Dropout speeds up convergence in shallow linear NNs, with a rate bound.
problem Analyzing convergence rate of Dropout in shallow linear NNs.
method Gradient flow analysis and Hessian examination.
result Bound on convergence rate depends on data, dropout probability, and NN width.
The paper characterizes functions of shallow ReLU NN denoisers under minimal norm constraints.
problem Understanding the theoretical success of neural network denoisers.
method Characterization of functions realized by shallow ReLU NN denoisers under minimal norm constraints.
result The functions realized by shallow ReLU NN denoisers are contractive toward clean data points and generalize better than the empirical MMSE estimator at low noise levels.
This paper explores how kernel methods can explain data effects on neural collapse.
problem Understanding how data affects neural collapse in neural networks.
method Formulating NC1 as a function of kernel, specializing to NNGP and NTK, and exploring a data-aware Gaussian Process kernel.
result The NTK does not represent more collapsed features than the NNGP for Gaussian data, highlighting the limitations of data-independent kernels.
We reparametrize ReLU NNs as splines to understand their learning dynamics.
problem Understanding the learning dynamics and inductive bias of neural networks.
method Reparametrize ReLU NNs as continuous piecewise linear splines to study learning dynamics.
result Standard weight initializations yield very flat functions, leading to strength and type of implicit regularization.
We propose a one-class neural network (OC-NN) model to detect anomalies in complex data sets. OC-NN combines the ability of deep networks to extract a progressively rich representation of data with the one-class objective of creating a tight envelope around normal data. The OC-NN approach breaks new ground for the foll…
Recent theoretical work has demonstrated that deep neural networks have superior performance over shallow networks, but their training is more difficult, e.g., they suffer from the vanishing gradient problem. This problem can be typically resolved by the rectified linear unit (ReLU) activation. However, here we show th…
The paper certifies neural network-based control barrier functions efficiently.
problem Certifying neural network-based barrier functions for safety in autonomous systems.
method Combines NN reachability and hyperplane arrangement enumeration for efficient certification.
result Soundly finds regions where neural networks are certified as barrier functions.
Paper studies the expressivity of Convolutional Neural Networks (CNNs).
problem Understanding the expressivity of CNNs and their superiority in deep learning.
method Mathematical analysis of linear regions in one-layer and multi-layer ReLU CNNs.
result Deeper CNNs and CNNs have more expressivity per parameter than fully-connected NNs.
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.
Shallow water environments create a challenging channel for communications. In this paper, we focus on the challenges posed by the frequency-selective signal distortion called the Doppler effect. We explore the design and performance of machine learning (ML) based demodulation methods --- (1) Deep Belief Network-feed f…
Neural networks estimate statistical divergences with performance guarantees.
problem Estimating statistical divergences with theoretical performance guarantees.
method Parametrizing empirical variational form by a neural network and optimizing over parameter space.
result Established non-asymptotic absolute error bounds for neural estimators of four f-divergences. Wide neural networks converge linearly to zero loss with feature learning.
problem Optimizing wide neural networks with feature learning guarantees.
method Gradient flow analysis for wide shallow and multi-layer NNs.
result Training loss converges linearly to zero for wide NNs under GF, demonstrating feature learning and better generalization.
We present a neural network (NN) approach to fit and predict implied volatility surfaces (IVSs). Atypically to standard NN applications, financial industry practitioners use such models equally to replicate market prices and to value other financial instruments. In other words, low training losses are as important as g…
This paper characterizes how randomized neural networks generalize well in multi-dimensional tasks.
problem Understanding the generalization of randomized neural networks in multi-dimensional tasks.
method Characterizes RSNs as an IGAM formalized by an optimization problem with a regularization functional and loss.
result RSNs generalize well in multi-dimensional tasks, akin to spline regression under certain conditions.
In this work, we propose a new training method for finding minimum weight norm solutions in over-parameterized neural networks (NNs). This method seeks to improve training speed and generalization performance by framing NN training as a constrained optimization problem wherein the sum of the norm of the weights in each…
NeuraLUT maps neural networks to lookup tables, reducing latency and improving expressivity.
problem Reducing latency in deep neural networks for FPGA accelerators.
method Mapping entire sub-networks to a single lookup table, introducing skip connections.
result Up to 4.3x lower latency for the same accuracy.
