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The derivative of the performance index Q taken with respect to the given weight wji comes in the form

We use the notation

Then, obviously, see Fig. 2.12

Thus

that assumes the form similar to that exploited in the delta learning rule.

The evident difference in the realization of the learning scheme depends whether zj is situated in the output or one of the hidden layers. If zj concerns the output layer, this signal can be compared with the target value, targetj. Then we obtain

so the overall expression is nothing but the delta rule. Now, if zj is not directly confronted to the target value, we employ the standard chaining rule of differential calculus. Let us start with δj,

where the above summation is taken over all the units (k) placed at the layer close to the output of the network. Calculate now

Plugging this into the previous formula, we get

The BP proceeds in two passes: the input x propagates through the network up starting from the input layer; the error signal necessary to update the weights propagates back from the output layer down to the input one.

2.5.5. Hebbian learning

The Hebbian learning results from the classic synaptic modification (Hebb, 1949) stating that a connection changes in proportion to strength between pre - and postsynaptic signals. This form of learning is a well-known example of unsupervised learning. For a single linear unit, y = xTw, the update rule articulating this modification scheme (Stent, 1973; see also Kosko, 1992) reads as

where x(k) and y(k) form an input - output pair of data; the learning rate is denoted by α. To understand what the Hebbian learning really means, let us insert the expression of the neuron into the learning rule. The increment Δw becomes equal

Taking the expected value of the weight increment E(Δ w) one has

Assuming that x and w are independent,

the increment depends upon the autocorrelation matrix of the input data, C = E(xxT).

2.5.6. Competitive learning

The learning mechanism of competition is easily explained in the simple neural architecture formed by a single layer of linear units. For any input x we make a single unit active at a given time. This neuron is called the winner. The winning node is the one that produces the highest output to the given input x, namely

where wi is the vector of the connections of the i-th unit. The winning rule is straightforward (winner-takes-all competition):

  node “i” becomes declared the winning node if

The update formula driving the changes of the connections of the winning node (i) is expressed as (Rumelhart and Zipser, 1985)

The connections of the remaining neurons are left intact.

The network implementation of the competitive learning is anticipated by incorporating inhibitory and excitatory connections between the neurons, Fig. 2.13. Each neuron excites itself (connection set to 1) while it suppresses the others with a strength e (usually e is chose from the range 0 to 1/m with m being the number of the competing units).


Figure 2.13  Neural network with competitive learning; marked only the connections for a single output neuron

2.5.7. Self-organizing maps

The mechanism of competitive learning can be realized through a so-called self-organizing feature map (Kohonen, 1984). This network is usually built as a two-dimensional grid of linear units (neurons), Fig. 2.14, where each of them receives the same input x situated in Rn.


Figure 2.14  Self-organizing map as an array of linear units

When the input x is provided, the neurons start to compete. Let the winning node will be the one with the coordinates i*j*. It occurs for the unit for which the distance between the input and the respective vector of the connections attains a minimal value over the entire grid of the neurons,

While in the previous model of competitive learning, only the winning node updates its connections. In this approach we allow a certain neighborhood of the winning node, say (i*j*) , to become affected and modify their weights, yet to a lesser degree as the winner itself. This interaction between the neurons occurring at the learning level is conveyed by the so-called neighbor function Φ(ij, i*j*). The updates of the connections are governed by the expression

with “ij” being the coordinates of the ij-th unit in the grid.

The neighborhood function Φ(ij, i*j*) preserves topological properties of the map in such a way that

  Φ(ij, i*j*) attains maximum (1) for the equal pairs of the coordinates, Φ(i*j*, i*j*)=1
  Φ(ij, i*j*) is a decreasing function of the coordinates; the more distant i and i* as well as j and j* are, the lower the value of the neighborhood function.

What this really means is that Δwij at this node is made lower up to the point where the increment attains zero expressing that the node is too far from the winner to become affected during the learning process. Additionally, the local field formed by the neighborhood function Φ is also changed over the training time - it usually shrinks (eventually in a linear fashion) once the learning progresses over time.

2.5.8. Learning in presence directly and indirectly labeled patterns

Here we discuss a problem of a hybrid learning when we are provide with some fully labeled and partially labeled data. The data that are fully labeled are denoted by (direct learning). The data whose class assignment is not made explicit but rather specified in a relational mode are collected in the second family of training data denoted by (referential learning). One can easily envision this type of mixed data in classification problems when one is given with data fully described in terms of class membership values along with some others with referential labeling only. This referential labeling works on pairs of data and quantifies their similarity, see Fig. 2.15.


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