# Almeida-Pineda recurrent backpropagation

**Almeida-Pineda recurrent backpropagation** is an error-driven learning technique developed in 1987 by Luis B. Almeida^{[1]} and Fernando J. Pineda.^{[2]}^{[3]} It is a *supervised* learning technique, meaning that the desired outputs are known beforehand, and the task of the network is to learn to generate the desired outputs from the inputs.

As opposed to a feedforward network, a recurrent network is allowed to have connections from any neuron to any neuron in any direction.

## Contents

## Model

Given a set of k-dimensional inputs with values between 0 and 1 represented as a column vector:

and a nonlinear neuron with (initially random, uniformly distributed between -1 and 1) synaptic weights from the inputs:

then the output *y* of the neuron is defined as follows:

where is a sigmoidal function such as that used in ordinary feedforward backpropagation (we will use the logistic function from that page), and *n* is the net input of the neuron, calculated as follows. Assuming *N* neurons where *k* of the neurons are simple inputs to the network, with the weight of the connection from neuron *i* to neuron *j* being , the net of neuron *j* (where *j* is not an input neuron) is computed using a discrete time approximation to the following equation, iteratively applied to all neurons until the nets settle to some equilibrium state. Initially set to 0 for all non-input neurons.

Note that if the weights between pairs of neurons are symmetric, that is, , then the network is guaranteed to settle to an equilibrium state.^{[4]} If symmetry is not held, the network will often settle.^{[5]} Of course, if *i* is an input, then does not exist.

Once the nets of the neurons are determined, an error phase is run to determine error terms for all neurons *solely for the purpose of weight modification*. As above, these weight modification error terms are computed using a discrete time approximation to the following equation, iteratively applied to all neurons until the error terms settle to some equilibrium state. Initially set for all neurons.

where is an error term for neurons which are outputs and have targets :

The weights are then updated according to the following equation:

where η is some small learning rate.

## Derivation

The error terms are considered estimates of during the derivation of the equations for feedforward backpropagation:

## Objections

While mathematically sound, the Almeida-Pineda model is biologically implausible, like feedforward backpropagation, because the model requires that neurons communicate error terms backwards through connections for weight updates.

## References

- ↑ Almeida, Luis B. (June 1987). "A learning rule for asynchronous perceptrons with feedback in a combinatorial environment."
*Proceedings of the IEEE First International Conference on Neural Networks* - ↑ "Generalization of backpropagation to recurrent neural networks". In Anderson, Dana Z.
*Neural Information Processing Systems*Springer (1988). pp. 602-611. ISBN 978-0883185698}} - ↑ Pineda, Fernando J. (1989). "Recurrent backpropagation and the dynamical approach to adaptive neural computation".
*Neural Computation***1**: 161-172 - ↑ Hopfield, J. J. (May 1984). "Neurons with graded response have collective computational properties like those of two-state neurons".
*Proceedings of the National Academy of Sciences of the United States of America***81**: 3088-3092 - ↑ "Deterministic Boltzmann learning in networks with asymmetric connectivity". In Touretzky, D. S.;Elman, J. L.; Sejnowski, T. J.; Hinton G. E.
*Connectionist Models: Proceedings of the 1990 Summer School*Morgan Kaufmann Publishers (1991). pp. 3-9. ISBN 978-1558601567