Supervised learning

Supervised learning fits a prediction rule from training data containing both inputs and associated target responses. Regression analysis predicts numerical responses; classification in statistical learning predicts class labels. A loss function specifies which prediction errors matter, and generalization concerns performance beyond the observations used to fit the rule.

Learning driven by a discrepancy between a target and the network’s actual output. The architecture must supply both a meaningful teaching signal and a mechanism for assigning that signal to the connections responsible for the error. This is distinct from error correction during retrieval in a fixed associative memory.

For a linear output $y=Wx$ and squared loss $L=\Vert t-y{\Vert }^2/2$, gradient descent gives $\Delta W=\eta (t-y)x^{\mathsf{T}}$. Indeed $\partial L/\partial W_{ij}=-(t_i-y_i)x_j$. On the same example, $e^{\prime }=(1-\eta \Vert x{\Vert }^2)e$, proving strict error reduction when $e\ne 0$ and $0<\eta \Vert x{\Vert }^2<2$.