1,612 research outputs found
A two step algorithm for learning from unspecific reinforcement
We study a simple learning model based on the Hebb rule to cope with
"delayed", unspecific reinforcement. In spite of the unspecific nature of the
information-feedback, convergence to asymptotically perfect generalization is
observed, with a rate depending, however, in a non- universal way on learning
parameters. Asymptotic convergence can be as fast as that of Hebbian learning,
but may be slower. Moreover, for a certain range of parameter settings, it
depends on initial conditions whether the system can reach the regime of
asymptotically perfect generalization, or rather approaches a stationary state
of poor generalization.Comment: 13 pages LaTeX, 4 figures, note on biologically motivated stochastic
variant of the algorithm adde
Phase Transitions of Neural Networks
The cooperative behaviour of interacting neurons and synapses is studied
using models and methods from statistical physics. The competition between
training error and entropy may lead to discontinuous properties of the neural
network. This is demonstrated for a few examples: Perceptron, associative
memory, learning from examples, generalization, multilayer networks, structure
recognition, Bayesian estimate, on-line training, noise estimation and time
series generation.Comment: Plenary talk for MINERVA workshop on mesoscopics, fractals and neural
networks, Eilat, March 1997 Postscript Fil
Multifractality and percolation in the coupling space of perceptrons
The coupling space of perceptrons with continuous as well as with binary
weights gets partitioned into a disordered multifractal by a set of random input patterns. The multifractal spectrum can be
calculated analytically using the replica formalism. The storage capacity and
the generalization behaviour of the perceptron are shown to be related to
properties of which are correctly described within the replica
symmetric ansatz. Replica symmetry breaking is interpreted geometrically as a
transition from percolating to non-percolating cells. The existence of empty
cells gives rise to singularities in the multifractal spectrum. The analytical
results for binary couplings are corroborated by numerical studies.Comment: 13 pages, revtex, 4 eps figures, version accepted for publication in
Phys. Rev.
Multilayer neural networks with extensively many hidden units
The information processing abilities of a multilayer neural network with a
number of hidden units scaling as the input dimension are studied using
statistical mechanics methods. The mapping from the input layer to the hidden
units is performed by general symmetric Boolean functions whereas the hidden
layer is connected to the output by either discrete or continuous couplings.
Introducing an overlap in the space of Boolean functions as order parameter the
storage capacity if found to scale with the logarithm of the number of
implementable Boolean functions. The generalization behaviour is smooth for
continuous couplings and shows a discontinuous transition to perfect
generalization for discrete ones.Comment: 4 pages, 2 figure
Training a perceptron in a discrete weight space
On-line and batch learning of a perceptron in a discrete weight space, where
each weight can take different values, are examined analytically and
numerically. The learning algorithm is based on the training of the continuous
perceptron and prediction following the clipped weights. The learning is
described by a new set of order parameters, composed of the overlaps between
the teacher and the continuous/clipped students. Different scenarios are
examined among them on-line learning with discrete/continuous transfer
functions and off-line Hebb learning. The generalization error of the clipped
weights decays asymptotically as / in the case of on-line learning with binary/continuous activation
functions, respectively, where is the number of examples divided by N,
the size of the input vector and is a positive constant that decays
linearly with 1/L. For finite and , a perfect agreement between the
discrete student and the teacher is obtained for . A crossover to the generalization error ,
characterized continuous weights with binary output, is obtained for synaptic
depth .Comment: 10 pages, 5 figs., submitted to PR
- …