Study on infinite neural networks for generalization.
problem Understanding generalization in infinitely-wide neural networks.
method Analytical and empirical investigations of infinite ensembles.
result Signals correlating with generalization identified.
Neural Tangents simplifies infinite-width neural networks for research.
problem Training and studying infinite-width neural networks.
method High-level API for specifying complex architectures, analytical or gradient-based training, and automatic distribution.
result Analytical training of infinite-width networks and automatic parallelization.
VC dimensions of group CNNs are infinite for certain kernels and groups.
problem Estimating the generalization capacity of group convolutional neural networks.
method Identifying precise VC dimension estimates for simple sets of group CNNs.
result Two-parameter families of convolutional neural networks have an infinite VC dimension for infinite groups and certain kernels.
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.
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.
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 infinite-depth limits of neural networks with fixed width.
problem Understanding the behavior of neural networks as depth increases with fixed width.
method Analyzing finite-width residual networks with random Gaussian weights, focusing on the infinite-depth limit.
result The pre-activations converge to a zero-drift diffusion process, differing from the infinite-width limit.
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.
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.
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.
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.
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.
The study of universal approximation of arbitrary functions f:X→Y by neural networks has a rich and thorough history dating back to Kolmogorov (1957). In the case of learning finite dimensional maps, many authors have shown various forms of the universality of both fixed depth and fixed width…
Stochastic neural networks with infinite width become deterministic, reducing training variance.
problem Understanding how stochasticity in neural networks affects learning and regularization.
method Theoretical analysis of stochastic neural networks with infinite width.
result As the width of an optimized stochastic neural network increases, its predictive variance on the training set decreases to zero.
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.
Recent work has argued that neural networks can be understood theoretically by taking the number of channels to infinity, at which point the outputs become Gaussian process (GP) distributed. However, we note that infinite Bayesian neural networks lack a key facet of the behaviour of real neural networks: the fixed kern…
Study on neural network initialization with shaped infinite depth-and-width networks.
problem Understanding the distribution of random covariance matrices in shaped infinite-depth-and-width networks.
method Introduced the Neural Covariance SDE to model the distribution of the random covariance matrix.
result Identified the precise scaling of the activation function necessary for a non-trivial limit.
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.
This paper shows how infinitely wide Tensor Networks converge to Gaussian Processes.
problem Understanding the relationship between Tensor Networks and Gaussian Processes.
method Analyzing the infinite-width limit of Tensor Networks and comparing them to Gaussian Processes.
result Infinitely wide Tensor Networks converge to Gaussian Processes, proving their equivalence.
Empirical study compares wide neural networks to kernel methods, resolving open questions.
problem Understanding the relationship between wide neural networks and kernel methods.
method Large-scale empirical study using various neural network architectures and kernel methods.
result Wide neural networks outperform fully-connected finite-width networks in some cases, but underperform convolutional finite-width networks.
Bayesian neural networks approximate Student-t processes in the infinite-width limit.
problem Modeling uncertainty in neural networks with greater flexibility.
method Extending asymptotic properties of Gaussian processes to Student-t processes in the infinite-width limit of BNNs.
result Posterior BNNs converge to Student-t processes in the infinite-width limit.
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.
Improved bounds for neural network approximations of functions.
problem Bounding the width of neural networks for function approximation.
method Extending Radon-based norms to bounded open sets and deriving new approximation bounds.
result Improved sparse approximation bounds for neural networks.
New framework for understanding infinite-width neural networks.
problem Understanding the infinite-width limit behavior of neural networks.
method General framework to study limit behavior of neural models based on hyperparameter scaling.
result Derives scaling for existing mean-field and neural tangent kernel limits and introduces new dynamically stable limits.
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.
Wide CNNs outperform infinite width networks, revealing scaling laws.
problem Understanding the performance difference between finite and infinite width convolutional networks.
method Diagrammatic approach to derive asymptotic width dependence for various quantities.
result The difference in performance between finite and infinite width models vanishes at a definite rate with respect to model width.
Adaptive kernels from neural networks improve model performance.
problem Improving neural network performance through adaptive kernels.
method Deriving adaptive kernels from infinite-width neural networks using feature learning and gradient flow training.
result Adaptive kernels achieve lower test loss compared to traditional kernels.
