SGD converges globally to logistic loss minima for two-layer nets.
problem Global convergence of SGD for logistic loss on two-layer neural nets.
method Demonstrates existence of Frobenius norm regularized logistic loss functions as Villani functions, proving convergence and exponential rate.
result SGD converges globally to the global minima of appropriately regularized logistic empirical risk of depth 2 nets.
Global convergence of SGD proven for two-layer neural nets with regularization.
problem Proving global convergence of SGD for two-layer neural nets.
method Regularized empirical risk, SGD iterates, Villani functions.
result Global convergence of SGD for a special class of initializations.
This work improves sample efficiency in neural function approximation for reinforcement learning.
problem Improving sample efficiency in reinforcement learning with neural function approximation.
method Study of function approximation with two-layer neural networks (ReLU and polynomial activations) under generative and realizability models.
result Significant improvement in sample complexity compared to linear methods.
Local convergence theory for mildly over-parameterized neural nets.
problem Understanding why over-parameterization works in neural networks.
method Developed a local convergence theory for two-layer neural nets, showing neuron convergence under certain conditions.
result All student neurons converge to one of teacher neurons when the loss is below a threshold.
We derive generalization and excess risk bounds for neural nets using a family of complexity measures based on a multilevel relative entropy. The bounds are obtained by introducing the notion of generated hierarchical coverings of neural nets and by using the technique of chaining mutual information introduced in Asadi…
Gradient descent learns over-param neural nets better than NTK.
problem Learning over-parametrized neural networks with ReLU activations.
method Gradient descent from random initialization on a Gaussian input distribution.
result Gradient descent achieves population loss o(1/d), while NTK achieves Ω(1/d). Feature selection, as a critical pre-processing step for machine learning, aims at determining representative predictors from a high-dimensional feature space dataset to improve the prediction accuracy. However, the increase in feature space dimensionality, comparing to the number of observations, poses a severe challe…
Catapult phase in neural nets shows exponential loss growth before quick decrease.
problem Understanding phase transitions in neural networks during training.
method Analyzing weight norm and loss behavior for super-critical learning rates.
result Proven existence of catapult phase in quadratic models and two-layer nets.
Recent works have shown that on sufficiently over-parametrized neural nets, gradient descent with relatively large initialization optimizes a prediction function in the RKHS of the Neural Tangent Kernel (NTK). This analysis leads to global convergence results but does not work when there is a standard ℓ2 regulari…
Recent works have cast some light on the mystery of why deep nets fit any data and generalize despite being very overparametrized. This paper analyzes training and generalization for a simple 2-layer ReLU net with random initialization, and provides the following improvements over recent works: (i) Using a tighter char…
This work finds a point with small test error in polynomial time for mildly overparameterized neural nets.
problem Achieving small test error in mildly overparameterized neural networks.
method The work shows that the landscape of loss functions with explicit regularization has a property that all local minima and certain stationary points achieve small test error. It also proves the existence of polynomial time algorithms for finding such points in convolutional and fully connected neural nets.
result Polynomial time algorithms exist for finding points with small test error in mildly overparameterized neural nets.
A simple function shows how neural nets can converge despite high sharpness.
problem Understanding why neural nets converge with high sharpness.
method Constructed a minimal example function and analyzed its training dynamics rigorously.
result Final converging point has sharpness close to 2/η. Sobolev training helps neural nets fit function values and derivatives.
problem Training neural nets to match function values and derivatives accurately.
method Using Sobolev loss with gradient flow for overparameterized networks.
result Gradient flow from random initialization can fit any function and its derivatives.
In this paper, we analyze efficacy of the fast gradient sign method (FGSM) and the Carlini-Wagner's L2 (CW-L2) attack. We prove that, within a certain regime, the untargeted FGSM can fool any convolutional neural nets (CNNs) with ReLU activation; the targeted FGSM can mislead any CNNs with ReLU activation to classify a…
Complexity measures for neural nets with general activations using path-based norms.
problem Control complexity of neural networks with arbitrary activation functions.
method Approximate general activations with ReLU networks and derive path-based norms for complexity control.
result Preliminary analyses of function spaces and regularized estimators.
