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.
Stable processes emerge as limits of deep neural networks with symmetric stable distributions.
problem Understanding the behavior of deep neural networks as they become infinitely wide.
method Analyzing fully connected feed-forward deep neural networks with symmetric stable distributions and showing the limit as a stable process.
result The infinite wide limit of the network is a stable process with multivariate stable distributions.
There has recently been much work on the "wide limit" of neural networks, where Bayesian neural networks (BNNs) are shown to converge to a Gaussian process (GP) as all hidden layers are sent to infinite width. However, these results do not apply to architectures that require one or more of the hidden layers to remain n…
Study of deep Stable neural networks with various activation functions.
problem Characterizing the infinitely wide limits of deep Stable neural networks.
method Investigation of large-width properties of deep Stable NNs with a generalized central limit theorem for heavy tails.
result Extension of characterization to a broader class of activation functions, including sub-linear, asymptotically linear, and super-linear functions.
Wide stochastic networks show Gaussian behavior and improve training with PAC-Bayesian methods.
problem Analyzing and training over-parameterised neural networks with large width.
method Establishing Gaussian behavior for a stochastic architecture, applying PAC-Bayesian training.
result PAC-Bayesian training on large but finite-width networks outperforms standard methods.
Wide deep neural networks with Gaussian weights approximate Gaussian processes closely.
problem Understanding the approximation of deep neural networks with Gaussian weights to Gaussian processes.
method Established novel rates for the Gaussian approximation of random deep neural networks with Gaussian parameters and Lipschitz activation functions in the wide limit.
result The distance between the network output and the Gaussian approximation scales inversely with the width of the network.
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.
Infinite CNNs lose spatial correlations, but can be restored by correlated weights.
problem Infinite CNNs lose spatial correlations, which are crucial for their performance.
method Introduced correlated weights to restore spatial correlations in infinite CNNs.
result Optimal performance is achieved with a moderate level of weight correlation.
Understanding the asymptotic behavior of wide networks is of considerable interest. In this work, we present a general method for analyzing this large width behavior. The method is an adaptation of Feynman diagrams, a standard tool for computing multivariate Gaussian integrals. We apply our method to study training dyn…
Embedded ensembles improve neural network performance efficiently.
problem Improving neural network performance with fewer resources.
method Analyzing the wide network limit of gradient descent dynamics using Neural-Tangent-Kernel.
result Embedded ensembles exhibit two regimes: independent and collective, affecting performance.
A longstanding goal in deep learning research has been to precisely characterize training and generalization. However, the often complex loss landscapes of neural networks have made a theory of learning dynamics elusive. In this work, we show that for wide neural networks the learning dynamics simplify considerably and…
The paper studies neural networks with wide layers and finds a deformed semicircle law.
problem Investigating spectral distributions of neural networks in the ultra-wide regime.
method Analyzes empirical kernel matrices, proves deformed semicircle law, provides nonlinear Hanson-Wright inequality.
result Emergence of a deformed semicircle law in the ultra-wide neural network regime.
Elman-type RNNs converge to globally optimal solutions in the mean-field regime.
problem Optimizing feature learning in wide RNNs.
method Analysis of gradient descent dynamics and mean-field limits.
result Fixed points of infinite-width dynamics are globally optimal.
Wide neural networks learn features under μP, identifying weights and decomposing support.
problem Feature learning in wide neural networks under μP. method Proving mean-field limit, characterizing identifiability, sparse-dictionary decomposition, and feature-learning-error decomposition.
result The triple (w∗,Dorb∗,S∗) identifies the natural learning cell of the architecture-data pair (σ,ρ). Gradient descent dynamics in wide neural networks are analyzed using a dynamical CLT.
problem Understanding the fluctuations in wide shallow neural networks trained via gradient descent.
method Dynamical Central Limit Theorem (CLT) applied to neural network dynamics.
result Asymptotic fluctuations remain bounded in mean square throughout training.
