New neural networks combine additive regression with traditional architectures.
problem Performance limitations and high parameter requirements of traditional neural networks.
method Introduce hybrid deep additive neural networks with simpler activation and basis functions.
result Hybrid neural networks achieve better performance with fewer parameters.
A mathematical framework connects neural networks and polynomial regression for better model understanding.
problem Neural networks are black boxes with challenges in dimensioning and prediction error evaluation.
method Developed a mathematical framework using Taylor expansion to relate neural networks and polynomial regression.
result Polynomial approximations from neural networks trained on polynomial data are accurate locally.
Gradient descent trains neural networks to match kernel regression's sharp generalization rate.
problem Training over-parameterized neural networks for nonparametric regression.
method Gradient descent with early stopping on over-parameterized two-layer neural networks.
result Trained neural networks achieve sharp generalization rate of O ( ε n 2 ) \mathcal{O}(ε_n^2) O ( ε n 2 ) . Deep neural networks with adversarial training achieve sup-norm convergence for nonparametric regression.
problem Achieving sup-norm convergence for deep neural network estimators in nonparametric regression.
method Developed an adversarial training scheme to address the sup-norm convergence issue.
result Deep neural network estimators achieve optimal sup-norm convergence with the proposed adversarial training.
Study shows trained neural networks can overfit without bias or variance issues.
problem Understanding overfitting in trained two-layer ReLU networks.
method Analysis of gradient flow in the neural tangent kernel regime, decomposition of excess risk.
result Trained networks can overfit benignly without bias or variance issues.
Enhances neural network regression performance by modeling weight and variance uncertainty.
problem Improving predictive performance of neural networks for regression tasks.
method Extended Blundell's framework to include variance uncertainty, using a full posterior distribution over variance parameters.
result Explicitly modeling variance uncertainty improves generalization of Bayesian neural networks.
New results show neural networks generalize well due to polynomial regression, not just overparametrization.
problem Generalization of neural networks despite overparametrization.
method Teacher/Student model with polynomial regression.
result Student networks interpolating teacher-generated data generalize well with a minimal sample size.
Tensor Neural Networks improve regression accuracy and efficiency.
problem Nonparametric regression problems with complex, high-dimensional functions.
method Integrates statistical regression and numerical integration within a tensor neural network framework.
result Superior performance in approximation accuracy and generalization capacity compared to FFNs and RBNs.
Proposes a deep neural network for spatial data regression.
problem Regression of spatial data using deep neural networks.
method Localized two-layer deep neural network for spatial data, proving consistency and asymptotic convergence.
result Asymptotic convergence rate is faster than existing methods, demonstrating effectiveness on temperature estimation.
We analyze how kernel machines and neural networks learn different frequency modes of the target function as data size increases.
problem Understanding how kernel machines and neural networks learn different frequency modes of the target function as data size increases.
method Theoretical methods from Gaussian processes and statistical physics, combined with simulations on synthetic data and MNIST dataset.
result Kernel machines and neural networks fit successively higher spectral modes of the target function as the size of the training set grows.
Proposes a neural network for contextual regression.
problem Improving model efficiency and interpretability in regression with contextual features.
method Simple contextual neural network (SCtxtNN) that separates context identification from context-specific regression.
result SCtxtNN achieves lower excess mean squared error and more stable performance than feed-forward neural networks.
There is emerging interest in performing regression between distributions. In contrast to prediction on single instances, these machine learning methods can be useful for population-based studies or on problems that are inherently statistical in nature. The recently proposed distribution regression network (DRN) has sh…
Neural networks can be simplified to linear regression for easier understanding by statisticians.
problem Introducing neural networks to statisticians who are not familiar with them.
method Describing neural networks that approximate linear regression and discussing customizations.
result Statisticians can now understand neural networks by focusing on linear regression.
Consider the multivariate nonparametric regression model. It is shown that estimators based on sparsely connected deep neural networks with ReLU activation function and properly chosen network architecture achieve the minimax rates of convergence (up to log n \log n log n -factors) under a general composition assumption on the re…
Paper develops neural network for distribution regression.
problem Regression with probability measures.
method Develops a novel fully connected neural network (FNN) for distribution inputs.
result Almost optimal learning rates for distribution regression derived.
Adapting robust statistics to neural networks, researchers found neural networks can be more robust with certain loss functions.
problem The robustness of neural networks in complex learning tasks.
method Adapting the regression breakdown point from robust statistics to neural networks and comparing different configurations and contamination settings.
result Neural networks can benefit from robust loss functions, as demonstrated in extensive simulations.
