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.
Shallow neural networks can represent polynomials efficiently.
problem Representing polynomials using shallow neural networks.
method Using shallow neural networks of width 2(R+d)d to represent d-variate polynomials of degree R. result Derives minimax optimal convergence rate for shallow networks to unknown univariate regression functions.
Wide and shallow networks approximate convex functions well.
problem Understanding why wide and shallow neural networks perform well.
method Analyzing the epigraph of the input-output map of shallow and wide neural networks.
result The epigraph of the input-output map approximates a convex function.
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.
New bounds for shallow neural networks with deterministic parameters.
problem Developing generalisation bounds for shallow neural networks.
method PAC-Bayesian theory applied to shallow neural networks with deterministic parameters.
result Empirical non-vacuous bounds for shallow neural networks trained with vanilla SGD.
Sharp lower bounds on shallow neural networks' approximation rates are derived.
problem The efficiency of shallow neural networks in approximating functions.
method Lower bounding the L2-metric entropy and Kolmogorov n-widths of the convex hull of neural network basis functions. result Sharp lower bounds on the approximation rates for shallow neural networks are provided.
Study shows limits on deep and shallow neural networks for approximating compact sets.
problem Understanding the limitations of deep and shallow neural networks in approximating compact sets.
method Proved Carl's type inequalities for approximation error, using Lipschitz widths.
result Lower bounds on approximation error for neural network outputs.
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 hidden neurons. result Shows ReLU shallow neural networks can uniformly approximate Hölder functions with rates close to optimal.
Study shows shallow ReLU networks struggle with high-dimensional Lipschitz functions.
problem Expressing high-dimensional Lipschitz functions with shallow ReLU networks.
method Established lower bounds on shallow network complexity for polynomial approximation.
result Shallow ReLU networks suffer from the curse of dimensionality for Lipschitz functions.
Gradient descent trains shallow neural networks to approximate functions in 1D.
problem Approximating functions in 1D with shallow neural networks trained by gradient descent.
method Gradient descent optimization of non-convex weight space for finite width networks in 1D.
result Gradient descent can approximate functions in 1D with a minimal number of weights, balancing practical performance and theoretical capabilities.
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.
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.
A new algorithm for learning shallow neural networks with infinite width.
problem Learning shallow over-parameterized neural networks.
method Sinkhorn proximal algorithm approximating mean field learning dynamics.
result The algorithm performs gradient descent of the free energy associated with the risk functional.
Complex-valued neural networks avoid spurious local minima.
problem Finding spurious local minima in neural networks.
method Proved no spurious local minima for shallow complex neural networks with quadratic activations.
result Complex-valued weights eliminate spurious local minima in neural networks.
Neural networks minimize error with shallow ReLU models for function estimation.
problem Estimating unknown functions from noisy data.
method Minimizing squared errors plus weight decay regularization.
result Neural network estimators are minimax optimal up to logarithmic factors.
New method trains shallow neural networks with subquadratic width scaling.
problem Training shallow neural networks with optimal width scaling.
method Polyak-Lojasiewicz condition, smoothness, standard data assumptions, random matrix theory.
result Subquadratic scaling on network width with standard initialization strategies.
It is well established that neural networks with deep architectures perform better than shallow networks for many tasks in machine learning. In statistical physics, while there has been recent interest in representing physical data with generative modelling, the focus has been on shallow neural networks. A natural ques…
Study shows overparameterization helps shallow neural networks recover signals in high dimensions.
problem Signal recovery in shallow neural networks with overparameterization.
method Gradient flow on population risk, Gaussian distribution assumption, high-dimensional limit analysis.
result Minimal overparameterization is sufficient for strong recovery of signals.
This paper reports the performances of shallow word-level convolutional neural networks (CNN), our earlier work (2015), on the eight datasets with relatively large training data that were used for testing the very deep character-level CNN in Conneau et al. (2016). Our findings are as follows. The shallow word-level CNN…
Deep networks can learn functions approximated by shallow networks, but not all functions.
problem The learnability of functions by deep neural networks and the approximation capacity of simpler classes.
method Study the connection between learnability and approximation capacity of functions by deep neural networks and simpler classes.
result A necessary condition for a function to be learnable by deep neural networks is to be approximable by shallow networks.
