Study of infinitely deep but narrow neural networks using NTK theory.
problem Analyzing the role of depth in deep learning with overparameterized networks.
method Infinite-depth limit analysis of MLP and CNN using Neural Tangent Kernel (NTK) theory.
result Established trainability guarantee for infinitely deep but narrow neural networks.
Deep neural networks' infinite-width behavior approximated by Gaussian models.
problem Understanding the behavior of deep neural networks in the limit of infinite width.
method Using the Lindeberg exchange principle to approximate weights by Gaussian random variables.
result Quantitative bounds on the 2-Wasserstein distance between deep neural networks and Gaussian limits.
Lectures on deep learning properties in infinite and large-width networks.
problem Understanding deep neural networks in extreme width conditions.
method Analysis of random deep neural networks, connections to linear models, kernels, and Gaussian processes, perturbative and non-perturbative treatments.
result Properties and behaviors of deep neural networks in the infinite-width limit and large-width regime.
Study deep maxout networks and their equivalence to Gaussian processes.
problem Understanding neural networks with infinite width.
method Derive equivalence between deep maxout networks and Gaussian processes, characterize maxout kernel, and provide efficient numerical implementation.
result Bayesian inference based on deep maxout network kernel leads to competitive results compared to finite-width counterparts and deep neural network kernels.
The paper examines how deep linear neural networks behave as they become infinitely wide.
problem Understanding the behavior of deep linear neural networks as they approach infinite width.
method Analyzes the infinite-width limit of deep linear neural networks, proving convergence to deterministic models and providing precise laws for random weights.
result The training dynamics of deep linear neural networks converge to those of a deterministic model, and the weights' behavior is precisely described.
Infinitely deep neural networks can be modeled as diffusion processes to avoid undesirable properties.
problem Desirable properties are lost as neural networks increase in depth.
method Parameter distributions shrink as depth increases, leading to well-behaved stochastic processes.
result Limiting processes do not suffer from vanishing dependency and restrictive function families issues.
Infinite CNNs lose spatial correlations, but can be restored by correlated weights.
problem Infinite CNNs lose spatial correlations, which are crucial for their performance.
method Introduced correlated weights to restore spatial correlations in infinite CNNs.
result Optimal performance is achieved with a moderate level of weight correlation.
Stable processes emerge as limits of deep neural networks with symmetric stable distributions.
problem Understanding the behavior of deep neural networks as they become infinitely wide.
method Analyzing fully connected feed-forward deep neural networks with symmetric stable distributions and showing the limit as a stable process.
result The infinite wide limit of the network is a stable process with multivariate stable distributions.
Infinite neural networks lack key flexibility, finite ones learn better.
problem Theoretical limitations of infinite neural networks and their inferior performance.
method Analytic results and empirical evidence on finite deep linear networks and SOTA architectures.
result Finite deep linear networks perform better and learn representations, unlike infinite networks.
Study of deep linear neural networks with proportional width and depth.
problem Lack of descriptive power in Gaussian limit of deep linear neural networks.
method Proportional infinite-width infinite-depth limit for deep linear neural networks.
result Characterization of limiting distribution as a nontrivial mixture of Gaussians.
Wide neural networks converge to Gaussian processes, improving generalization.
problem Understanding the generalization of wide neural networks, especially deep equilibrium models.
method Investigation of deep equilibrium models (DEQs) with infinite-depth layers, focusing on their convergence to Gaussian processes as width and depth approach infinity.
result Wide DEQs converge to Gaussian processes, maintaining generalization performance.
Deep kernel processes unify various models using Gram matrices and kernel functions.
problem Unified representation of various deep learning models.
method Defining deep kernel processes with progressively transformed Gram matrices and sampling from inverse Wishart distributions.
result Deep Gaussian processes, BNNs, infinite BNNs, and infinite BNNs with bottlenecks can all be written as deep kernel processes.
UDN adapts depth to data complexity, outperforming standard neural networks.
problem Adapting neural network depth to data complexity.
method Variational inference for infinitely deep neural networks with a novel algorithm.
result UDN outperforms standard neural networks and other infinite-depth approaches.
Deep neural nets approximate random dynamical system trajectories uniformly in time.
problem Approximating trajectories of random dynamical systems over infinite time horizons.
method Recurrent neural networks with simple feedback structures.
result Certain random trajectories can be approximated uniformly in time to any desired accuracy.
This paper removes the finite variance assumption for deep convolutional neural networks.
problem Removing the finite variance assumption for deep convolutional neural networks.
method Assuming iid parameters distributed according to a stable distribution, the paper shows that the infinite-channel limit of a deep feed-forward convolutional neural network is a multivariate stable stochastic process.
result The infinite-channel limit of a deep feed-forward convolutional neural network, under suitable scaling, is a multivariate stable stochastic process.
