arXiv research

George Cybenko's landmark 1989 paper showed that there exists a feedforward neural network, with exactly one hidden layer (and a finite number of neurons), that can arbitrarily approximate a given continuous function $f$ on the unit hypercube. The paper did not address how to find the weight/parameters of such a networ…

This paper extends depth separation results to piece-wise oscillatory functions.

We study the problem of learning one-hidden-layer neural networks with Rectified Linear Unit (ReLU) activation function, where the inputs are sampled from standard Gaussian distribution and the outputs are generated from a noisy teacher network. We analyze the performance of gradient descent for training such kind of n…

A neural network with a single hidden layer can't represent certain multivariable functions.

Study on neural networks' sample complexity with one hidden layer.

We study the sample complexity of learning one-hidden-layer convolutional neural networks (CNNs) with non-overlapping filters. We propose a novel algorithm called approximate gradient descent for training CNNs, and show that, with high probability, the proposed algorithm with random initialization grants a linear conve…

The paper analyzes GNNs with one hidden layer, proving their generalizability and convergence rate.

For one-hidden-layer ReLU networks, we prove that all differentiable local minima are global inside differentiable regions. We give the locations and losses of differentiable local minima, and show that these local minima can be isolated points or continuous hyperplanes, depending on an interplay between data, activati…

This paper explores loss landscapes of sparse neural networks, finding unique characteristics compared to dense networks.

We propose a new algorithm to learn a one-hidden-layer convolutional neural network where both the convolutional weights and the outputs weights are parameters to be learned. Our algorithm works for a general class of (potentially overlapping) patches, including commonly used structures for computer vision tasks. Our a…

New algorithm for learning ReLU networks with Gaussian noise, improving previous results.

We study model recovery for data classification, where the training labels are generated from a one-hidden-layer neural network with sigmoid activations, also known as a single-layer feedforward network, and the goal is to recover the weights of the neural network. We consider two network models, the fully-connected ne…

Hardness proven for learning neural networks with polynomial size and Gaussian inputs.

We use smoothed analysis techniques to provide guarantees on the training loss of Multilayer Neural Networks (MNNs) at differentiable local minima. Specifically, we examine MNNs with piecewise linear activation functions, quadratic loss and a single output, under mild over-parametrization. We prove that for a MNN with …

Although deep learning has shown its powerful performance in many applications, the mathematical principles behind neural networks are still mysterious. In this paper, we consider the problem of learning a one-hidden-layer neural network with quadratic activations. We focus on the under-parameterized regime where the n…

We show that the gradient descent algorithm provides an implicit regularization effect in the learning of over-parameterized matrix factorization models and one-hidden-layer neural networks with quadratic activations. Concretely, we show that given $\tilde{O}(dr^{2})$ random linear measurements of a rank $r$ positive s…

The paper analyzes how minority group imbalance affects neural network performance.

New method approximates neural network training for robustness.

Consider a feedforward neural network $ψ: \mathbb{R}^d\rightarrow \mathbb{R}^d$ such that $ψ\approx \nabla f$, where $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a smooth function, therefore $ψ$ must satisfy $\partial_j ψ_i = \partial_i ψ_j$ pointwise. We prove a theorem that a $ψ$ network with more than one hidden layer…

We consider the problem of learning a one-hidden-layer neural network: we assume the input $x\in \mathbb{R}^d$ is from Gaussian distribution and the label $y = a^\top σ(Bx) + ξ$, where $a$ is a nonnegative vector in $\mathbb{R}^m$ with $m\le d$, $B\in \mathbb{R}^{m\times d}$ is a full-rank weight matrix, and $ξ$ is a n…

The study analyzes local minima in ReLU networks and finds low probability of bad local minima.

We present a framework to derive risk bounds for vector-valued learning with a broad class of feature maps and loss functions. Multi-task learning and one-vs-all multi-category learning are treated as examples. We discuss in detail vector-valued functions with one hidden layer, and demonstrate that the conditions under…

We consider the problem of learning a one-hidden-layer neural network with non-overlapping convolutional layer and ReLU activation, i.e., $f(\mathbf{Z}, \mathbf{w}, \mathbf{a}) = \sum_j a_jσ(\mathbf{w}^T\mathbf{Z}_j)$, in which both the convolutional weights $\mathbf{w}$ and the output weights $\mathbf{a}$ are paramete…

In order to classify linearly non-separable data, neurons are typically organized into multi-layer neural networks that are equipped with at least one hidden layer. Inspired by some recent discoveries in neuroscience, we propose a new model of artificial neuron along with a novel activation function enabling the learni…

The paper establishes limits of transfer learning with neural networks.

