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.
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.
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.
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 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.
Paper analyzes infinite-width attention layers using Tensor Programs.
problem Capturing the infinite-width limit of attention layers.
method Tensor Programs framework to rigorously identify the limit distribution.
result Derives exact form of infinite-width limit distribution without Gaussian approximations.
We prove that the palindromic width of HNN extension of a group by proper associated subgroups is infinite. We also prove that the palindromic width of the amalgamated free product of two groups via a proper subgroup is infinite (except when the amalgamated subgroup has index two in each of the factors). Combining thes…
ResNets approximate log-Gaussian at initialization, improving network performance.
problem Understanding the initialization behavior of deep neural networks like ResNets.
method Analyzing ReLU ResNets in the infinite-depth-and-width limit, showing log-Gaussian behavior.
result ResNets at initialization exhibit hypoactivation and interlayer correlations, which are not captured by Gaussian limits.
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.
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.
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.
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.
We show that the verbal width is infinite for acylindrically hyperbolic groups, which include hyperbolic groups, mapping class groups and Out(Fn).
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 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…
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.
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.
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.
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.
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.
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.
There are currently two parameterizations used to derive fixed kernels corresponding to infinite width neural networks, the NTK (Neural Tangent Kernel) parameterization and the naive standard parameterization. However, the extrapolation of both of these parameterizations to infinite width is problematic. The standard p…
Neural Tangents is a library designed to enable research into infinite-width neural networks. It provides a high-level API for specifying complex and hierarchical neural network architectures. These networks can then be trained and evaluated either at finite-width as usual or in their infinite-width limit. Infinite-wid…
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.
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.
This work establishes the equivalence between neural networks and support vector machines.
problem Establishing the equivalence between neural networks and support vector machines.
method Proposed a method to establish the equivalence between infinitely wide neural networks trained by soft margin loss and standard soft margin SVMs with NTK trained by subgradient descent.
result The equivalence between NN and SVM is established, enabling practical applications such as non-vacuous generalization bounds and robustness certificates.
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.
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.
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.
New framework connects two neural network theories, improving finite-width approximations.
problem Theoretical guarantees for neural network training in general cases.
method Developed a general framework linking mean-field and constant kernel theories.
result Discrete-time MF limit provides better approximation for finite-width nets.
This paper studies large-width asymptotics for ReLU neural networks with α-Stable initializations.
problem Characterizing the large-width behavior of ReLU neural networks with α-Stable initializations.
method Analysis of the large-width distributions and training dynamics of ReLU neural networks initialized with α-Stable distributions.
result For ReLU neural networks with α-Stable initializations, the large-width training dynamics achieve zero training error at a linear rate, characterized by a random kernel.
Large learning rates work surprisingly well in standard parameterization, contrary to theory.
problem Theoretical limits of large learning rates do not match practical network behavior.
method Fine-grained analysis of learning rates and network behavior under cross-entropy loss.
result There are two distinct sub-regimes of unstable learning rates, with a controlled divergence regime where features continue to evolve.
Study shows polynomial-width neural networks can closely approximate infinite-width networks in polynomial time.
problem Approximating dynamics of polynomial-width neural networks with infinite-width networks.
method Bounding approximation gap through a differential equation governed by mean-field dynamics, considering local Hessian.
result Polynomially many neurons are sufficient to closely approximate mean-field dynamics.
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.
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.
Proof of learning rate transfer in MLPs with μP parameterization.
problem Understanding and optimizing learning rates in neural networks with different parameterizations.
method Theoretical analysis and empirical validation of learning rate transfer in MLPs with μP, SP, and NTP parameterizations. result The optimal learning rate converges to a non-zero constant as width goes to infinity under μP, explaining learning rate transfer. Bayesian neural networks fail at OOD detection, revealing fundamental issues.
problem The ability of Bayesian neural networks to detect out-of-distribution data.
method Empirical study of Bayesian inference in neural networks, including infinite-width and finite-width cases using Hamiltonian Monte Carlo.
result Bayesian inference in common neural network architectures does not lead to good OOD detection.
New Transformer architecture prevents rank degeneracy in deep attention models.
problem Rank degeneracy in deep attention models.
method Modified Softmax-based attention model with skip connections, centered at identity, and scaled logits.
result Existence of a stable SDE implies well-behaved covariance structure, preventing rank degeneracy.
The Rosenberg index vanishes if a manifold admits a wide Riemannian band or cube-like domain.
problem Proving the Rosenberg index does not vanish for certain manifolds.
method Analyzing isometric immersions of wide Riemannian bands and cube-like domains on spin manifolds.
result Closed spin manifolds with infinite KO-width have non-vanishing Rosenberg index. 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.
In "Width complexes for knots and 3-manifolds," Jennifer Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in the 3-sphere and the isotopies between these positions. She poses several questions about these width complexes; in particular, she asks whether the…
Khovanov homology is a bigraded Z-module that categorifies the Jones polynomial. The support of Khovanov homology lies on a finite number of slope two lines with respect to the bigrading. The Khovanov width is essentially the largest horizontal distance between two such lines. We show that it is possible to generate in…
While studying the existence of closed geodesics and minimal hypersurfaces in compact manifolds, the concept of width was introduced in different contexts. Generally, the width is realized by the energy of the closed geodesics or the volume of minimal hypersurfaces, which are found by the Minimax argument. Recently, Ma…
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.
Study examines dependence properties of Bayesian neural network units in finite-width networks.
problem Understanding dependence properties of hidden units in practical finite-width Bayesian neural networks.
method Theoretical analysis and empirical evaluation of depth and width impacts.
result Hidden units in finite-width Bayesian neural networks are dependent, contrary to the infinite-width limit assumption.
Deep networks can be biased to learn top eigenfunctions of the kernel outside the training set.
problem Spectral bias of deep networks in the kernel regime.
method Quantitative bounds on L2 difference between finite-width and infinite-width network trajectories. result Deep networks learn top eigenfunctions of the Neural Tangent Kernel over the entire input space, not just the training set.
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.
Extends width estimates to family case using index theory.
problem Sharp width estimates for Riemannian bands with positive scalar curvature.
method Dirac operators and family index theory.
result Proves width estimate for fiber bundles with infinite A-hat area.