1,654 research outputs found
Numerical analysis of least squares and perceptron learning for classification problems
This work presents study on regularized and non-regularized versions of
perceptron learning and least squares algorithms for classification problems.
Fr'echet derivatives for regularized least squares and perceptron learning
algorithms are derived. Different Tikhonov's regularization techniques for
choosing the regularization parameter are discussed. Decision boundaries
obtained by non-regularized algorithms to classify simulated and experimental
data sets are analyzed
Perceptron learning with random coordinate descent
A perceptron is a linear threshold classifier that separates examples with a hyperplane. It is perhaps the simplest learning model that is used standalone. In this paper, we propose a family of random coordinate descent algorithms for perceptron learning on binary classification problems. Unlike most perceptron learning algorithms which require smooth cost functions, our algorithms directly minimize the training error, and usually achieve the lowest training error compared with other algorithms. The algorithms are also computational efficient. Such advantages make them favorable for both standalone use and ensemble learning, on problems that are not linearly separable. Experiments show that our algorithms work very well with AdaBoost, and achieve the lowest test errors for half of the datasets
Multifractal analysis of perceptron learning with errors
Random input patterns induce a partition of the coupling space of a
perceptron into cells labeled by their output sequences. Learning some data
with a maximal error rate leads to clusters of neighboring cells. By analyzing
the internal structure of these clusters with the formalism of multifractals,
we can handle different storage and generalization tasks for lazy students and
absent-minded teachers within one unified approach. The results also allow some
conclusions on the spatial distribution of cells.Comment: 11 pages, RevTex, 3 eps figures, version to be published in Phys.
Rev. E 01Jan9
Statistical Mechanics of Nonlinear On-line Learning for Ensemble Teachers
We analyze the generalization performance of a student in a model composed of
nonlinear perceptrons: a true teacher, ensemble teachers, and the student. We
calculate the generalization error of the student analytically or numerically
using statistical mechanics in the framework of on-line learning. We treat two
well-known learning rules: Hebbian learning and perceptron learning. As a
result, it is proven that the nonlinear model shows qualitatively different
behaviors from the linear model. Moreover, it is clarified that Hebbian
learning and perceptron learning show qualitatively different behaviors from
each other. In Hebbian learning, we can analytically obtain the solutions. In
this case, the generalization error monotonically decreases. The steady value
of the generalization error is independent of the learning rate. The larger the
number of teachers is and the more variety the ensemble teachers have, the
smaller the generalization error is. In perceptron learning, we have to
numerically obtain the solutions. In this case, the dynamical behaviors of the
generalization error are non-monotonic. The smaller the learning rate is, the
larger the number of teachers is; and the more variety the ensemble teachers
have, the smaller the minimum value of the generalization error is.Comment: 13 pages, 9 figure
Stability of the replica symmetric solution in diluted perceptron learning
We study the role played by the dilution in the average behavior of a
perceptron model with continuous coupling with the replica method. We analyze
the stability of the replica symmetric solution as a function of the dilution
field for the generalization and memorization problems. Thanks to a Gardner
like stability analysis we show that at any fixed ratio between the
number of patterns M and the dimension N of the perceptron (),
there exists a critical dilution field above which the replica symmetric
ansatz becomes unstable.Comment: Stability of the solution in arXiv:0907.3241, 13 pages, (some typos
corrected
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