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.
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.
New activations improve deep network reproducibility without sacrificing accuracy.
problem Deep networks' reproducibility issues, especially on distributed systems.
method Developed SmeLU activations, smoother than ReLU, to enhance reproducibility.
result SmeLU activations provide better accuracy-reproducibility tradeoffs.
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.
Deep ReLU networks can approximate and learn smooth functions efficiently.
problem Efficiently approximating and learning smooth functions using deep ReLU neural networks.
method Extending recent results to anisotropic and mixed smooth function classes, establishing approximation rates.
result Deep ReLU networks achieve minimax optimal rates up to logarithmic factors for various smooth function classes.
Randomly initialized ReLU networks of depth two can approximate smooth functions well.
problem Approximation power of two-layer networks of random ReLUs.
method Harmonic analysis and ridgelet representation theory for upper bounds, dimensionality arguments for lower bounds.
result Near-matching upper and lower bounds for L2-approximation and Sobolev norms. 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.
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.
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.
Optimal rates for shallow ReLU networks in nonparametric regression.
problem Approximating smooth and non-smooth functions with shallow ReLU networks.
method Analysis of shallow ReLUk neural networks, using variation norms and deep learning theory. result Optimal approximation rates for shallow ReLU networks in nonparametric regression.
In spite of several notable efforts, explaining the generalization of deterministic non-smooth deep nets, e.g., ReLU-nets, has remained challenging. Existing approaches for deterministic non-smooth deep nets typically need to bound the Lipschitz constant of such deep nets but such bounds are quite large, may even incre…
This paper establishes the (nearly) optimal approximation error characterization of deep rectified linear unit (ReLU) networks for smooth functions in terms of both width and depth simultaneously. To that end, we first prove that multivariate polynomials can be approximated by deep ReLU networks of width $\mathcal{O}(N…
Sharp representation theorems show depth benefits for ReLU networks.
problem Representing functions with ReLU networks of varying depths.
method Proved sharp representation results capturing depth benefits.
result Approximation rates improve with depth for less smooth functions.
Characterizes neural kernel and NNGP for various activations.
problem Understanding neural kernels and NNGP for non-RELU activations.
method Characterization of RKHS for various activation functions.
result Broad class of non-infinitely smooth activations generate equivalent RKHSs at different depths.
ConquerNet smooths quantile regression for deep learning with minimax guarantees.
problem Optimization challenges in quantile regression for deep models.
method ConquerNet uses convolution-smoothed quantile ReLU neural networks.
result ConquerNet provides minimax guarantees and outperforms standard quantile neural networks.
We propose Top-N-Rank, a novel family of list-wise Learning-to-Rank models for reliably recommending the N top-ranked items. The proposed models optimize a variant of the widely used discounted cumulative gain (DCG) objective function which differs from DCG in two important aspects: (i) It limits the evaluation of DCG …
Smooth DNNs mitigate the curse of dimensionality in uniform convergence for various regression tasks.
problem The curse of dimensionality in uniform convergence of ReLU networks.
method Analysis of smoothly activated deep neural networks (smooth DNNs), establishing pseudo-dimension bounds and non-asymptotic approximation guarantees.
result Smooth DNNs achieve non-asymptotic uniform convergence rates across multiple statistical contexts, mitigating the curse of dimensionality.
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.
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/(2s+1) convergence rate and asymptotic normality. New findings show learning deeper neural networks is hard even with Gaussian inputs and non-degenerate weights.
problem The computational complexity of learning neural networks, especially deeper ones.
method Smoothed analysis framework and local pseudorandom generators.
result Learning depth-3 ReLU networks under Gaussian input distribution is hard even if weight matrices are non-degenerate.
We study the necessary and sufficient complexity of ReLU neural networks---in terms of depth and number of weights---which is required for approximating classifier functions in L2. As a model class, we consider the set Eβ(Rd) of possibly discontinuous piecewise Cβ functions $f : [-1/2, 1/2]^…
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.
Spike-and-Slab Deep Learning (SS-DL) is a fully Bayesian alternative to Dropout for improving generalizability of deep ReLU networks. This new type of regularization enables provable recovery of smooth input-output maps with unknown levels of smoothness. Indeed, we show that the posterior distribution concentrates at t…
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.
Guarantees uniform convergence for square-root Lipschitz losses.
problem Uniform convergence guarantees for square-root Lipschitz losses.
method Using Rademacher complexity and square root of scalar loss function Lipschitz constant.
result Generalizes previous results and handles non-smooth loss functions.
Neural networks with ReLU^k approximate Sobolev functions efficiently via Radon transform.
problem Approximating functions from Sobolev spaces using shallow ReLU^k neural networks.
method Utilizing the Radon transform and discrepancy theory, we provide nearly optimal approximation rates.
result Optimal approximation rates for smoothness up to order s = k + (d+1)/2.
A new method uses Coordinate Descent to optimize ResNet networks for private inference.
problem Reducing ReLU count in ResNet networks for private inference.
method Directly optimizing in the discrete domain using Coordinate Descent.
result Our method yields a sparse solution and is state-of-the-art.
Deep neural networks with specific parameter sets can approximate smooth functions efficiently.
problem Approximating smooth functions with deep neural networks.
method Deep neural networks with ReLU activation and specific parameter sets {0,±21,±1,2} are used to approximate Cβ-smooth functions. result The constructed networks can approximate Cβ-smooth functions with parameters {0,±21,±1,2} efficiently, achieving the same convergence rate as sparse networks with parameters in [−1,1]. New activation function BrownianReLU improves LSTM network performance on financial time series.
problem Gradient instability in noisy financial time series data.
method Introduces BrownianReLU, a stochastic activation function based on Brownian motion.
result Significantly improved predictive accuracy and generalization on financial datasets.