In this paper, we introduce transformations of deep rectifier networks, enabling the conversion of deep rectifier networks into shallow rectifier networks. We subsequently prove that any rectifier net of any depth can be represented by a maximum of a number of functions that can be realized by a shallow network with a …
Sharp bounds on neural network approximation rates and widths.
problem Estimating approximation rates, metric entropy, and n-widths of shallow neural networks.
method Introducing smoothly parameterized dictionaries and providing upper and lower bounds.
result Sharp bounds on approximation rates, metric entropy, and n-widths for neural networks with various activation functions.
This work explains neural collapse in shallow neural networks and its impact on generalization.
problem Understanding neural collapse in shallow neural networks and its effect on generalization.
method Analysis of two and three-layer ReLU neural networks, focusing on data dimension, sample size, and signal-to-noise ratio.
result Neural collapse occurs in shallow ReLU networks under certain conditions related to data properties and network architecture.
GD with early stopping trains shallow neural nets for nonparametric regression robustly.
problem Learning Lipschitz regression functions with noisy labels.
method Overparameterized shallow neural networks trained by GD with early stopping.
result Optimal rates of convergence for nonparametric regression.
The paper analyzes stability and generalization of shallow neural networks using gradient methods.
problem Understanding the generalization of overparameterized shallow neural networks.
method The paper uses gradient descent and stochastic gradient descent to study shallow neural networks, developing consistent excess risk bounds.
result The analysis improves on existing methods by providing a refined estimation of iterates and Hessian eigenvalues, leading to better excess risk bounds.
PICN learns physical fields from shallow neural networks, improving AI in multi-physical systems.
problem Challenges in modeling and forecasting multi-physical systems due to data scarcity and noise.
method Physics-informed convolutional network (PICN) combining CNN and physical laws, using deconvolution and convolution layers.
result PICN effectively solves and estimates nonlinear physical operator equations and recovers physical information from noisy observations.
GD-trained shallow ReLU nets learn Lipschitz functions with noise.
problem Learning Lipschitz functions with additive noise in overparameterized neural networks.
method Gradient Descent (GD) with early stopping, focusing on the Neural Tangent Kernel (NTK).
result Early-stopped GD achieves minimax optimal rates for learning Lipschitz functions.
Improved generative models using overparametrized shallow neural networks.
problem Improving generative models for data with hidden low-dimensional structure.
method Using energy-based models with overparametrized shallow neural networks as approximators.
result Models trained in the 'active' regime outperform those in the 'lazy' or kernel regime, leading to better adaptivity to hidden structure.
There is some theoretical evidence that deep neural networks with multiple hidden layers have a potential for more efficient representation of multidimensional mappings than shallow networks with a single hidden layer. The question is whether it is possible to exploit this theoretical advantage for finding such represe…
Revisits shallow neural networks using Lipschitz norms and measures.
problem Existence and compactness of minimizers in neural network formulations.
method Mean field parametrization, signed measures, duality pairings, Kantorovich-Rubinstein norms.
result Compactness results and uniform large data limits for empirical risk minimization.
Proposes a learning rate method for shallow nets based on gradient Lipschitz constant.
problem Finding optimal learning rates for shallow neural networks.
method Associates learning rate with gradient Lipschitz constant and proposes a search algorithm.
result The proposed method significantly outperforms existing tuning methods.
Deep neural networks can grok better than shallow ones, showing multi-stage generalization.
problem Understanding the generalization behavior of deep neural networks.
method Empirical replication and analysis of grokking phenomenon in deep MLP models.
result Deep neural networks exhibit multi-stage generalization, with a secondary surge in test accuracy.
New bound on neural nets complexity for approximating functions.
problem Approximating continuous functions with shallow neural networks.
method Inspired by Stone-Weierstrass theorem, constructive proof.
result General upper bound on neuron count for accuracy.
