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.
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.
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.
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.
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.
We analyze double descent in finite-width neural networks using influence functions.
problem Understanding double descent in finite-width neural networks.
method Using influence functions to derive population loss bounds and investigate loss function effects.
result Derived bounds exhibit double descent behavior at the interpolation threshold.
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.
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.
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…
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.
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.
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.
Gaussian processes are ubiquitous in nature and engineering. A case in point is a class of neural networks in the infinite-width limit, whose priors correspond to Gaussian processes. Here we perturbatively extend this correspondence to finite-width neural networks, yielding non-Gaussian processes as priors. The methodo…
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.
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.
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 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.
Wide networks with polynomial activations have proven asymptotic behavior.
problem Understanding the behavior of neural networks in the large width limit.
method Proving a conjecture for deep networks with polynomial activation functions.
result Tight bounds on the behavior of wide networks during stochastic gradient descent and derivation of their finite-width dynamics.
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.
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.
The paper provides non-asymptotic Edgeworth expansions for neural network outputs.
problem Approximating deviations of finite-width neural networks from their Gaussian limit.
method Multidimensional Edgeworth expansions of arbitrary order for neural network outputs.
result Established a bound on the total variation distance between neural network output and its Edgeworth approximation.
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 …
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.
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.
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…
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.
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.
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.
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…
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.
The evolution of a deep neural network trained by the gradient descent can be described by its neural tangent kernel (NTK) as introduced in [20], where it was proven that in the infinite width limit the NTK converges to an explicit limiting kernel and it stays constant during training. The NTK was also implicit in some…
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.
Ensembles of neural networks improve training dynamics and performance.
problem Improving neural network performance through model size increase.
method Defining collegial ensembles (CE) as multiple independent models trained as a single model, and using theoretical results on NTK to optimize architecture search.
result CE dynamics simplify and scale favorably, resembling wide models, and can be efficiently implemented using group convolutions and block diagonal layers.
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.
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.
Convex geometry explains optimal neural network parameters.
problem Understanding optimal parameters in over-parameterized neural networks.
method Convex geometry, extreme points, linear spline interpolation, kernel matrix, cutting-plane algorithm.
result Optimal network parameters can be characterized as interpretable closed-form formulas.
Renormalization in neural networks linked to quantum field theory.
problem Implementing renormalization in neural networks.
method Mapping neural networks to quantum field theory, applying renormalization techniques.
result Changing weight standard deviation corresponds to a renormalization flow.
Gradient descent converges linearly in finite-width networks with positive NTK and compatible conditions.
problem Local convergence of gradient descent in finite-width networks.
method Positive Neural Tangent Kernel (NTK), local Polyak-Łojasiewicz inequality, fixed-step containment in Locally Quasi-Convex Region (LQCR).
result Linear convergence achieved under specific conditions.
This paper tightens bounds on the smallest eigenvalue of NTK for deep ReLU networks.
problem Analyzing the smallest eigenvalue of Neural Tangent Kernel for deep ReLU networks.
method Analyzing various quantities of independent interest, including lower bounds on the smallest singular value of hidden feature matrices and upper bounds on the Lipschitz constant of input-output feature maps.
result Tight bounds on the smallest eigenvalue of NTK matrices for deep ReLU nets, both in the limiting case of infinite widths and for finite widths.
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.
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.
The paper establishes principles for initializing and designing GNNs with ReLU activations to avoid oversmoothing and correlation collapse.
problem Oversmoothing and correlation collapse in deep ReLU GNNs.
method The paper derives and validates three principles for initialization and architecture selection in finite width graph neural networks with ReLU activations.
result Correct initialization, residual aggregation operators, and residual connections significantly improve early training dynamics in deep ReLU GNNs.
This work uses sampling theory to analyze smoothness and error bounds of finite neural networks.
problem Analyzing the function space of finite neural networks and providing error bounds.
method Applying sampling theory to finite neural networks with non-expansive activation functions, considering both deterministic and random sampling.
result Novel error bounds for univariate neural networks under band-limited input assumption, highlighting the advantage of deterministic uniform sampling.
SGD converges with positive probability for non-convex deep neural networks under specific conditions.
problem Convergence of SGD for non-convex deep neural networks.
method Established local convergence with positive probability under local Łojasiewicz condition and additional structural assumption.
result SGD converges with positive probability for non-convex deep neural networks under specific conditions.
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…
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.
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.