This paper extends ResNet theory to infinitely deep networks, linking them to diffusion processes.
problem Training infinitely deep ResNets with i.i.d. initializations leads to undesirable properties.
method Introduced doubly infinite ResNets with i.i.d. initializations, linking to diffusion processes.
result The dynamics of quantities of interest converge to deterministic limits in the limit of infinite depth.
To help understand the underlying mechanisms of neural networks (NNs), several groups have, in recent years, studied the number of linear regions ℓ of piecewise linear functions generated by deep neural networks (DNN). In particular, they showed that ℓ can grow exponentially with the number of network paramet…
Neural networks learn higher-order derivatives for physics problems.
problem Lack of higher-order derivatives in neural networks for theoretical physics.
method Graph-theoretical approach to assign diagrams to partial derivatives, iterative NN perturbation theory.
result NNs can learn higher-order derivatives, improving machine-learned approximations.
Two-layer NN with channel attention learns low-degree spherical polynomials efficiently.
problem Learning low-degree spherical polynomials with over-parameterized neural networks.
method Two-layer neural network with channel attention, vanilla gradient descent, learnable channel selection.
result Minimally improved sample complexity of $n \asymp Θ(d^{\ell_0}/\eps)$ for learning low-degree polynomials.
We investigated the feature map inside deep neural networks (DNNs) by tracking the transport map. We are interested in the role of depth (why do DNNs perform better than shallow models?) and the interpretation of DNNs (what do intermediate layers do?) Despite the rapid development in their application, DNNs remain anal…
We propose a simple approach which, given distributed computing resources, can nearly achieve the accuracy of k-NN prediction, while matching (or improving) the faster prediction time of 1-NN. The approach consists of aggregating denoised 1-NN predictors over a small number of distributed subsamples. We show, bot…
Randomly trained neural networks can generalize well if there's a simpler underlying teacher model.
problem Why randomly trained neural networks generalize well despite interpolating training data.
method Examined a random neural network that interpolates training data and showed it generalizes well if there's a simpler underlying teacher model.
result Randomly trained neural networks can generalize well if there's a simpler underlying teacher model.
We analyze symmetries in overparametrized neural networks using a mean-field approach.
problem Understanding symmetries in overparametrized neural networks under data symmetry.
method Developed a Mean-Field view of learning dynamics for overparametrized neural networks under data symmetry.
result Symmetric models under data symmetry converge to the space of invariant laws and minimize population risk.
This paper analyzes deep Stable neural networks, showing convergence rates under different growth settings.
problem Analyzing the behavior of deep Stable neural networks as width increases.
method Large-width asymptotic analysis and convergence rates for fully connected feed-forward deep Stable NNs.
result The rescaled deep Stable NN converges weakly to a Stable SP under joint growth, with sup-norm convergence rates established.
Study bridges GARCH and NN models for volatility forecasting.
problem Lack of interaction between GARCH and NN approaches for volatility forecasting.
method Established equivalence between GARCH and NN models, introduced GARCH-NN approach.
result GARCH-NN approach enhances volatility forecasting compared to standalone models.
Paper proposes MS-k-NN for improved convergence rate in k-NN classification.
problem Improving convergence rate of k-NN classification methods.
method Proposes MS-k-NN that extrapolates unweighted k-NN estimators to k=0.
result MS-k-NN achieves improved convergence rate under certain conditions.
A fast method for LOOCV in k-NN regression reduces computation time.
problem Efficient computation of LOOCV for k-NN regression.
method Identical LOOCV estimate to (k+1)-NN MSE on training data.
result LOOCV computation can be done with (k+1)-NN regression once.
Adds layers to NNs to protect them from reverse engineering.
problem Extracting the underlying model of a Neural Network.
method Introducing parasitic layers that approximate a noisy identity mapping with a Convolutional NN.
result The protected NN's predictions remain mostly unchanged while making reverse-engineering more complex.
Artificial neural networks (NN) are instrumental in realizing highly-automated driving functionality. An overarching challenge is to identify best safety engineering practices for NN and other learning-enabled components. In particular, there is an urgent need for an adequate set of metrics for measuring all-important …
Prototype rules simplify multiclass classification in metric spaces, achieving consistency and reduced complexity.
problem Multiclass classification in metric spaces, focusing on universal consistency and convergence rates.
method Novel Proto-NN and hybrid rules for multiclass classification in metric spaces, analyzing convergence rates.
result Proto-NN is universally consistent and simpler to implement, with similar computational advantages.
