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.
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 …
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.
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.
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 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.
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.
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.
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 efficient neural operator learning using variation spaces.
problem Operator learning using encoder-decoder neural networks.
method Introduce variation space for nonlinear operators, establish approximation bounds.
result Algebraic approximation and learning rates for polynomially decaying input and output encoding errors.
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.
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.
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.
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 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…
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.
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…
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.
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.
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. 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 work approximates finite neural networks with Gaussian processes, providing error bounds and applications in prior selection.
problem Approximating finite neural networks with Gaussian processes for error bounds and uncertainty quantification.
method Iterative approximation of neural network layers as mixtures of Gaussian processes, using optimal transport and Gaussian processes.
result The ability to return a mixture of Gaussian processes that is ε-close to the neural network at a finite set of input points.
Study loop corrections in random feature models affecting training and test errors.
problem Analyzing loop corrections in random feature models to understand training and test errors.
method Statistical physics and effective field theory approach to study loop corrections.
result Derived loop corrections to training error, test error, and generalization gap.
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.
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.
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 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.
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 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…
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.
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 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 shows trained neural networks can overfit without bias or variance issues.
problem Understanding overfitting in trained two-layer ReLU networks.
method Analysis of gradient flow in the neural tangent kernel regime, decomposition of excess risk.
result Trained networks can overfit benignly without bias or variance issues.
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.
Characterizes test error in learning with deep, structured feature maps.
problem Characterizing test error in learning with deep, structured feature maps.
method Asymptotic analysis of feature covariance and population covariance.
result Closed-form formula for feature covariance in Gaussian rainbow neural networks.
GD-trained shallow ReLU nets learn Lipschitz functions with noise.
problem Learning Lipschitz functions with additive noise in overparameterized neural networks.
method Gradient Descent (GD) with early stopping, focusing on the Neural Tangent Kernel (NTK).
result Early-stopped GD achieves minimax optimal rates for learning Lipschitz functions.
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 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.
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.
It has been noted in existing literature that over-parameterization in ReLU networks generally improves performance. While there could be several factors involved behind this, we prove some desirable theoretical properties at initialization which may be enjoyed by ReLU networks. Specifically, it is known that He initia…
This study reveals fundamental trade-offs between memorization and robustness in neural networks.
problem Understanding the balance between memorization and robustness in neural networks.
method Analyzes two-layer neural networks in various high-dimensional linearized regimes, focusing on Sobolev-seminorm.
result Establishes fundamental trade-offs between memorization and robustness, with lower bounds on Sobolev-seminorm.
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.
We study the concentration of NTK for MLPs at EOC, proving finite-width approximation of gradient independence.
problem Understanding the concentration of Neural Tangent Kernel (NTK) for MLPs at the Edge of Chaos (EOC).
method Proved approximate gradient independence holds at finite width, using maximal inequalities to show NTK matrix concentrates around its infinitely wide limit.
result The NTK matrix of MLPs at EOC concentrates around its infinitely wide limit, requiring hidden layer widths to grow quadratically.
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.