Almeida-Pineda recurrent backpropagation

Almeida-Pineda recurrent backpropagation is an error-driven learning technique developed in 1987 by Luis B. Almeida and Fernando J. Pineda. 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.

Model


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

$$\vec{x} = [x_1, x_2, \cdots, x_k]^T$$

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

$$\vec{w} = [w_1, w_2, \cdots, w_k]^T$$

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

$$\begin{align} y &= \varphi \left ( n \right ) \end{align}$$

where $$\varphi \left ( \cdot \right )$$ 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 $$w_{ij}$$, the net $$n_j$$ 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 $$n_j$$ to 0 for all non-input neurons.

$$\frac{\mathrm{d} n_j}{\mathrm{d} t} = -n_j + \sum_{i=1}^N w_{ij} \begin{cases} y_i & \text{ if } i \text{ is not an input } \\ x_i & \text{ if } i \text{ is an input } \end{cases}$$

Note that if the weights between pairs of neurons are symmetric, that is, $$w_{ij} = w_{ji}$$, then the network is guaranteed to settle to an equilibrium state. If symmetry is not held, the network will often settle. Of course, if $$i$$ is an input, then $$w_{ji}$$ 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 $$e_j = 0$$ for all neurons.

$$\begin{align} \frac{\mathrm{d} e_j}{\mathrm{d} t} &= -e_j + \frac{\mathrm{d} \varphi \left ( n_j \right ) }{\mathrm{d} n_j} \sum_{i=1}^N w_{ij} e_i + J_j\\ &= -e_j + \varphi \left ( n_j \right ) \left ( 1 - \varphi \left ( n_j \right ) \right ) \sum_{i=1}^N w_{ij} e_i + J_j\\ &= -e_j + y_j \left ( 1 - y_j \right ) \sum_{i=1}^N w_{ij} e_i + J_j \end{align}$$

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

$$J_j = t_j - y_j$$

The weights are then updated according to the following equation:

$$\Delta w_{ij} = \eta e_j y_i$$

where $$\eta$$ is some small learning rate.

Derivation
The error terms $$e_j$$ are considered estimates of $$-\mathrm{d} E / \mathrm{d} n_j$$ during the derivation of the equations for feedforward backpropagation:

$$\begin{align} \frac{\partial E }{\partial w_{ij}} &= \frac{\mathrm{d} E }{\mathrm{d} n_j} \frac{\partial n_j}{\partial w_{ij}} \\ &= - e_j y_i \\ \Delta w_{ij} &= - \eta \frac{\partial E}{\partial w_{ij}} \\ &= \eta e_j y_i \end{align}$$

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.