The paper extends infinite-width analysis to neural network Jacobians, revealing convergence to Gaussian processes and linear ODEs.
problem Understanding the training dynamics of neural networks in the infinite-width limit.
method Extending infinite-width analysis to Jacobians, characterizing convergence to Gaussian processes and linear ODEs.
result The evolution of MLPs under robust training in the infinite-width limit is described by a linear ODE.
Deep neural nets estimate operators between infinite-dimensional spaces with fast rates.
problem Estimating operators between infinite-dimensional spaces.
method Deep neural networks for nonparametric estimation of Lipschitz operators.
result Error bounds decay with fast rates depending on intrinsic dimension.
It has long been known that a single-layer fully-connected neural network with an i.i.d. prior over its parameters is equivalent to a Gaussian process (GP), in the limit of infinite network width. This correspondence enables exact Bayesian inference for infinite width neural networks on regression tasks by means of eva…
Study of deep Stable neural networks with various activation functions.
problem Characterizing the infinitely wide limits of deep Stable neural networks.
method Investigation of large-width properties of deep Stable NNs with a generalized central limit theorem for heavy tails.
result Extension of characterization to a broader class of activation functions, including sub-linear, asymptotically linear, and super-linear functions.
Study of deep neural networks' NTK evolution during training.
problem Understanding the performance gap between deep neural networks and kernel regression.
method Derive an infinite hierarchy of ordinary differential equations (NTH) to capture gradient descent dynamics of deep neural networks.
result Truncated NTH approximates the dynamic of the NTK up to arbitrary precision under certain conditions.
The paper studies deep neural networks with Gaussian weights and finds their asymptotic behavior.
problem Understanding the behavior of deep neural networks with large width.
method Function-space perspective, Gaussian process analysis, weak convergence in large-width limit.
result Deep neural networks with large width converge to a continuous Gaussian process.
This paper shows how deep neural networks can learn rich, independent features that significantly deviate from initialization.
problem Understanding how deep neural networks achieve meaningful feature learning and global convergence.
method Investigation of infinitely wide, L L L -layer neural networks using the tensor program framework under Maximal Update parametrization. result SGD enables these networks to learn linearly independent features that substantially deviate from their initial values, capturing relevant data information.
Bayesian deep neural networks converge to processes with α-stable marginals under infinite variance weights.
problem Representation learning in deep kernel processes is hindered by deterministic covariance kernels.
method Showed convergence to α-stable processes with conditionally Gaussian representations in infinite-width networks.
result Conditional random covariance kernels can be recursively linked, even if the process is α-stable.
Innovative PGMs match neural networks, revealing precise approximations during forward propagation.
problem Lack of precise semantics and probabilistic interpretation in neural networks.
method Constructing infinite tree-structured PGMs that correspond to neural networks.
result DNNs perform precise approximations of PGM inference during forward propagation.
Wide neural networks can benefit from multi-task learning in their infinite-width limit.
problem The generalization behavior of wide neural networks in multi-task learning settings.
method Optimizing wide ReLU neural networks with L2-regularization promotes multi-task learning in the infinite-width limit.
result An exact quantitative characterization of multi-task learning in the infinite-width limit of wide ReLU neural networks.
New principle reveals how neural networks learn complex interactions.
problem Understanding neural networks' success and complexity.
method Infinite-width networks, focusing on frequency and space.
result Fine-grained eigenstructure improves network learnability.
This paper extends ResNet theory to infinitely deep networks, linking them to diffusion processes.
problem Training infinitely deep ResNets with i.i.d. initializations leads to undesirable properties.
method Introduced doubly infinite ResNets with i.i.d. initializations, linking to diffusion processes.
result The dynamics of quantities of interest converge to deterministic limits in the limit of infinite depth.
New framework analyzes deep learning optimization with finite width networks, revealing generalization gaps and excess risks.
problem Analyzing generalization error of deep learning with finite width networks.
method Formulating neural network training as transportation map estimation and analyzing via infinite dimensional Langevin dynamics.
result Achieves fast learning rate and minimax optimal rates for classification and regression problems.
Study of deep neural networks with dependent weights leading to new model limits and properties.
problem Characterizing deep neural networks with dependent weights in the infinite-width limit.
method Modeling weights as a mixture of Gaussian distributions and analyzing the infinite-width limit.
result Characterization of neural network layers by scalar parameters and Lévy measures, leading to new model limits.
Deep neural networks with heavy-tailed weights converge to stable distributions.
problem Understanding the convergence of heavy-tailed weights in infinitely-wide neural networks.
method Analyzing infinitely-wide multi-layer perceptrons with i.i.d. symmetric α α α -stable weight distributions. result The vector of pre-activation values converges to i.i.d. symmetric α α α -stable distributions. 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.
Theoretical limits of deep residual networks show consistent covariance structures.
problem Understanding the limits of deep residual networks.
method Analyzing the behavior of deep residual networks with skip connections as width and depth approach infinity.
result Theoretical analysis confirms that the covariance structure remains consistent regardless of the order of width and depth.
