Other Decision based Neural Networks

The Linear Perceptron is applicable only when the classes of patterns are known to be separable by linear decision boundaries. in contrast, the nonlinear perceptron offers a much greater domain of practical applications. In training a complex network, the key lies in the following distributive decision based credit-assigment principle:

When to update In the decision-based learning rule, weight updating is performed only when misclassification occurs.
Which subnets to update? The learning rule is distributive and localized. It applies reinforced learning to the subnet corresponding to the correct class and antireinforced learning to the (unduly) winning subnet.
How to update? Because the decision boundary depends on the discriminant function , it is natural to adjust the boundary by adjusting the weight vector w either in the direction of the gradient of the discriminat function (i.e., reinforced learning) or opposite to that direction (i.e., antireinforced learning):

The gradient vector of the function

with respect to w is denoted

Decision Based Learning Rule

Suppose that

is a set of given training patterns, each corresponding to one of the L classes

. Each class is modeled by a subnet with discriminant functions, say,

i= 1, ..., L. Suppose that the mth training pattern

is known to belong to class

and

That is, the winning class for the pattern is the j-th class (subnet).

When j = i, then the pattern is already correctly classified and no update is needed.
When , that is, is still misclassified, then the following update is perfomed:

Reinforced Learning:

Antireinforced Learning:

Note that for all

and

that is, those weights remain unchanged. Just like the LP, the M training patterns will be repeatedly used for as many sweeps as required for convergence.

In this learning rule, the reinforced learning moves w along the positive gradient direction, so the value of discriminant function will increase, enhancing the chance of the pattern's future selection. The antireinforced learning moves w along the negative gradient direction, so the value of discriminant function will decrease, suppressing the chance of its future selection.

In the special linear case, the discriminant function adopted is based on the Linear Basis Function (LBF)

Then the gradient in the updating formula, is simply

which leads to the linear perceptron rule.

Radial Basis Function This is used in a example of nonlinear decision based learning rule. A RBF discriminant function is a function of the radius betwenn the pattern and a centroid, :

is used for each subnet l. So the centroid (

) closest to the present pattern is the winner. By applying the decision-based learning formula to last equation and noting that

, the following learning rules can be derived:

Reinforced Learning:

Antireinforced Learning:

Elliptic Basis Function The basic RBF version of the DBNN discussed before is based on the asumption that the feature space is uniformly weighted in all directions. In practice, however, different features may have varying degrees of importance depending on the way they are measured. This leads to the adoption of a more versatile elliptic discriminant function. The most general form of a second order basis functions is the (skewed) hyperelliptic basis function. In practice and for most applications, the EBF discriminant function is confined to the following (upright) version: The discriminant function (for each subnet l) can be generalized to an (upright) elliptic function:

where N is the dimension of the input patterns, and

is the vector comprising all the weight parameters

. The learning formula can be derived from the equation.

Hierarquical DBNN Structure

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Artificial Neural Networks

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