In this paper, we consider one dimensional (shallow) ReLU neural networks in which weights are chosen randomly and only the terminal layer is trained. First, we mathematically show that for such networks L2-regularized regression corresponds in function space to regularizing the estimate's second derivative for fairly …
We show that deep networks are better than shallow networks at approximating functions that can be expressed as a composition of functions described by a directed acyclic graph, because the deep networks can be designed to have the same compositional structure, while a shallow network cannot exploit this knowledge. Thu…
This paper improves neural network approximation for analytic functions with adjustable depth and width.
problem Approximating analytic functions using neural networks with depth and width parameters.
method Characterizes approximation rates as a joint function of width (N) and depth (L) for ReLU networks.
result Establishes upper bounds for analytic function approximation rates of O(N^(-CL^τ)) with τ influenced by N and L.
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.
Existence of minimizers proven for residual ANNs with ReLU activation.
problem Existence of minimizers in neural network optimization landscapes.
method Proof using closure of search space containing ANNs and additional discontinuous responses.
result Existence of minimizers proven for residual ANNs with ReLU activation.
In this paper, we develop an alternating direction method of multipliers (ADMM) for deep neural networks training with sigmoid-type activation functions (called \textit{sigmoid-ADMM pair}), mainly motivated by the gradient-free nature of ADMM in avoiding the saturation of sigmoid-type activations and the advantages of …
A new IPM uses ReLU networks to measure probability discrepancies.
problem Measuring the difference between two probability distributions in high dimensions.
method Proposes a new parametric IPM using ReLU neural networks to optimize and distinguish between distributions.
result The proposed IPM has good convergence rates and can be used as a surrogate for other IPMs.
Neural networks approximate unit spheres as polytopes.
problem Approximating unit spheres with neural networks.
method Using ReLU activation in neural networks to generate polytopes.
result Neural networks can approximate unit spheres as polytopes.
ResNets promote smoother interpolations than MLPs, enhancing generalization.
problem Understanding the difference in smoothness between ResNets and MLPs.
method Neural Tangent Kernel (NTK) analysis during gradient descent training.
result ResNet's NTK results in smoother interpolations than MLPs.
Revisits the connection between neural networks and the Kolmogorov-Arnold theorem.
problem Explains the limitations of using the Kolmogorov-Arnold theorem to explain neural networks with multiple hidden layers.
method Derives modifications of the Kolmogorov-Arnold representation that transfer smoothness properties to the outer function and can be well approximated by ReLU networks.
result Shows that a deep neural network with most layers approximating the interior function is a more natural interpretation of the Kolmogorov-Arnold representation.
Neural networks can approximate high-dimensional classifiers with ReLU networks under margin conditions.
problem Approximating high-dimensional discontinuous classifiers with neural networks.
method Using ReLU neural networks with three hidden layers, approximating a classifier with a Barron-regular decision boundary.
result High-dimensional discontinuous classifiers can be approximated with a rate of n−1 under strong margin conditions. High-dimensional models trained on smooth manifolds achieve optimal rates in Wasserstein metrics.
problem Training score-based generative models on complex, low-dimensional manifolds.
method Proves optimal rates for SGMs on smooth manifolds, separating into noise regimes and using ReLU nearest-projection coordinates.
result Optimal intrinsic Wasserstein rates are achieved, with polynomial ambient dependence for families with controlled geometry and density.
This paper finds sparsest ReLU networks for interpolating data.
problem Finding the sparsest neural network that fits a dataset.
method Proposes a continuous, differentiable objective function based on ℓp quasinorms. result Global minimizers of the proposed objective correspond to sparsest ReLU networks.
Whereas recovery of the manifold from data is a well-studied topic, approximation rates for functions defined on manifolds are less known. In this work, we study a regression problem with inputs on a d∗-dimensional manifold that is embedded into a space with potentially much larger ambient dimension. It is shown tha…
Deep learning has shown high performances in various types of tasks from visual recognition to natural language processing, which indicates superior flexibility and adaptivity of deep learning. To understand this phenomenon theoretically, we develop a new approximation and estimation error analysis of deep learning wit…
In this paper, we consider regression problems with one-hidden-layer neural networks (1NNs). We distill some properties of activation functions that lead to local strong convexity in the neighborhood of the ground-truth parameters for the 1NN squared-loss objective. Most popular nonlinear activation function…
Deep neural networks approximate analytic functions in high dimensions with exponential rates.
problem Approximating analytic functions in high-dimensional spaces using neural networks.
method Analyzing convergence rates of ReLU and ReLU^k activations in L2(Rd,γd) for d∈N∪{∞}. result Exponential convergence rates for analytic functions in L2(Rd,γd) for d∈N, and dimension-independent bounds for d=∞. In this paper, we consider parameter recovery for non-overlapping convolutional neural networks (CNNs) with multiple kernels. We show that when the inputs follow Gaussian distribution and the sample size is sufficiently large, the squared loss of such CNNs is locally strongly convex in a basin of attraction…
We develop a corrective mechanism for neural network approximation: the total available non-linear units are divided into multiple groups and the first group approximates the function under consideration, the second group approximates the error in approximation produced by the first group and corrects it, the third gro…