Neural Networks (NN) have recently emerged as backbone of several sensitive applications like automobile, medical image, security, etc. NNs inherently offer Partial Fault Tolerance (PFT) in their architecture; however, the biased PFT of NNs can lead to severe consequences in applications like cryptography and security …
Shallow neural networks can represent polynomials efficiently.
problem Representing polynomials using shallow neural networks.
method Using shallow neural networks of width 2(R+d)d to represent d-variate polynomials of degree R. result Derives minimax optimal convergence rate for shallow networks to unknown univariate regression functions.
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.
In the k-nearest neighborhood model (k-NN), we are given a set of points P, and we shall answer queries q by returning the k nearest neighbors of q in P according to some metric. This concept is crucial in many areas of data analysis and data processing, e.g., computer vision, document retrieval and machi…
Estimates generalization error for two-layer ReLU NNs through minimum norm solutions.
problem Estimating generalization error for two-layer ReLU NNs trained by mean squared error.
method Uses minimum norm solutions and Neural Tangent Kernel (NTK) regime to derive generalization error bounds.
result Derives an a priori generalization error bound for two-layer ReLU NNs without requiring exponentially large number of neurons.
Machine Learning (ML) is making a strong resurgence in tune with the massive generation of unstructured data which in turn requires massive computational resources. Due to the inherently compute- and power-intensive structure of Neural Networks (NNs), hardware accelerators emerge as a promising solution. However, with …
This paper studies large-width asymptotics for ReLU neural networks with α-Stable initializations.
problem Characterizing the large-width behavior of ReLU neural networks with α-Stable initializations.
method Analysis of the large-width distributions and training dynamics of ReLU neural networks initialized with α-Stable distributions.
result For ReLU neural networks with α-Stable initializations, the large-width training dynamics achieve zero training error at a linear rate, characterized by a random kernel.
Qualitative analysis of MC dropout for NN model uncertainty.
problem Measuring uncertainty in neural network models.
method Mathematical formulation of Monte Carlo dropout and its benefits/costs in NN models.
result Potential benefits and associated costs of using MC dropout in NN models.
Training a Neural Network (NN) with lots of parameters or intricate architectures creates undesired phenomena that complicate the optimization process. To address this issue we propose a first modular approach to NN design, wherein the NN is decomposed into a control module and several functional modules, implementing …
The paper optimizes k-NN for distributed learning with minimax optimal performance.
problem Minimizing error rates in classification, regression, and density estimation.
method Optimal aggregation of fixed-k nearest neighbors from multiple subsets of data.
result Achieves minimax optimal error rates up to a logarithmic factor.
Despite the success of neural networks (NNs), there is still a concern among many over their "black box" nature. Why do they work? Here we present a simple analytic argument that NNs are in fact essentially polynomial regression models. This view will have various implications for NNs, e.g. providing an explanation for…
Neural Networks (NNs) have been extensively used for a wide spectrum of real-world regression tasks, where the goal is to predict a numerical outcome such as revenue, effectiveness, or a quantitative result. In many such tasks, the point prediction is not enough: the uncertainty (i.e. risk or confidence) of that predic…
Study on dropout in neural networks using percolation theory.
problem Understanding dropout's effect on neural network training.
method Investigates percolation models mimicking dropout in neural networks.
result Dropout can cause a breakdown in neural networks.
We derive high-probability finite-sample uniform rates of consistency for k-NN regression that are optimal up to logarithmic factors under mild assumptions. We moreover show that k-NN regression adapts to an unknown lower intrinsic dimension automatically. We then apply the k-NN regression rates to establish new …
Wide and shallow networks approximate convex functions well.
problem Understanding why wide and shallow neural networks perform well.
method Analyzing the epigraph of the input-output map of shallow and wide neural networks.
result The epigraph of the input-output map approximates a convex function.
Ensembles of neural networks (NNs) have long been used to estimate predictive uncertainty; a small number of NNs are trained from different initialisations and sometimes on differing versions of the dataset. The variance of the ensemble's predictions is interpreted as its epistemic uncertainty. The appeal of ensembling…
A garland based on a manifold P is a finite set of manifolds homeomorphic to P with some of them glued together at marked points. Fix a manifold M and consider a space $\NN$ of all smooth mappings of garlands based on P into M. We construct operations ∙ and [−,−] on the bordism groups $\bor_*(\NN)$ …