural networks.
problem Understanding and mitigating the dying ReLU problem in neural networks.
method Empirical analysis and theoretical modeling of a discrete-time linear autonomous system.
result Target variance plays a crucial role in the dying ReLU phenomenon, and momentum exacerbates this issue.
We study algebraic varieties of ReLU networks to understand their representable functions.
problem Understanding the functions that ReLU neural networks can represent.
method We introduce algebraic varieties associated with ReLU networks and derive polynomial equations to characterize representable functions.
result Conditions under which ReLU networks attain their expected dimension, providing insight into their structural properties.
Gradient descent converges to minimum Bayes risk for two-layer ReLU networks in mean field regime.
problem Training two-layer ReLU networks using gradient descent in the mean field regime.
method Describes a condition for convergence to minimum Bayes risk, extending previous results to ReLU-activated networks.
result The condition for convergence does not depend on initialization and concerns weak convergence of network realization.
The paper calculates upper bounds on ReLU network Lipschitz constants.
problem Determining the maximum perturbation size for robustness of neural networks.
method Analyzing ReLU, affine-ReLU, and max pooling functions; combining results; tracking zero elements; using a computational approach.
result The method produces the largest known bounds on minimum adversarial perturbations for large networks.
Gradient descent biases towards stable rank networks for nearly-orthogonal data.
problem Understanding implicit bias in non-smooth neural networks trained by gradient descent.
method Analysis of two-layer ReLU and leaky ReLU networks trained by gradient descent on nearly-orthogonal data.
result Gradient descent biases towards networks with stable rank and uniform margin for nearly-orthogonal data.
Bayesian approach fixes overconfidence in ReLU networks, even slightly.
problem Overconfidence in ReLU networks far from training data.
method Theoretical analysis of approximate Gaussian distributions on ReLU weights, and empirical validation.
result Even a simplistic Bayesian approximation fixes overconfidence issues.
New bounds on ReLU networks for low-regular functions.
problem Bounding approximation error for ReLU networks on low-regular functions.
method Complexity analysis of Fourier features residual networks to ReLU networks.
result Approximation error bound proportional to target function norm and inversely proportional to network width and depth.
The paper examines topological features of ReLU networks and their relation to decision boundaries and training loss.
problem Understanding the topological structure of ReLU neural network activation patterns.
method Polytope decomposition of feature space, Fiedler partition of dual graph, homology computation of cellular decomposition.
result The Fiedler partition of the dual graph correlates with decision boundaries in binary classification tasks, and similar patterns in training loss and polyhedral cell-count emerge in regression tasks.
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 underlying data distribution being e.g. Gaussian, or require the network size an…
ReLU networks implicitly favor low-rank solutions, but not as strongly as linear networks.
problem Understanding implicit regularization in ReLU networks for rank minimization.
method Analysis of gradient flow on ReLU networks, empirical testing.
result Gradient flow on ReLU networks does not necessarily minimize ranks, unlike in linear networks.
Polynomial time algorithm learns depth-2 neural networks with ReLU activations.
problem Learning depth-2 neural networks with non-zero bias terms and general ReLU activations.
method Robust tensor decomposition of Hermite expansions.
result Polynomial time and sample efficient learning of depth-2 networks with ReLU activations.
The paper reveals how clustering in ReLU networks enhances interpretability.
problem Improving interpretability of deep learning models.
method Clustering of ReLU neuron patterns to reveal underlying affine maps.
result The network's predictions can be explained by feature importance within clusters.
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.
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.
Unified approach to verify NN properties using ReLU's unique polytope structure.
problem Lack of robustness and interpretability in ReLU NNs for risk-sensitive applications.
method Identifying and traversing the local polytopes of ReLU NNs, developing an algorithm to verify properties.
result Unified approach to examine network behavior in risk-sensitive settings.
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…
Leaky ReLU activations improve the calibration of Bayesian neural networks.
problem Bayesian neural networks struggle with mean-field variational inference for ReLU activations.
method Investigated the effect of activation functions on the calibration of Bayesian neural networks.
result Leaky ReLU activations lead to more Gaussian-like weight posteriors and lower expected calibration error.
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.
