Develops a new theory for neural systems stability and width effects.
problem Stability and finite-width effects in deep neural systems.
method Gauge-covariant stochastic effective field theory using classical commuting fields.
result Predicts the edge of chaos and low-frequency spectral deformation.
Study on fluctuations in neural network kernels and predictions, focusing on finite width effects.
problem Characterizing fluctuations in finite width neural networks.
method Dynamical mean field theory analysis of wide but finite feature learning neural networks.
result Fluctuations in kernels and predictions are dynamically coupled, leading to reduced variance in feature learning regimes.
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.
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.
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.
Paper characterizes gradient descent dynamics for neural networks with finite width.
problem Characterize gradient descent dynamics for multi-layer neural networks.
method Non-asymptotic state evolution theory for finite-width networks.
result Gradient descent dynamics provide precise distributional characterization.
New approach predicts generalization of deep neural networks in proportional-width regime.
problem Predicting generalization of deep neural networks in proportional-width regime.
method Equivalent Wishart Ansatz for hierarchical empirical kernels, renormalized NNGP kernel.
result Renormalized NNGP kernel captures dominant stochastic fluctuations in deep neural networks.
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.
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 proves deep narrow RNNs can approximate any function, with minimum width independent of data length.
problem Proving universality of deep narrow RNNs with bounded widths.
method Analyzing RNNs as dynamical systems, proving universality for deep narrow structures with specific widths.
result Minimum width for universality of deep narrow RNNs is independent of data length.
Finite-width neural networks use non-Gaussian priors, extending Gaussian process theory.
problem Understanding the behavior of neural networks with finite width.
method Perturbative extension of Gaussian process theory to finite-width neural networks, tracking preactivation distributions.
result Non-Gaussian processes as priors in finite-width neural networks.
Novel algorithms improve efficiency of finite width NTK computation.
problem Efficiency of computing finite width Neural Tangent Kernel (NTK).
method Leveraging neural network structure, propose two novel algorithms.
result Significantly improved efficiency in compute and memory requirements.
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.
The paper proves finite Urysohn 1-width for 4 and 5 manifolds with positive biRicci curvature.
problem Proving finite Urysohn 1-width for 4 and 5 manifolds with positive biRicci curvature.
method Using positive biRicci curvature and uniform scalar curvature bounds, the paper shows that the Urysohn 1-width is finite and depends only on the curvature bounds.
result Closed 4 and 5 manifolds with positive biRicci curvature have finite Urysohn 1-width, which depends only on the curvature bounds.
Wider neural networks perform better than deeper ones with the same number of parameters.
problem Understanding the role of network width versus the number of parameters in neural network performance.
method Comparing models with different ways of increasing width while keeping the number of parameters constant, analyzing their performance and using Gaussian Process kernels for analysis.
result Network width is the determining factor for good performance, while the number of weights is secondary as long as trainability is ensured.
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.
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.
Deep and wide ReLU networks learn data-dependent features even in the lazy training regime.
problem Understanding the behavior of neural networks with finite depth and width.
method Analyzing the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network.
result The NTK has a non-trivial evolution during training, with the mean of its first SGD update being exponential in the ratio of depth to width.
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.
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.
ResNets and DenseNets converge to NTK with depth and width, offering advantages for kernel regression.
problem Understanding convergence of ResNets and DenseNets to Neural Tangent Kernel (NTK).
method Analysis of finite width and depth corrections for NTK of ResNets and DenseNets.
result ResNets and DenseNets can converge to NTK with depth and width, unlike vanilla networks.
Study Gaussian-process limits of neural networks using tensor programs.
problem Understanding the behavior of neural networks as they approach infinite width.
method Quantitative analysis through tensor programs and Wasserstein distance.
result Explicit finite-width error bounds, showing convergence to Gaussian-process limits.
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.
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.
Empirical study shows standard CNNs deviate from NTK predictions.
problem Understanding how standard finite-width CNNs behave compared to their infinite-width NTK counterparts.
method Empirical analysis of AlexNet and LeNet architectures.
result Standard CNNs deviate significantly from their NTK counterparts, but deviation decreases with wider networks.
There are many "minimax" complexity functions in mathematics: width of a tree or a link, Heegaard genus of a 3-manifold, the Cheeger constant of a Riemannian manifold. We define such a function w, "width", on countable (or finite) groups and show w(Z^k) = k-1.
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.
