Multilayer neural networks with extensively many hidden units

It has been shown \citep{broeck90:physicalreview,patarnello87:europhys} that feedforward Boolean networks can learn to perform specific simple tasks and generalize well if only a subset of the learning examples is provided for learning. Here, we extend this body of work and show experimentally that random Boolean networks (RBNs), where both the interconnections and the Boolean transfer functions are chosen at random initially, can be evolved by using a state-topology evolutio...

In this short note, we reify the connection between work on the storage capacity problem in wide two-layer treelike neural networks and the rapidly-growing body of literature on kernel limits of wide neural networks. Concretely, we observe that the "effective order parameter" studied in the statistical mechanics literature is exactly equivalent to the infinite-width Neural Network Gaussian Process Kernel. This correspondence connects the expressivity and trainability of wide ...

We consider feed-forward neural networks with one hidden layer, tree architecture and a fixed hidden-to-output Boolean function. Focusing on the saturation limit of the storage problem the influence of replica symmetry breaking on the distribution of local fields at the hidden units is investigated. These field distributions determine the probability for finding a specific activation pattern of the hidden units as well as the corresponding correlation coefficients and therefo...

We describe an approach to understand the peculiar and counterintuitive generalization properties of deep neural networks. The approach involves going beyond worst-case theoretical capacity control frameworks that have been popular in machine learning in recent years to revisit old ideas in the statistical mechanics of neural networks. Within this approach, we present a prototypical Very Simple Deep Learning (VSDL) model, whose behavior is controlled by two control parameters...

We analyze the loss of information and the loss of learning capability inside an arrangement of neural networks. Our method is new and based on the formulation of non-unitary Bogoliubov transformations in order to connect the information between different points of the arrangement. This can be done after expanding the activation function in a Fourier series and then assuming that its information is stored inside a Quantum scalar field.

This paper considers the problem of information capacity of a random neural network. The network is represented by matrices that are square and symmetrical. The matrices have a weight which determines the highest and lowest possible value found in the matrix. The examined matrices are randomly generated and analyzed by a computer program. We find the surprising result that the capacity of the network is a maximum for the binary random neural network and it does not change as ...

Recurrent neural networks (RNN) are simple dynamical systems whose computational power has been attributed to their short-term memory. Short-term memory of RNNs has been previously studied analytically only for the case of orthogonal networks, and only under annealed approximation, and uncorrelated input. Here for the first time, we present an exact solution to the memory capacity and the task-solving performance as a function of the structure of a given network instance, ena...

In this preliminary work, we study the generalization properties of infinite ensembles of infinitely-wide neural networks. Amazingly, this model family admits tractable calculations for many information-theoretic quantities. We report analytical and empirical investigations in the search for signals that correlate with generalization.

This paper underscores the conjecture that intrinsic computation is maximal in systems at the "edge of chaos." We study the relationship between dynamics and computational capability in Random Boolean Networks (RBN) for Reservoir Computing (RC). RC is a computational paradigm in which a trained readout layer interprets the dynamics of an excitable component (called the reservoir) that is perturbed by external input. The reservoir is often implemented as a homogeneous recurren...

The inductive bias and generalization properties of large machine learning models are -- to a substantial extent -- a byproduct of the optimization algorithm used for training. Among others, the scale of the random initialization, the learning rate, and early stopping all have crucial impact on the quality of the model learnt by stochastic gradient descent or related algorithms. In order to understand these phenomena, we study the training dynamics of large two-layer neural n...