A new method combines deep kernels with Gaussian processes to avoid overfitting.
problem Losing Bayesian benefits in deep kernel learning due to kernel optimization.
method Using Infinite-width neural networks and Neural Network Gaussian Process (NNGP) as a guide for DKL optimization.
result Robustness to overfitting and good predictive performance on various datasets.
Graph convolutional deep kernel machine learns representations for graph tasks.
problem Limited representation learning in infinite-width neural networks.
method Developed a graph convolutional deep kernel machine as an infinite-width limit.
result Representation learning improves performance for heterophilous node classification tasks.
Improves deep generative models to generate images of any size.
problem Fixed-sized output images from deep generative models.
method Integrates spatial noise vectors into fully convolutional neural networks.
result Theoretical interpretation of infinite spatial generation using spatial stochastic processes.
Network degeneracy affects training performance, especially in deep networks.
problem Degeneracy in deep neural networks leads to poor training performance.
method Predicted degeneracy level correlates with training dynamics using finite and infinite width networks.
result Degeneracy in neural networks correlates with training performance and can be predicted.
New kernels from ELU and GELU networks reveal non-trivial fixed points.
problem Understanding fixed-point dynamics in deep neural networks with ELU and GELU activations.
method Deriving covariance functions and analyzing fixed-point dynamics of ELU and GELU networks.
result ELU and GELU networks exhibit non-trivial fixed-point dynamics, explaining implicit regularization in overparameterized models.
The paper examines when NTK theory applies to real finite-width neural networks.
problem Understanding when NTK theory accurately predicts the behavior of finite-width neural networks.
method Empirical study of fully-connected ReLU and sigmoid DNNs with various hyperparameters and depths.
result NTK theory does not always apply to sufficiently deep networks with exploding gradients, and the kernel changes significantly during training.
Study how depth affects inference in deep Bayesian neural networks.
problem Understanding how depth impacts inference in overparameterized linear Bayesian neural networks.
method Interpreting finite deep linear Bayesian neural networks as scale mixtures of Gaussian process predictors.
result Advances analytical understanding of how depth affects inference in a simple class of Bayesian neural networks.
Choosing appropriate architectures and regularization strategies for deep networks is crucial to good predictive performance. To shed light on this problem, we analyze the analogous problem of constructing useful priors on compositions of functions. Specifically, we study the deep Gaussian process, a type of infinitely…
Bayesian deep ensembles improve prediction accuracy in various settings.
problem Improving prediction accuracy of deep ensembles in out-of-distribution settings.
method Introducing a randomised, untrainable function to each ensemble member, enabling a posterior predictive distribution interpretation.
result Bayesian deep ensembles make more conservative predictions and outperform standard ensembles in various tasks.
Deep neural nets solve complex insurance math equations.
problem Optimal control problems in insurance math.
method Deep neural network algorithm for elliptic PDEs.
result Solves high-dimensional semilinear elliptic PDEs.
Establishes connection between MTDNN and multitask GP, revealing weight correlation as key to task sharing.
problem Limited theoretical understanding of information sharing in MTDNN.
method Derives multitask GP kernels for MTDNN and MTBNN, showing shared hyper-parameters and last layer weights.
result Information sharing in MTDNN is due to weight correlation, not intermediate layer weights.
This paper analyzes convergence rates of neural networks in the deep learning regime.
problem Understanding convergence rates of neural networks in the deep learning regime.
method Analyzing the Neural Tangent Kernel (NTK) convergence rates in the large depth limit.
result Quantifies the impact of initialization and activation function on NTK convergence rates.
Study reveals hidden null components in overparametrized neural networks.
problem Hidden null components in overparametrized neural networks.
method Structure theorem of null space for neural networks using ridgelet transforms.
result Null components can be uniquely written as linear combinations of ridgelet transforms.
This work investigates training infinite mixtures with maximum likelihood for improved uncertainty quantification.
problem Improving uncertainty quantification in neural networks.
method Investigates training infinite mixtures with maximum likelihood instead of variational inference.
result The proposed method leads to stochastic networks with increased predictive variance, improved robustness, and higher entropy on out-of-distribution data.
Random deep neural networks are robust to adversarial examples, scaling with input size and dimension.
problem Adversarial examples challenge the reliability of deep learning algorithms.
method Analysis of random deep neural networks with Gaussian process equivalence and experiments on MNIST and CIFAR10.
result The ℓ p \ell^p ℓ p distance of adversarial examples scales as 1 / d i m e s ℓ p 1/\sqrt{d} imes \ell^p 1/ d im es ℓ p norm of the input. We develop a scalable method for Bayesian neural networks with stochastic differential equations.
problem Uncertainty quantification in deep neural networks.
method Gradient-based stochastic variational inference in continuous-depth Bayesian neural networks.
result Gradient estimator with zero variance as the approximation improves.