Optimal feature learning strength improves generalization in deep networks.
problem Understanding how feature learning strength affects generalization in practical settings.
method Empirical studies and theoretical analysis of gradient flow dynamics in two-layer ReLU nets.
result Optimal feature learning strength yields substantial generalization gains, contrary to the prevailing intuition.
Deep neural nets on 1-D data are convex Lasso models with reflection features.
problem Training neural networks on 1-D data.
method Proving equivalence to convex Lasso problems with discrete, explicitly defined dictionary matrices.
result Reflection features in neural networks with certain activations.
Adversarial training leads to clean data generalization with significant robust overfitting gap.
problem Significant robust generalization gap in adversarial training.
method Two theoretical views: representation complexity and training dynamics.
result ReLU nets with O(ND) extra parameters can achieve CGRO. 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.
We consider the problem of function approximation by two-layer neural nets with random weights that are "nearly Gaussian" in the sense of Kullback-Leibler divergence. Our setting is the mean-field limit, where the finite population of neurons in the hidden layer is replaced by a continuous ensemble. We show that the pr…
Gradient descent with logistic loss can make two-layer networks interpolate binary classification data.
problem Training two-layer networks for binary classification.
method Gradient descent with logistic loss applied to two-layer networks.
result Gradient descent can drive training loss to zero under certain conditions.
Two-layer neural networks can approximate functions with fractal singularities.
problem Characterizing functions that can be represented by infinitely wide two-layer neural networks.
method Representation formulas and pointwise properties analysis.
result Functions with fractal or curved singularities cannot be represented by two-layer networks with finite path-norm.
Analytical method finds deeper optima in two-layer ReLU networks.
problem Training two-layer ReLU networks with analytical methods.
method Analytically finding critical points of the loss function for one layer while keeping the other fixed.
result Significantly smaller training loss values on real datasets compared to gradient descent methods.
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.
Three-layer networks learn more complex features than two-layer networks.
problem Understanding feature learning in deep neural networks.
method Analysis of three-layer neural networks trained with gradient descent.
result Three-layer networks can learn functions that two-layer networks cannot.
We study cascades on a two-layer multiplex network, with asymmetric feedback that depends on the coupling strength between the layers. Based on an analytical branching process approximation, we calculate the systemic risk measured by the final fraction of failed nodes on a reference layer. The results are compared with…
Two-layer neural networks learn efficiently using kernel methods in mean-field analysis.
problem Feature learning ability of two-layer neural networks in the mean-field regime.
method Mean-field analysis through kernel methods, focusing on dynamics of the first layer's kernel.
result Two-layer neural networks can learn a union of multiple reproducing kernel Hilbert spaces more efficiently than kernel methods.
Two-layer networks trained on low-dimensional subspaces are vulnerable to adversarial examples.
problem Vulnerability of two-layer neural networks to adversarial examples on low-dimensional subspaces.
method Analysis of gradient behavior and effect of initialization scale and regularization.
result Decreasing initialization scale or adding L2 regularization can improve robustness to adversarial perturbations orthogonal to the data.
Two-layer neural networks must be robust, even with arbitrary weights.
problem Proving the robustness of two-layer neural networks with arbitrary weights.
method Developed a new function-space covering method to prove the robustness law, replacing parameter-space covering.
result Proved the conjectured law for two-layer networks with arbitrary real weights, biases, and affine skip connections.
This work shows linear convergence for two-layer neural networks in mean-field regime.
problem Optimizing two-layer neural networks in the mean-field regime.
method Mean-field analysis and continuous-time noisy gradient descent.
result Establishes linear convergence rate for two-layer neural networks.
We focus on estimating \emph{a priori} generalization error of two-layer ReLU neural networks (NNs) trained by mean squared error, which only depends on initial parameters and the target function, through the following research line. We first estimate \emph{a priori} generalization error of finite-width two-layer ReLU …
We study the limits and methods of training two-layer autoencoders.
problem Understanding the limits and methods of training two-layer autoencoders.
method Focus on non-linear two-layer autoencoders trained in the proportional regime, using gradient methods.
result Gradient methods achieve the minimizers of the population risk and reveal the structure of the features.