A recent line of work has shown that an overparametrized neural network can perfectly fit the training data, an otherwise often intractable nonconvex optimization problem. For (fully-connected) shallow networks, in the best case scenario, the existing theory requires quadratic over-parametrization as a function of the …
New method identifies parameters of wider shallow neural networks with biases.
problem Identifying parameters of wide shallow neural networks with biases from finite samples.
method Two-step pipeline: direction of weights via second order information, signs via algebraic evaluations, biases via gradient descent.
result Constructive methods and theoretical guarantees of finite sample identification for wider shallow networks with biases.
We prove that the global minimum of the backpropagation (BP) training problem of neural networks with an arbitrary nonlinear activation is given by the ridgelet transform. A series of computational experiments show that there exists an interesting similarity between the scatter plot of hidden parameters in a shallow ne…
Hierarchical neural networks are exponentially more efficient than their corresponding "shallow" counterpart with the same expressive power, but involve huge number of parameters and require tedious amounts of training. By approximating the tangent subspace, we suggest a sparse representation that enables switching to …
Guiding the design of neural networks is of great importance to save enormous resources consumed on empirical decisions of architectural parameters. This paper constructs shallow sigmoid-type neural networks that achieve 100% accuracy in classification for datasets following a linear separability condition. The separab…
We provide several new depth-based separation results for feed-forward neural networks, proving that various types of simple and natural functions can be better approximated using deeper networks than shallower ones, even if the shallower networks are much larger. This includes indicators of balls and ellipses; non-lin…
Deep neural networks are over-parameterized, which implies that the number of parameters are much larger than the number of samples used to train the network. Even in such a regime deep architectures do not overfit. This phenomenon is an active area of research and many theories have been proposed trying to understand …
Mirror flow in shallow neural networks shows similar implicit bias to gradient flow, with key differences in curvature penalties.
problem Analyzing implicit bias in shallow neural networks with mirror flow.
method Characterization through variational problems and scaled potentials.
result Mirror flow with scaled potentials induces a rich class of biases not captured by RKHS norms.
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.
New stability bounds for GD in overparameterised shallow nets without NTK assumptions.
problem Generalisation and excess risk bounds for shallow neural networks.
method Oracle inequalities and stability analysis of GD without kernelisation.
result Oracle type bounds reveal GD's generalisation is controlled by an interpolating network with shortest GD path.
Solves approximation problems for zonoids and neural networks, closing gaps in dimensions 2 and 3.
problem Approximating zonoids and shallow neural networks in uniform norm.
method Combines techniques to solve both problems, closing gaps in dimensions 2 and 3.
result Completes the solution for zonoid approximation in all dimensions and improves neural network approximation rates.
Wide neural networks with asymmetrical node scaling converge globally and learn features.
problem Global convergence and feature learning in over-parameterised shallow networks.
method Gradient-based optimisation of wide, shallow neural networks with asymmetrical node scaling.
result Gradient flow and gradient descent converge to a global minimum and learn features, unlike in the NTK parameterisation.
LogDet estimator improves entropy estimation in neural networks.
problem Inconsistent observations and diversified interpretation in neural networks.
method Proposes LogDet estimator for reliable entropy approximation.
result LogDet estimator overcomes distributional diversity issues.
New neural network rates for unbounded domains with weighted Sobolev spaces.
problem Improving neural network approximation rates for unbounded domains.
method Embedding results for weighted Fourier-Lebesgue spaces in weighted Sobolev spaces, followed by asymptotic approximation rates.
result Asymptotic approximation rates for shallow neural networks without curse of dimensionality for unbounded domains and Muckenhoupt weights.
Study variation spaces for neural networks, linking them to approximation theory.
problem Understanding the variation spaces of shallow neural networks.
method Examined variation spaces defined by convex hulls and integral representations for a dictionary of functions.
result Found that Barron space, spectral Barron space, and Radon BV space are variation spaces for certain neural networks.