Machine Learning - DISCo

354 MACHINE LEARNING
hold when x = xi. In the more general case where the target function has multiple
output units, the gradient is computed for each of these outputs. This matrix of
gradients is called the Jacobian of the target function.
To see the importance of these training derivatives in helping to learn the
target network, consider the derivative ,E~~~i,,e. If the domain theory encodes
the knowledge that the feature Expensive is irrelevant to the target function Cup,
then the derivative ,E~~e~i,,e extracted from the explanation will have the value
zero. A derivative of zero corresponds to the assertion that a change in the feature
Expensive will have no impact on the predicted value of Cup. On the other
hand, a large positive or negative derivative corresponds to the assertion that the
feature is highly relevant to determining the target value. Thus, the derivatives
extracted from the domain theory explanation provide important information for
distinguishing relevant from irrelevant features. When these extracted derivatives
are provided as training derivatives to TANGENTPROP for learning the target network
Cup,,,,,,, they provide a useful bias for guiding generalization. The usual
syntactic inductive bias of neural network learning is replaced in this case by the
bias exerted by the derivatives obtained from the domain theory.
Above we described how the domain theory prediction can be used to generate
a set of training derivatives. To be more precise, the full EBNN algorithm
is as follows. Given the training examples and domain theory, EBNN first creates
a new, fully connected feedforward network to represent the target function.
This target network is initialized with small random weights, just as in BACK-
PROPAGATION. Next, for each training example (xi, f (xi)) EBNN determines the
corresponding training derivatives in a two-step process. First, it uses the domain
theory to predict the value of the target function for instance xi. Let A(xi) denote
this domain theory prediction for instance xi. In other words, A(xi) is the
function defined by the composition of the domain theory networks forming the
explanation for xi. Second, the weights and activations of the domain theory networks
are analyzed to extract the derivatives of A(xi) 'with respect to each of the
components of xi (i.e., the Jacobian of A(x) evaluated at x = xi). Extracting these
derivatives follows a process very similar to calculating the 6 terms in the BACK-
PROPAGATION algorithm (see Exercise 12.5). Finally, EBNN uses a minor variant
of the TANGENTPROP algorithm to train the target network to fit the following error
function
where
Here xi denotes the ith training instance and A(x) denotes the domain theory
prediction for input x. The superscript notation xj denotes the jth component of
the vector x (i.e., the jth input node of the neural network). The coefficient c is
a normalizing constant whose value is chosen to assure that for all i, 0 5 pi 5 1.