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.
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.
We give a new algorithm for learning a two-layer neural network under a general class of input distributions. Assuming there is a ground-truth two-layer network y = A σ ( W x ) + ξ , y = A σ(Wx) + ξ, y = A σ ( W x ) + ξ , where A , W A,W A , W are weight matrices, ξ ξ ξ represents noise, and the number of neurons in the hidden layer is no larger than the input or outp…
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.
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.
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.
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…
New algorithm optimizes smooth functions with Hölder exponent > 1.
problem Optimizing smooth functions with unknown Hölder exponent > 1.
method Two-layer algorithms using misspecified linear/polynomial bandit algorithms in bins.
result Regret bound of O ~ ( T d + α d + 2 α ) \tilde{O}(T^{\frac{d+\alpha}{d+2\alpha}}) O ~ ( T d + 2 α d + α ) for α > 1 \alpha > 1 α > 1 . 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.
Stochastic gradient methods converge for training wide PINNs.
problem Convergence of stochastic gradient descent in training over-parameterized PINNs.
method Established linear convergence of stochastic gradient descent/flow in training over-parameterized two-layer PINNs.
result Linear convergence with high probability for general activation functions.
Algorithm learns two-layer residual units using ReLU activations from samples.
problem Learning two-layer residual units from samples.
method Design layer-wise objectives as functionals, formulate ERM as QP, solve using LP, prove statistical consistency.
result Strong statistical consistency and robustness of the algorithm.
This paper extends stability and generalization analysis of GD for multi-layer NNs.
problem Understanding the generalization of multi-layer neural networks trained by GD.
method Comprehensive stability and generalization analysis of GD for multi-layer NNs, focusing on two-layer and three-layer networks.
result Derives excess risk rates of O ( 1 / n ) O(1/\sqrt{n}) O ( 1/ n ) for GD in two-layer and three-layer NNs under specific conditions. We establish connections between the problem of learning a two-layer neural network and tensor decomposition. We consider a model with feature vectors x ∈ R d \boldsymbol x \in \mathbb R^d x ∈ R d , r r r hidden units with weights { w i } 1 ≤ i ≤ r \{\boldsymbol w_i\}_{1\le i \le r} { w i } 1 ≤ i ≤ r and output y ∈ R y\in \mathbb R y ∈ R , i.e., $y=\sum_{i=1}^r σ( \boldsymbol w_i…
Heuristic tools from statistical physics have been used in the past to locate the phase transitions and compute the optimal learning and generalization errors in the teacher-student scenario in multi-layer neural networks. In this contribution, we provide a rigorous justification of these approaches for a two-layers ne…
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.
Study shows how large neural networks avoid overfitting through decoupling of feature learning and complexity growth.
problem Understanding inductive bias and generalization in large neural networks.
method Dynamical mean field theory applied to large two-layer networks.
result Training dynamics of large networks exhibit a separation of timescales, decoupling feature learning and overfitting.
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.
A two-layer classifier improves smartphone transportation mode recognition.
problem Improving accuracy of transportation mode classification.
method Two-layer hierarchical classifier combining time and frequency domain features.
result Maximum classification accuracy of 97.02%.
We solve the optimization of two-layer ReLU networks using convex math.
problem Optimizing two-layer ReLU neural networks.
method Exact characterization of optimal solutions via convex optimization.
result We prove that all globally optimal solutions can be found via convex optimization.
Recent studies show overparameterized neural networks behave like convex systems.
problem Understanding the behavior of overparameterized neural networks.
method Analysis of two-layer neural networks, focusing on restricted settings and neural tangent kernel space.
result Overparameterized neural networks behave like convex systems under certain conditions.
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 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.
DP-GD improves CNN training accuracy with privacy, especially with low signal-to-noise ratios.
problem Privacy-preserving training of neural networks with crowdsourced data.
method Differentially private gradient descent (DP-GD) algorithm applied to two-layer CNNs.
result DP-GD can achieve superior generalization performance compared to GD, especially with low signal-to-noise ratios.
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.
Generalizes neural tangent kernel analysis for two-layer networks with noise and regularization.
problem Limitations of NTK analysis in deep learning practice.
method Generalized NTK analysis for two-layer neural networks with weight decay and gradient noise.
result Noisy gradient descent with weight decay exhibits 'kernel-like' behavior and converges linearly.
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.
We consider the problem of inferring the input and hidden variables of a stochastic multi-layer neural network from an observation of the output. The hidden variables in each layer are represented as matrices. This problem applies to signal recovery via deep generative prior models, multi-task and mixed regression and …
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.
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…
Gradient descent dynamics in neural networks show quenching and activation phases.
problem Understanding training dynamics in neural networks.
method Numerical and phenomenological study of gradient descent algorithm for two-layer neural networks.
result Gradient descent dynamics exhibit quenching and activation phases in under-parametrized networks.
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.
Two-layer networks favor simple features, especially in complex datasets.
problem Simplicity bias in neural networks over-reliing on simple features.
method Characterization of two-layer neural networks with small weights and gradient flow.
result Features learned in middle training stages are more useful for out-of-distribution transfer.
New estimates for the population risk are established for two-layer neural networks. These estimates are nearly optimal in the sense that the error rates scale in the same way as the Monte Carlo error rates. They are equally effective in the over-parametrized regime when the network size is much larger than the size of…
Gradient descent struggles with high-dimensional data fitting.
problem Gradient descent struggles with high-dimensional data fitting.
method Gradient descent training of a two-layer neural network on empirical or population risk.
result Gradient descent training may not decrease population risk faster than t − 4 / ( d − 2 ) t^{-4/(d-2)} t − 4/ ( d − 2 ) under mean field scaling. Develops a two-layer model to design mortgage assistance products.
problem Designing effective mortgage assistance products to improve household resilience.
method Two-layer approach: simulation and optimization.
result Shows how the approach can design and evaluate mortgage assistance products.
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.
Nowadays, online learning is an appealing learning paradigm, which is of great interest in practice due to the recent emergence of large scale applications such as online advertising placement and online web ranking. Standard online learning assumes a finite number of samples while in practice data is streamed infinite…