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.
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.
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.
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.
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 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.
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…
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.
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.
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.
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.
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…
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 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 …
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.
Study on Bayesian deep linear networks with multiple outputs and convolutional layers.
problem Characterize feature learning in finite-width Bayesian deep linear networks.
method Exact and analytical formulas for joint and posterior distributions, using large deviation theory.
result Quantitative description of feature learning in infinite-width regime.
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.
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.
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.
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.
This is an exposition of aspects of the result of Daskalopoulos and Sesum that any 2-dimensional complete noncompact ancient solution to Ricci flow with bounded positive scalar curvature and finite width must be the cigar soliton.
We propose \emph{Taylorized training} as an initiative towards better understanding neural network training at finite width. Taylorized training involves training the k-th order Taylor expansion of the neural network at initialization, and is a principled extension of linearized training---a recently proposed theory …
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.
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.
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.
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…
We show that a finite collection of stable subgroups of a finitely generated group has finite height, finite width and bounded packing. We then use knowledge about intersections of conjugates to characterize finite families of quasimorphisms on hyperbolically embedded subgroups that can be to simultaneously extended to…
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.
Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: "When is the Laplace-Beltrami operator Δ:Hk+1(M)∩H01(M)→Hk−1(M), k∈N0, invertible?" We consider also the case of mixed boundary conditions. The study of this main question lea…
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.
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…
Develops EFT for ResNets, revealing limitations of kernel-only approach.
problem Limitations of kernel-only approach in deep neural networks.
method Collective kernel EFT for pre-activation ResNets based on G-only closure hierarchy. result Numerical findings show V4 equation residual accumulates to an O(1) error. 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.
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.
The paper argues against the inefficiency of explaining deep learning phenomena.
problem Explaining deep learning phenomena often leads to ad hoc hypotheses.
method Analyzing recent literature to assess the relevance of these phenomena.
result Many phenomena do not appear in real-world applications and may be inefficient.
We develop a convex analytic approach to analyze finite width two-layer ReLU networks. We first prove that an optimal solution to the regularized training problem can be characterized as extreme points of a convex set, where simple solutions are encouraged via its convex geometrical properties. We then leverage this ch…
In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of non-negative 1-Lipschitz functions with Dirichlet boundary condition around zero, which is called boundary concentration phenomena. We first examine relation between boundary concen…
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.
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.
It is well-known that the distribution over functions induced through a zero-mean iid prior distribution over the parameters of a multi-layer perceptron (MLP) converges to a Gaussian process (GP), under mild conditions. We extend this result firstly to independent priors with general zero or non-zero means, and secondl…
In this paper, we shall use the Kähler geometry formulation to study the global behavior of the Ricci flow on R2. The geometric feature of our Ricci flow is that it has finite width. Our aim is to determine the limiting metric (which corresponds an eternal Ricci flow) obtained by L.F.Wu. We can use the classificatio…
Sharp changes in time series representing market dynamics are studied by means of the self--similar analysis suggested earlier by the authors. These sharp changes are market booms and crashes. Such crises phenomena in markets are analogous to critical phenomena in physics. A simple classification of the market crisis p…
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 on Lane-Emden equation on curved spaces, revealing new existence and non-existence phenomena.
problem Existence and non-existence of positive solutions for the Lane-Emden equation on Riemannian models.
method Analysis of the subcritical Lane-Emden equation on various Riemannian manifolds with polynomial volume growth.
result Subcritical regime divides into three ranges with distinct existence and non-existence phenomena.
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.
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.