We consider a diffuse interface approximation for the lipid phases of rotationally symmetric two-phase bilayer membranes and rigorously derive its Γ-limit. In particular, we prove that limit vesicles are C1 across interfaces, which justifies a regularity assumption that is widely made in formal asymptotic and nume…
We propose a model for the dynamics of a limit order book in a liquid market where buy and sell orders are submitted at high frequency. We derive a functional central limit theorem for the joint dynamics of the bid and ask queues and show that, when the frequency of order arrivals is large, the intraday dynamics of the…
Tensor programs prove neural network limits for any architecture.
problem Understanding the limits of neural networks of any architecture.
method Prove convergence of neural network's Tangent Kernel (NTK) to a deterministic limit as network widths increase.
result Identify conditions for correct NTK limit calculation based on gradient independence assumption.
This paper provides a mathematical foundation for deep neural networks solving PDEs.
problem Mathematical foundation for deep neural networks solving high-dimensional PDEs.
method Decomposed generalization error into approximation and training errors; derived gradient flow in the wide network limit.
result Generalization error tends to zero as the number of neurons and training time tend to infinity.
Study on MC dropout in wide neural networks and its convergence to Gaussian processes.
problem Understanding the behavior of Monte Carlo dropout in wide neural networks.
method Rigorously studied the limiting distribution of wide untrained NNs under dropout, proving convergence to Gaussian processes. Investigated correlations and non-Gaussian behavior in finite width NNs.
result Wide untrained neural networks under dropout converge to Gaussian processes for fixed sets of weights and biases.
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.
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.
New insights into tSNE for large datasets.
problem Limitations of tSNE in handling large datasets.
method Identified continuum limit of tSNE objective function, proposed rescaled model.
result Rescaled model has a consistent limit for large datasets.
Randomly initialized wide neural networks with zero-mean activations are nearly independent, potentially solving AI interpretability limits.
problem Measuring the limits of AI interpretability.
method Randomly initialized neural networks with large width and zero-mean activation functions.
result Neural networks with zero-mean activations are nearly independent, solving the computational no-coincidence conjecture.
New research on limits of transfer learning, proving key selection and dependence requirements.
problem Insufficient theoretical foundation for transfer learning.
method Proved novel results on transfer learning, emphasizing selection of information and dependence between domains.
result Upper bound on improvement possible with transfer learning, highlighting the need for careful selection.
Paper proves a Central Limit Theorem for Random Forest Permutation Importance Measure.
problem Lack of theoretical analysis of Random Forest Permutation Importance Measure (RFPIM).
method Formal proof using U-Statistics theory, deviating from conventional Random Forest model.
result Established a Central Limit Theorem for RFPIM.
GNNs learn graph representations, with new theory on their power and limitations.
problem Understanding the capabilities and limitations of GNNs.
method Theoretical analysis of GNNs, focusing on approximation and learning properties.
result New insights into the representation, generalization, and extrapolation of GNNs.
We study the limiting behaviour of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that the limiting dynamics is given by a McKean-Vlasov evolution equation. Moreover, we show that in a wide range of cases the evolution of the cum…
Stabilizes GAN training with limited data.
problem Overfitting in GANs with scarce data.
method Adaptive discriminator augmentation.
result Good results possible with few thousand images.
This paper evaluates LLMs on large graph property estimation tasks.
problem Limited context length of LLMs limits their evaluation on large graphs.
method Developed EstGraph dataset and introduced four tasks for LLMs to estimate large graph properties.
result LLMs perform better on graph property estimation tasks when provided with context-rich prompts based on random walks.
We prove that a single-layer neural network trained with the Q-learning algorithm converges in distribution to a random ordinary differential equation as the size of the model and the number of training steps become large. Analysis of the limit differential equation shows that it has a unique stationary solution which …
Newly discovered Eguchi-Hanson metric arises from edge metrics.
problem Understanding limits of compact singular Einstein spaces.
method Constructing Kahler-Einstein edge metrics on Calabi-Hirzebruch manifolds.
result Eguchi-Hanson metric emerges as a Gromov-Hausdorff limit.
Study reveals three limiting regimes for neural network functionals.
problem Understanding the behavior of functionals of random neural networks.
method Central and non-central limit theorems, Hermite expansions, Diagram Formula, Stein-Malliavin techniques.
result Three distinct limiting regimes based on fixed points of covariance function.
