Dynamics of Supervised Learning with Restricted Training Sets

We study the dynamics of supervised learning in layered neural networks, in the regime where the size $p$ of the training set is proportional to the number $N$ of inputs. Here the local fields are no longer described by Gaussian probability distributions. We show how dynamical replica theory can be used to predict the evolution of macroscopic observables, including the relevant performance measures, incorporating the old formalism in the limit $\alpha=p/N\to\infty$ as a special case. For simplicity we restrict ourselves to single-layer networks and realizable tasks.Comment: 36 pages, latex2e, 12 eps figures (to be publ in: Proc Newton Inst Workshop on On-Line Learning '97

Artificial Neural Network Methods in Quantum Mechanics

In a previous article we have shown how one can employ Artificial Neural Networks (ANNs) in order to solve non-homogeneous ordinary and partial differential equations. In the present work we consider the solution of eigenvalue problems for differential and integrodifferential operators, using ANNs. We start by considering the Schr\"odinger equation for the Morse potential that has an analytically known solution, to test the accuracy of the method. We then proceed with the Schr\"odinger and the Dirac equations for a muonic atom, as well as with a non-local Schr\"odinger integrodifferential equation that models the $n+\alpha$ system in the framework of the resonating group method. In two dimensions we consider the well studied Henon-Heiles Hamiltonian and in three dimensions the model problem of three coupled anharmonic oscillators. The method in all of the treated cases proved to be highly accurate, robust and efficient. Hence it is a promising tool for tackling problems of higher complexity and dimensionality.Comment: Latex file, 29pages, 11 psfigs, submitted in CP

Do Campaign Contribution Limits Curb the Influence of Money in Politics?

Over 40% of countries around the world have adopted limits on campaign contributions to curb the influence of money in politics. Yet, we have limited knowledge on whether and how these limits achieve this goal. With a regression discontinuity design that uses institutional rules on contribution limits in Colombian municipalities, we show that looser limits increase the number and value of public contracts assigned to the winning candidate’s donors. The evidence suggests that this is explained by looser limits concentrating influence over the elected candidate among top donors and not by a reduction in electoral competition or changes in who runs for office. We further show that looser limits worsen the performance of donor-managed contracts: they are more likely to run over costs and require time extensions. Overall, this paper demonstrates a direct link between campaign contribution limits, donor kickbacks, and worse government contract performance