This work explains how large neural networks generalize well despite overparameterization.
problem Understanding the generalization behavior of large neural networks.
method Theoretical analysis of approximation and generalization errors in regression and classification tasks.
result Deep overparameterized neural networks are statistically consistent across different tasks when regularization is applied.
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.
Analyzes neural networks using linear models to understand their behavior.
problem Understanding multi-layer neural networks through linear models.
method Recalls and reviews four models: linear regression with concentrated features, kernel ridge regression, random feature model, and neural tangent model.
result Highlights limitations of linear theory and discusses approaches to overcome them.
Neural networks improve nonparametric regression with measurement errors.
problem Nonparametric regression with measurement errors.
method Proposes a neural network design using FNN, normalizing flow, and inference network.
result Neural network approach is more flexible and superior or comparable to classical methods.
This paper characterizes how randomized neural networks generalize well in multi-dimensional tasks.
problem Understanding the generalization of randomized neural networks in multi-dimensional tasks.
method Characterizes RSNs as an IGAM formalized by an optimization problem with a regularization functional and loss.
result RSNs generalize well in multi-dimensional tasks, akin to spline regression under certain conditions.
Bayesian Additive Regression Networks use neural networks for regression tasks.
problem Regression tasks with small neural networks and ensemble learning.
method Bayesian Additive Regression Tree principles applied to small neural networks, Gibbs sampling for ensemble learning.
result BARN provides more consistent and often more accurate results than shallow neural networks, BART, and ordinary least squares.
Generalizes leverage score sampling for neural networks, accelerating kernel methods and deep learning.
problem Accelerating kernel methods and deep learning training.
method Generalizes leverage score sampling to neural networks and proves equivalence to neural tangent kernel ridge regression.
result Equivalence between regularized neural network and neural tangent kernel ridge regression under leverage score sampling initialization.
New method extracts aleatoric and epistemic uncertainties from regression-based neural networks.
problem Need for principled uncertainty reasoning in machine learning systems.
method Learning evidential distributions for aleatoric and epistemic uncertainties.
result Allows for the simultaneous extraction of both uncertainties without sampling or out-of-distribution data.
Neural networks trained with PGD achieve sharp regression rates in interpolation spaces.
problem Nonparametric regression using over-parameterized neural networks in interpolation spaces.
method Over-parameterized two-layer neural networks trained with Preconditioned Gradient Descent (PGD) and early stopping.
result Achieves a sharp regression rate of \(\cO(n^{-\frac{2αs'}{2αs'+1}})\) in interpolation spaces \(\bth{\cH_K}^{s'}\).
Bayesian neural network models improve uncertainty quantification in multivariate regression.
problem Uncertainty quantification in multivariate regression models with heteroscedastic noise.
method Proposes Bayesian Last Layer neural network models and EM algorithms for parameter learning.
result Capable of disentangling aleatoric and epistemic uncertainty.
The paper bounds neural networks' approximation error and applies it to regression and GANs.
problem Bounding the approximation error of norm-constrained neural networks.
method Proved upper and lower bounds on approximation error using Rademacher complexity.
result Obtained convergence rates for over-parameterized neural networks and optimal GAN learning rates.
Sparse-penalized deep neural networks improve performance in weakly dependent processes.
problem Nonparametric regression and classification under weak dependence.
method Sparse-penalized deep neural networks with oracle inequalities and convergence rates established.
result The proposed estimators outperform non-penalized ones in simulations.
Extends deep learning with interpretable additive models.
problem Identifiability issues between neural networks and additive models.
method Orthogonalization cell to separate deep neural network and structured model parts.
result Stable estimation and interpretability of structured model parts.
NTK neural networks are robust to adversarial attacks in nonparametric regression.
problem Adversarial robustness of neural networks in nonparametric regression.
method Gradient flow with early stopping for NTK neural networks, proving robustness in Sobolev spaces.
result NTK neural networks achieve optimal adversarial robustness rates in Sobolev spaces.
We analyzed optimism in linear and kernel regression models.
problem Understanding predictive complexity in regression models.
method Derived closed-form asymptotic optimism for linear and kernel regression models.
result Scaled optimism is a useful measure for model complexity.
The paper explains generalization in kernel regression and deep neural networks using spectral bias and task-model alignment.
problem Understanding generalization in machine learning models, especially deep neural networks.
method Analytical expression for generalization error derived from statistical mechanics, applied to various kernels and data distributions.
result Spectral bias and task-model alignment explain generalization in kernel regression and deep neural networks.