Limited liability reduces leveraged risk in loan portfolio management models.
problem The impact of limited liability on risk in loan portfolio management models is not well understood.
method Formulated four models to analyze the effect of limited liability on risk and return in loan portfolio management.
result Including limited liability in loan portfolio management models produces better results in minimizing risk and maximizing expected return.
We prove the precise scaling, at finite depth and width, for the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network. The standard deviation is exponential in the ratio of network depth to width. Thus, even in the limit of infinite overparameterization, the NTK is not determinist…
Deep neural networks with heavy-tailed weights converge to stable distributions.
problem Understanding the convergence of heavy-tailed weights in infinitely-wide neural networks.
method Analyzing infinitely-wide multi-layer perceptrons with i.i.d. symmetric α-stable weight distributions. result The vector of pre-activation values converges to i.i.d. symmetric α-stable distributions. Novel mutual information bound improves statistical inference rates.
problem Improving statistical inference rates in Bayesian nonparametrics.
method Introduces a novel mutual information bound.
result Improved contraction rates for fractional posteriors.
Nonconvex optimization problems such as the ones in training deep neural networks suffer from a phenomenon called saddle point proliferation. This means that there are a vast number of high error saddle points present in the loss function. Second order methods have been tremendously successful and widely adopted in the…
New algorithm solves non-convex, non-differentiable min-max games.
problem Limited theoretical understanding of non-smooth min-max games.
method Proximal gradient descent-ascent algorithm for convex-strongly convex games.
result Algorithm converges to ε-Nash equilibrium with polynomial gradient evaluations.
Kernel methods are widespread in machine learning; however, they are limited by the quadratic complexity of the construction, application, and storage of kernel matrices. Low-rank matrix approximation algorithms are widely used to address this problem and reduce the arithmetic and storage cost. However, we observed tha…
A new VAE model identifies and estimates treatment effects with limited overlap.
problem Identifying and estimating treatment effects when subjects with certain features belong to a single treatment group.
method Developed a latent variable model to estimate a prognostic score, which is sufficient for treatment effects. The model is a new type of VAE called β-Intact-VAE.
result The model identifies individualized treatment effects and provides TE error bounds.
This paper explores loss landscapes of sparse neural networks, finding unique characteristics compared to dense networks.
problem Understanding the loss landscape of sparse neural networks, especially one-hidden-layer networks.
method Analyzes sparse networks with dense and sparse final layers, focusing on linear and non-linear models.
result Sparse networks can have no spurious valleys under certain conditions, but spurious valleys and minima can exist for wide sparse networks.
Deep learning methods are becoming widely used for restoration of defects associated with fluorescence microscopy imaging. One of the major challenges in application of such methods is the availability of training data. In this work, we propose a unified method for reconstruction of multi-defect fluorescence microscopy…
Variational inference with a factorized Gaussian posterior estimate is a widely used approach for learning parameters and hidden variables. Empirically, a regularizing effect can be observed that is poorly understood. In this work, we show how mean field inference improves generalization by limiting mutual information …
TAMD prevents degeneracy in finite mixtures, offering strong guarantees but modest practical improvements.
problem Degeneracy in maximum likelihood estimation of finite mixtures.
method Transcendental regularization with analytic barrier functions.
result Strong theoretical guarantees (identifiability, consistency, robustness) but modest practical improvements.
Deep Gaussian Processes are reinterpreted as deep trigonometric networks for tractable inference.
problem Challenging inference in DGPs due to intractable marginalization in latent function space.
method Viewing DGPs as deep trigonometric networks with Bochner's theorem, and using the wide limit with a bottleneck to translate DGPs into deep trigonometric networks.
result The weight space view yields the same effective covariance functions as obtained in function space, and varying prior distributions over network parameters is equivalent to employing different kernels.
The paper improves confidence regions for band-limited functions using tighter norm bounds and majority voting.
problem Constructing reliable confidence regions for band-limited functions from noisy data.
method Improved norm bounds using Hoeffding's inequality and empirical Bernstein bound, majority voting to aggregate intervals.
result Confidence intervals retain their simultaneous coverage guarantee even when aggregated from random subsamples.
Training of large-scale deep neural networks is often constrained by the available computational resources. We study the effect of limited precision data representation and computation on neural network training. Within the context of low-precision fixed-point computations, we observe the rounding scheme to play a cruc…