100 research outputs found
Storage Capacity of the Tilinglike Learning Algorithm
The storage capacity of an incremental learning algorithm for the parity
machine, the Tilinglike Learning Algorithm, is analytically determined in the
limit of a large number of hidden perceptrons. Different learning rules for the
simple perceptron are investigated. The usual Gardner-Derrida one leads to a
storage capacity close to the upper bound, which is independent of the learning
algorithm considered.Comment: Proceedings of the Conference Disordered and Complex Systems, King's
College, London, July 2000. 6 pages, 1 figure, uses aipproc.st
A Model for Scaling in Firms' Size and Growth Rate Distribution
We introduce a simple agent-based model which allows us to analyze three
stylized facts: a fat-tailed size distribution of companies, a `tent-shaped'
growth rate distribution, the scaling relation of the growth rate variance with
firm size, and the causality between them. This is achieved under the simple
hypothesis that firms compete for a scarce quantity (either aggregate demand or
workforce) which is allocated probabilistically. The model allows us to relate
size and growth rate distributions. We compare the results of our model to
simulations with other scaling relationships, and to similar models and relate
it to existing theory. Effects arising from binning data are discussed.Comment: accepted for publication in Physica
Learning curves for Soft Margin Classifiers
Typical learning curves for Soft Margin Classifiers (SMCs) learning both
realizable and unrealizable tasks are determined using the tools of Statistical
Mechanics. We derive the analytical behaviour of the learning curves in the
regimes of small and large training sets. The generalization errors present
different decay laws towards the asymptotic values as a function of the
training set size, depending on general geometrical characteristics of the rule
to be learned. Optimal generalization curves are deduced through a fine tuning
of the hyperparameter controlling the trade-off between the error and the
regularization terms in the cost function. Even if the task is realizable, the
optimal performance of the SMC is better than that of a hard margin Support
Vector Machine (SVM) learning the same rule, and is very close to that of the
Bayesian classifier.Comment: 26 pages, 10 figure
Rigorous Bounds to Retarded Learning
We show that the lower bound to the critical fraction of data needed to infer
(learn) the orientation of the anisotropy axis of a probability distribution,
determined by Herschkowitz and Opper [Phys.Rev.Lett. 86, 2174 (2001)], is not
always valid. If there is some structure in the data along the anisotropy axis,
their analysis is incorrect, and learning is possible with much less data
points.Comment: 1 page, 1 figure. Comment accepted for publication in Physical Review
Letter
Discrete Choices under Social Influence: Generic Properties
We consider a model of socially interacting individuals that make a binary
choice in a context of positive additive endogenous externalities. It
encompasses as particular cases several models from the sociology and economics
literature. We extend previous results to the case of a general distribution of
idiosyncratic preferences, called here Idiosyncratic Willingnesses to Pay
(IWP). Positive additive externalities yield a family of inverse demand curves
that include the classical downward sloping ones but also new ones with non
constant convexity. When j, the ratio of the social influence strength to the
standard deviation of the IWP distribution, is small enough, the inverse demand
is a classical monotonic (decreasing) function of the adoption rate. Even if
the IWP distribution is mono-modal, there is a critical value of j above which
the inverse demand is non monotonic, decreasing for small and high adoption
rates, but increasing within some intermediate range. Depending on the price
there are thus either one or two equilibria. Beyond this first result, we
exhibit the generic properties of the boundaries limiting the regions where the
system presents different types of equilibria (unique or multiple). These
properties are shown to depend only on qualitative features of the IWP
distribution: modality (number of maxima), smoothness and type of support
(compact or infinite). The main results are summarized as phase diagrams in the
space of the model parameters, on which the regions of multiple equilibria are
precisely delimited.Comment: 42 pages, 15 figure
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