We propose convex relaxations for convolutional neural nets with one hidden layer where the output weights are fixed. For convex activation functions such as rectified linear units, the relaxations are convex second order cone programs which can be solved very efficiently. We prove that the relaxation recovers the glob…

This work concerns testing the number of parameters in one hidden layer multilayer perceptron (MLP). For this purpose we assume that we have identifiable models, up to a finite group of transformations on the weights, this is for example the case when the number of hidden units is know. In this framework, we show that …

In this paper, we consider regression problems with one-hidden-layer neural networks (1NNs). We distill some properties of activation functions that lead to $\mathit{local~strong~convexity}$ in the neighborhood of the ground-truth parameters for the 1NN squared-loss objective. Most popular nonlinear activation function…

This work provides an additional step in the theoretical understanding of neural networks. We consider neural networks with one hidden layer and show that when learning symmetric functions, one can choose initial conditions so that standard SGD training efficiently produces generalization guarantees. We empirically ver…

Enhanced feature learning using neural networks and kernel methods with improved robustness.

Neural networks favor simple features over complex ones, even when complex features are available.

Meta-learning consists in learning learning algorithms. We use a Long Short Term Memory (LSTM) based network to learn to compute on-line updates of the parameters of another neural network. These parameters are stored in the cell state of the LSTM. Our framework allows to compare learned algorithms to hand-made algorit…

We present a variation of the Autoencoder (AE) that explicitly maximizes the mutual information between the input data and the hidden representation. The proposed model, the InfoMax Autoencoder (IMAE), by construction is able to learn a robust representation and good prototypes of the data. IMAE is compared both theore…

Giving provable guarantees for learning neural networks is a core challenge of machine learning theory. Most prior work gives parameter recovery guarantees for one hidden layer networks, however, the networks used in practice have multiple non-linear layers. In this work, we show how we can strengthen such results to d…

We prove a negative result for the approximation of functions defined on compact subsets of $\mathbb{R}^d$ (where $d \geq 2$) using feedforward neural networks with one hidden layer and arbitrary continuous activation function. In a nutshell, this result claims the existence of target functions that are as difficult to…

Upper bounds on fixed points in PWL neural networks with hyperplane analysis.

Study explores how neural networks and Transformers learn modular arithmetic with multiple inputs.

In this paper we propose a method to build a neural network that is similar to an ensemble of decision trees. We first illustrate how to convert a learned ensemble of decision trees to a single neural network with one hidden layer and an input transformation. We then relax some properties of this network such as thresh…

The study proves a quantitative functional CLT for neural networks with smooth activation functions.

Training deep neural networks with the error backpropagation algorithm is considered implausible from a biological perspective. Numerous recent publications suggest elaborate models for biologically plausible variants of deep learning, typically defining success as reaching around 98% test accuracy on the MNIST data se…

Study shows how neural networks generalize with minimal training data.

Based on the tree architecture, the objective of this paper is to design deep neural networks with two or more hidden layers (called deep nets) for realization of radial functions so as to enable rotational invariance for near-optimal function approximation in an arbitrarily high dimensional Euclidian space. It is show…

We establish, for the first time, connections between feedforward neural networks with ReLU activation and tropical geometry --- we show that the family of such neural networks is equivalent to the family of tropical rational maps. Among other things, we deduce that feedforward ReLU neural networks with one hidden laye…

Neural networks compress uninformative input directions, improving test error.

Study shows benefits of transfer learning with neural networks.

Deep learning models are often successfully trained using gradient descent, despite the worst case hardness of the underlying non-convex optimization problem. The key question is then under what conditions can one prove that optimization will succeed. Here we provide a strong result of this kind. We consider a neural n…

Overparametrized neural networks retain significant epistemic uncertainty even with sufficient data.

The paper analyzes deep ReLU CNNs' approximation properties in 2D space.