The neural tangent kernel equivalence theorem fails in practice.
problem Does the neural tangent kernel (NTK) equivalence theorem hold in practical neural network training?
method Rigorously derived NTK and conducted numerical experiments to evaluate the equivalence theorem.
result Adding a layer to a neural network and the corresponding updated NTK do not yield matching changes in predictor error.
The paper uses geometry to understand how neural networks learn.
problem Understanding the learning capability of neural networks.
method Statistical and differential geometric analysis of neural networks performing simple regression.
result Neural networks with higher generalization capability have a slower convergence rate.
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.
Neural networks simplify SDR in regression tasks.
problem Sufficient dimension reduction in regression problems.
method Applying neural networks with rank regularization to estimate the central mean subspace.
result Neural networks effectively perform SDR, consistent with theoretical estimations.
Twin neural network regression predicts differences between two data points.
problem Traditional regression methods are inaccurate for certain data sets.
method TNN regression predicts differences between two data points and averages predictions from an ensemble of all training data points.
result TNN regression yields more accurate predictions compared to other methods.
A new neural network model for ordinal regression.
problem Ordinal regression with non-proportional odds.
method Interpretable neural network for both continuous and discrete responses, training a non-linear neural network as a coefficient function.
result N 3 ^3 3 POM preserves interpretability while offering flexibility. Trained neural networks perform Bayesian reasoning for tasks beyond their initial scope.
problem Performing Bayesian reasoning for tasks outside the trained neural networks' initial scope.
method Used deep generative models as priors and classification/regression networks as constraints. Approximated Bayesian inference through variational or sampling techniques.
result The approach built on top of already trained networks, expanding the addressable questions.
DRE combines DNN with random feature regression for efficient neural network design.
problem Designing and training deep neural networks (DNN) efficiently and effectively.
method DRE architecture with two-layer neural networks, randomly drawn input and output weights trained with linear ridge regression.
result DRE outperforms state-of-the-art DNN in many data sets with lower computational cost.
Deep neural nets can estimate regression with dependent data without the curse of dimensionality.
problem Regression with dependent data and structural assumptions on the regression function.
method Deep recurrent neural network estimate under suitable structural assumptions.
result Deep neural nets can circumvent the curse of dimensionality for regression with dependent data.
Paper analyzes neural network complexity for planning problems.
problem Understanding neural network complexity for planning policies.
method Circuit complexity analysis for relational neural networks.
result Three classes of planning problems identified based on network complexity.
A neural network model tackles high-dimensional data with latent structures.
problem Modeling high-dimensional data with latent low-dimensional structures.
method Integrates PCA and Soft PCA layers into neural network architecture for factor modeling and non-linear transformations.
result Demonstrates improved performance in forecasting and nowcasting with real-world data.
This work improves online regression and contextual bandits using neural networks.
problem Improving online regression and contextual bandits using neural networks.
method Investigates neural networks for online regression, showing O ( log T ) \mathcal{O}(\log T) O ( log T ) regret for almost convex losses and KL loss. result Shows i l d e O ( K L ∗ + K ) ilde{\mathcal{O}}(\sqrt{KL^*} + K) i l d e O ( K L ∗ + K ) regret for NeuCB, outperforming existing algorithms. This paper explores adaptive methods in over-parameterized linear regression.
problem Understanding why neural networks generalize well in over-parameterized settings.
method Characterizes two sub-classes of adaptive methods and their generalization performance.
result Adaptive methods in over-parameterized linear regression converge to the minimum norm solution.
A neural network solves logistic regression with ℓ 1 \ell_1 ℓ 1 regularization efficiently.
problem Efficiently solving logistic regression with ℓ 1 \ell_1 ℓ 1 regularization due to non-differentiability of ℓ 1 \ell_1 ℓ 1 norm. method A simple projection neural network that avoids auxiliary variables and smooth approximations.
result The neural network converges to a solution of the problem with any initial value and outperforms existing methods.
Neural networks can approximate functionals on RKHS with error bounds.
problem Approximating functionals on RKHS using neural networks.
method Interpolating orthogonal projections in RKHS using point evaluations.
result Explicit error bounds for various kernels (inverse multiquadric, Gaussian, Sobolev).
The pricing of Bermudan options amounts to solving a dynamic programming principle, in which the main difficulty, especially in high dimension, comes from the conditional expectation involved in the computation of the continuation value. These conditional expectations are classically computed by regression techniques o…