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.
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.
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…
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.
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.
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 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.
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.
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.
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.
Study on double descent behavior in two-layer neural networks for binary classification.
problem Understanding the double descent phenomenon in model test error.
method Two-layer neural network with ReLU activation for binary classification. Quantified model size by sample-to-dimension ratio. Empirical risk minimization using Convex Gaussian Min Max Theorem.
result Observed and investigated the double descent behavior of model test error.
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.
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…
AdaLoss optimizes adaptive learning rates for efficient convergence in various models.
problem Efficiently optimizing adaptive learning rates for gradient descent methods.
method AdaLoss uses loss function information to dynamically adjust step sizes.
result AdaLoss achieves linear convergence in linear regression and robust global convergence in neural networks.
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…
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.
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 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.
The paper studies learning dynamics in two-layer neural networks.
problem Learning dynamics and time scales in two-layer neural networks.
method Gradient flow dynamics of a wide two-layer neural network in high-dimension, with data following a single-index model.
result The learning dynamics exhibit separation of timescales and intermittency.
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.
Batch normalization biases linear models towards uniform margins, improving performance in binary classification.
problem Understanding the implicit bias of batch normalization in linear models and neural networks.
method Analyzing gradient descent convergence on linear models and two-layer CNNs with batch normalization.
result Gradient descent with batch normalization in linear models converges to a uniform margin classifier with an exponential convergence rate.
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 …
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.
Two-layer CNNs can overfit without harm under certain conditions.
problem Understanding when and how overfitting occurs in neural networks.
method Theoretical analysis of a two-layer CNN trained by gradient descent.
result A sharp phase transition between benign and harmful overfitting based on signal-to-noise ratio.
The paper explores how linear neural networks can overfit without bias when data is well-behaved.
problem Understanding why linear neural networks can generalize well despite fitting noisy data.
method Analyzing two-layer linear neural networks trained with gradient flow, deriving bounds on excess risk.
result The excess risk depends on initialization quality and data covariance matrix properties.
Two-layer neural networks can overfit without increasing risk when data is noisy.
problem Understanding why neural networks can overfit without increasing risk in noisy data.
method Combining bias and variance analysis in a high-dimensional setting.
result The excess learning risk of the interpolator decays under mild conditions.
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.
Two-layer model studies reinsurance contracts and competition between insurer and reinsurers.
problem Modeling and analyzing reinsurance contracts and competition between insurer and reinsurers.
method Two-layer stochastic game model with insurer negotiating with reinsurers, and reinsurers competing for business.
result Existence and uniqueness of equilibrium strategies for the insurer and reinsurers, characterized in semiclosed form.
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.
New bounds for neural networks without loss boundedness assumption.
problem Generalization error bounds for two-layer neural networks.
method Wasserstein distance estimates and moment bounds for stochastic gradient method.
result Dimension-free rate of order O(n−1/2) for independent test data. 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.
Spectral analysis shows neural networks separate from linear methods in approximating functions.
problem Separating two-layer neural networks from linear methods in function approximation.
method Spectral-based approach using Kolmogorov width and kernel spectrum.
result Upper and lower bounds on separation, explicit hard functions identified.
Study on SignGD optimization of two-layer transformer on noisy data.
problem Understanding how SignGD optimizes transformers and its generalization.
method Analysis of a two-layer transformer with SignGD on a linearly separable noisy dataset.
result SignGD converges fast but has poor generalization on noisy data.
Study examines how two-layer networks learn features after one gradient step.
problem Understanding feature learning in neural networks after a single gradient descent step.
method Modeling the trained network as a spiked Random Features (sRF) model and leveraging Gaussian universality.
result Exact asymptotic description of the generalization error of the sRF in high-dimensional limit.
Study of two-layer ReLU neural network phase diagram at infinite-width limit.
problem Characterize the dynamical regimes of two-layer ReLU neural networks.
method Combining experimental and theoretical approaches, including phase diagram analogy.
result Identification of three regimes: linear, critical, and condensed.
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.
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.
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…
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.
Two-layer ReLU networks outperform kernel methods in teacher-student settings.
problem Understanding the excess risk of two-layer ReLU neural networks in teacher-student models.
method Investigated a two-phase training process for a student network, comparing it to kernel methods.
result The student network reaches near-global optimality and outperforms kernel methods in minimax optimal rate.
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 establish connections between the problem of learning a two-layer neural network and tensor decomposition. We consider a model with feature vectors x∈Rd, r hidden units with weights {wi}1≤i≤r and output y∈R, i.e., $y=\sum_{i=1}^r σ( \boldsymbol w_i…
Auto-regressive models learn latent states from partially observed linear dynamical systems.
problem Understanding how auto-regressive models learn latent representations from partially observed linear dynamical systems.
method Empirical risk minimization on partially observed linear dynamical systems.
result Two-layer linear auto-regressive models learn to approximate Kalman filtering, coinciding with optimal state estimates.