Gradient descent with logistic loss can interpolate deep networks with smoothed ReLU activations under certain conditions.
problem Conditions for gradient descent to drive logistic loss to zero in deep networks with smoothed ReLU activations.
method Gradient descent applied to fixed-width deep networks with smoothed ReLU approximations (e.g., Swish, Huberized ReLU).
result Gradient descent can drive logistic loss to zero under specific conditions, providing bounds on convergence rate.
Hamiltonian Monte Carlo on ReLU networks is inefficient due to large local error.
problem Inefficiency of Hamiltonian Monte Carlo on ReLU neural networks.
method Analysis of Hamiltonian Monte Carlo with leapfrog integrator for Bayesian neural network inference.
result Leapfrog HMC for ReLU networks has a large local error rate of Ω ( ε ) Ω(ε) Ω ( ε ) , leading to inefficiency. Quantile regression with ReLU networks achieves minimax rates for various function types.
problem Estimating quantiles from covariates with neural networks.
method Quantile regression with rectified linear unit (ReLU) neural networks.
result ReLU networks achieve minimax rates for broad collections of function types.
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.
A new neural network model for optimal treatment assignment.
problem Learning optimal treatment assignment from observational data.
method Prescriptive ReLU network (P-ReLU) model that balances performance and interpretability.
result P-ReLU can be converted into an equivalent prescriptive tree with hyperplane splits for interpretability.
The study examines Fisher information matrices and neural tangent kernels for simple ReLU networks with random weights.
problem Understanding the relationship between Fisher information matrices and neural tangent kernels for 2-layer ReLU networks.
method Analyzes Fisher information matrices and neural tangent kernels for 2-layer ReLU networks with random hidden weights, focusing on spectral decomposition and eigenfunctions.
result Obtained an approximation formula for functions represented by 2-layer neural networks.
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.
Novel approach analyzes ReLU networks' training dynamics and proposes GmP for improved optimization.
problem Stochastic optimization instability in ReLU networks impedes convergence and generalization.
method Characteristic activation boundaries analysis and Geometric Parameterization (GmP) technique.
result GmP resolves instability, leading to better optimization, convergence, and generalization.
Improved sample complexity for ReLU networks with norm constraints.
problem Estimating sample complexity for ReLU networks under norm constraints.
method Refined Rademacher complexity analysis for function class.
result Often no explicit depth-dependence in sample complexity bound.
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.
Study reveals properties of local minima in ReLU networks.
problem Understanding the loss landscape of neural networks.
method Theoretical analysis of one-hidden-layer ReLU networks.
result All differentiable local minima are global within certain regions.
Despite their prevalence in neural networks we still lack a thorough theoretical characterization of ReLU layers. This paper aims to further our understanding of ReLU layers by studying how the activation function ReLU interacts with the linear component of the layer and what role this interaction plays in the success …
Paper improves confidence intervals and variance estimation for deep learning models.
problem Improving confidence intervals and variance estimation in deep learning models.
method Residual-based framework for conditional variance estimation; robust bootstrap procedure for confidence intervals.
result First non-asymptotic bounds for variance estimation using ReLU networks.
The study examines if ReLU activation function is optimal for modularity in neural networks.
problem Finding the best activation function for modularity in neural networks.
method Comparing ReLU with other activation functions for modularity and performance.
result ReLU may not be the best choice for modularity, suggesting other functions could be more suitable.
We study the approximation properties of random ReLU features through their reproducing kernel Hilbert space (RKHS). We first prove a universality theorem for the RKHS induced by random features whose feature maps are of the form of nodes in neural networks. The universality result implies that the random ReLU features…
The paper proves neural networks with ReLU and softmax can approximate any function.
problem Approximating functions and class labels in neural networks.
method Extended universal approximator theory to neural networks with ReLU and softmax.
result Neural networks with ReLU and softmax can approximate any function and class labels.
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.
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 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 ) . New insights into how neural networks classify data.
problem Understanding the topological structure of decision regions in ReLU networks.
method Defining generic and transversal ReLU networks, and using linear complexes to identify obstructions.
result Generic, transversal ReLU networks have at most one bounded connected component in their decision regions.