Study shows neural networks trained with GD converge to Gaussian processes with polynomial decay.
problem Understanding convergence of neural networks to Gaussian processes during training.
method Explicit upper bounds on quadratic Wasserstein distance between trained networks and Gaussian approximations.
result Polynomial decay of approximation error with network width and training time.
Derives a family of hyperparameter scaling strategies for neural networks.
problem Optimizing hyperparameters for wide and deep neural networks.
method Introduces a one-parameter family of hyperparameter scaling strategies.
result Reveals proper scaling of depth with width for large-scale models.
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.
Quantitative CLTs show neural network distributions converge to Gaussian as width increases.
problem Understanding the distribution of fully connected neural networks with random weights and biases.
method Analyzing the distribution of a fully connected neural network with random Gaussian weights and biases, proving quantitative bounds on normal approximations.
result The distance between a random fully connected network and the corresponding infinite width Gaussian process scales like n−γ for γ>0. We show that finite-width deep ReLU neural networks yield rate-distortion optimal approximation (Bölcskei et al., 2018) of polynomials, windowed sinusoidal functions, one-dimensional oscillatory textures, and the Weierstrass function, a fractal function which is continuous but nowhere differentiable. Together with thei…
Simplified neural network EFTs reveal a single critical condition.
problem Understanding neuron statistics in neural networks at initialization.
method Diagrammatic approach to effective field theories (EFTs).
result A single condition governs criticality of all neuron preactivations.
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…
Analyzes DNNs trained with noisy gradients, finding FWCs negligible for large n.
problem Analyzing DNNs trained with noisy gradients.
method Introduced analytical framework to analyze non-Gaussian stochastic process.
result FWCs negligible for large n, improving CNN performance.
Taylorized training improves neural network training at finite width.
problem Understanding and improving neural network training at finite width.
method Training the k-th order Taylor expansion of the neural network at initialization.
result Taylorized training agrees with full neural network training better as k increases and can significantly close the performance gap.
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.
Theory captures feature learning effects in finite CNNs.
problem Feature learning in finite deep neural networks.
method Derive a self-consistent Gaussian Process theory.
result Good agreement with experiments and sharp transition between regimes.
Sharp feature transitions revealed in extensive-width networks.
problem Learning hierarchical features from noisy queries in large networks.
method Information-theoretic analysis and heuristic decoupling argument.
result Sequential phase transitions in feature learnability and effective width.
We study how finite Bayesian neural networks adapt their hidden representations.
problem Understanding how finite Bayesian neural networks differ from infinite ones.
method We analyze the asymptotics of learned feature kernels for various network architectures.
result The leading finite-width corrections to feature kernels have a universal form.
Wide neural networks can degrade performance, contrary to conventional wisdom.
problem Understanding the limitations of increasing network width in neural networks.
method Using Deep Gaussian Processes to decouple capacity and width, analyzing their effects on representational power and non-Gaussianity.
result Wide neural networks can become less adaptable and more Gaussian, leading to performance degradation.
To each knot K⊂S3 one can associated its knot Floer homology HFK^(K), a finitely generated bigraded abelian group. In general, the nonzero ranks of these homology groups lie on a finite number of slope one lines with respect to the bigrading. The width of the homology is, in essence, the largest horizo…
Paper studies ResNet dynamics using NTH, reducing width requirement.
problem Understanding ResNet dynamics and improving training efficiency.
method Uses Neural Tangent Hierarchy (NTH) to analyze ResNet dynamics.
result Reduces width requirement from quartic to cubic for ResNet.
Researchers derive exact priors for finite Bayesian neural networks.
problem Understanding non-Gaussian priors in finite Bayesian neural networks.
method Analytical derivation of function space priors for finite fully-connected feedforward networks.
result Exact solutions for priors of finite networks, including Meijer G-function for linear networks and mixtures for ReLU networks.
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.
Study Gaussian approximation for deep neural networks with random weights.
problem Understanding the distribution of deep neural networks with random weights.
method Established Gaussian approximation bounds in Wasserstein-1 norm.
result Convergence rates of order n−(1/6)L−1+ε for deep networks with proportional layer widths. 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…
Study on hidden units in finite Bayesian neural networks and their tail properties.
problem Understanding the behavior of hidden units in finite Bayesian neural networks.
method Introduced a generalized Weibull-tail property to describe hidden units tails.
result Unit priors become heavier-tailed going deeper, providing insights into finite Bayesian neural networks.