Study on SGD for overparameterized neural networks, focusing on convergence rates.
problem Understanding convergence rates of SGD in overparameterized two-layer neural networks.
method Combines NTK approximation with RKHS analysis to explore SGD dynamics.
result Established sharp convergence rates for SGD in overparameterized two-layer neural networks.
Deep networks can adapt to intrinsic dimensionality beyond domain constraints.
problem Approximating functions on low-dimensional manifolds with high-dimensional data.
method Two-layer compositions with ReLU activation, using dimensionality reducing feature maps.
result Near optimal approximation rates depend on the complexity of the dimensionality reducing map, not the ambient dimension.
Study shows infoGAN's generalization error bound for two-layer networks.
problem Understanding generalization error in infoGAN for two-layer neural networks.
method Analyzes the difference between empirical and population objective functions, derives Rademacher complexity bounds.
result Derives error bound for infoGAN's generalization error in a two-layer network.
This paper presents a phase diagram for two-layer neural networks under different initialization scales.
problem Understanding the behavior of neural networks under varying scales of initialization.
method Analysis of a phase diagram for two-layer neural networks.
result Condensation of weight vectors on isolated orientations during training.
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.
Two-layer CNNs can overfit well if initialized correctly.
problem Understanding the conditions for benign overfitting in over-parameterized CNNs.
method Extending analysis to fully trainable two-layer CNNs, examining initialization scaling effects.
result Initialization scaling of the output layer is crucial; large scales lead to fixed output behavior, small scales to complex interactions.
This paper explains double descent in linear neural networks, identifying new factors.
problem Understanding double descent in linear neural networks.
method Gradient flow derivation and necessary conditions for double descent.
result Singular values of input-output covariance matrix are important for double descent in two-layer models.
Two-layer networks learn hard GLMs with SGD in high dimensions.
problem Learning hard generalized linear models with SGD in high-dimensional settings.
method Reduction of SGD dynamics to a stochastic process in lower dimensions, focusing on the role of stochasticity.
result Overparameterization enhances convergence by a constant factor, suggesting minimal role of stochasticity.
Study of two-layer NNs under Gaussian mixtures data, proving polynomial models equivalent to neural networks.
problem Training and generalization performance of two-layer NNs under structured Gaussian mixture data.
method Asymptotic analysis of two-layer NNs after one gradient descent step under Gaussian mixture data assumption.
result High-order polynomial models equivalent to nonlinear neural networks under certain conditions.
The paper explores discrete isothermic nets using checkerboard patterns in quadrilateral nets.
problem Defining and understanding discrete isothermic nets in quadrilateral nets.
method Using checkerboard patterns and discrete differential geometry to define and analyze isothermic nets.
result The class of isothermic nets is invariant under dualization and Moebius transformations.
We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on larger parameter rectangles. For discrete planar quadrilateral nets, circular nets, Q∗-nets and conical…
It has been shown that gradient descent can yield the zero training loss in the over-parametrized regime (the width of the neural networks is much larger than the number of data points). In this work, combining the ideas of some existing works, we investigate the gradient descent method for training two-layer neural ne…
Recently, several studies have proven the global convergence and generalization abilities of the gradient descent method for two-layer ReLU networks. Most studies especially focused on the regression problems with the squared loss function, except for a few, and the importance of the positivity of the neural tangent ke…
Signal models based on sparsity, low-rank and other properties have been exploited for image reconstruction from limited and corrupted data in medical imaging and other computational imaging applications. In particular, sparsifying transform models have shown promise in various applications, and offer numerous advantag…
ARFF reduces spectral bias in SGD-trained neural networks.
problem Spectral bias in two-layer neural networks.
method Comparison of SGD and ARFF on spectral bias and robustness.
result ARFF yields a closer to zero spectral bias compared to SGD.
Analyzes SGD dynamics in two-layer networks, bridging different regimes.
problem Understanding SGD dynamics in high-dimensional and mean-field settings.
method Rigorous analysis via deterministic low-dimensional description of sufficient statistics.
result Infinite-width dynamics remains close to a low-dimensional subspace.