Improved standard parameterization yields well-defined neural tangent kernel.
problem Extrapolation of standard parameterization to infinite width is problematic.
method Proposed an improved extrapolation of the standard parameterization.
result Improved standard parameterization yields similar accuracy to NTK parameterization but with better correspondence to finite width networks.
This study uses neural networks to solve interpolation problems with sparse, infinitely wide layers.
problem Exact data interpolation using sparse, infinitely wide neural networks.
method Atomic norm framework to derive convex hulls and equivalent convex formulations.
result Simple characterizations of convex hulls for different constraints on network weights and biases.
New analysis shows a gap between Gaussian RKHS and neural networks on unbounded domains.
problem Understanding the function space bias of neural networks compared to Gaussian RKHS.
method Infinite-center asymptotic analysis of neural network Banach space and Gaussian RKHS on unbounded domains.
result Certain functions in Gaussian RKHS have infinite norm in neural network Banach space on unbounded domains.
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.
Multivariate splines linked to infinitely-wide neural networks with improved numerical performance.
problem Understanding the relationship between multivariate splines and neural networks.
method Showed multivariate splines can be represented as random features in infinitely-wide neural networks with a homogeneous activation function.
result The function space of multivariate splines is a Sobolev space on a Euclidean ball with explicit norm bounds on derivatives.
New findings on depth vs. width in neural networks, showing depth can improve learnability.
problem Understanding the role of depth in neural networks, especially when width is unbounded.
method Analyzing sample complexity for learnability in norm-controlled depth-2 and depth-3 ReLU networks.
result Depth can improve learnability of functions that are otherwise unlearnable with depth-2 networks.
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.
New method reduces over-parametrization in neural networks, ensuring sparsity and finite network size.
problem Over-parametrization leads to too many active neurons in neural networks, especially with large data.
method Investigates a nonconvex regularization method for shallow ReLU networks.
result Locally optimal networks are finite even with infinite data, maintaining approximation guarantees and network size bounds.
Empirical study compares finite- and infinite-width BNNs, revealing performance differences under model mismatch.
problem Comparing BNNs with different widths due to conflicting model properties and inference intractability.
method Empirical comparison of finite- and infinite-width BNNs, analyzing performance under model mismatch.
result Increasing width can hurt BNN performance when the model is mis-specified, and finite-width BNNs generalize better under model mismatch.
Generalizes neural networks for infinite-dimensional mappings, including PDE solutions.
problem Learning mappings between infinite-dimensional spaces and finite-dimensional approximations.
method Graph kernel network architecture with message passing for kernel integration.
result Competitive performance compared to state-of-the-art solvers for PDEs.
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.
Bayesian neural networks learn efficiently at infinite width, matching polynomial-width performance.
problem Understanding the inductive bias of infinite-width neural networks.
method Analyzing the reduced entropy and using subsampling techniques.
result The Bayesian mean-field learner generalizes exactly on polynomially-bounded targets.
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…
The study reveals a transition in neural network performance from infinite-width to variance-limited behavior as dataset size increases.
problem Understanding the transition from infinite-width to variance-limited behavior in neural networks.
method Empirical study of the transition from infinite-width to variance-limited behavior as a function of sample size and network width.
result The critical sample size \( P^* \) is approximately \( \sqrt{N} \) for polynomial regression with ReLU networks.
New reservoir computing approach handles infinite-dimensional systems.
problem Approximating and generalizing complex input/output systems.
method Randomly generated echo state networks with neural networks.
result Proves universal approximation properties for new class of systems.
Bayesian neural networks ignore data in infinite units limit.
problem Pathological behavior of posterior in over-parameterized networks.
method Mean-field variational inference in infinite hidden units limit.
result Posterior mean converges to zero, ignoring data.
This paper investigates the approximation power of three types of random neural networks: (a) infinite width networks, with weights following an arbitrary distribution; (b) finite width networks obtained by subsampling the preceding infinite width networks; (c) finite width networks obtained by starting with standard G…
Neural networks can find financial arbitrage opportunities without needing market models.
problem Finding arbitrage opportunities in financial markets without using market models.
method Used neural networks to solve convex semi-infinite programs and detect arbitrage opportunities.
result Neural networks can detect model-free static arbitrage strategies in financial markets.
Meta-learning neural networks for better clustering representations.
problem Improving clustering performance with appropriate representations.
method Meta-learning method that trains neural networks for representations using VB inference with an infinite Gaussian mixture model.
result The method achieves higher clustering performance than existing methods.
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.