Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / Section F / 47 / d / Solution

Compose affine maps with nonlinear hidden activation functions, for example $h(x)=W_2\mathrm{ReLU}(W_{1}x+b_1)+b_2$. Several hidden units can respond to different linear combinations of input features; later units combine their responses to represent curved or piecewise-linear decision boundaries and interactions. A product interaction can be understood from $xy=[(x+y{)}^2-(x-y{)}^2]/4$: units approximating nonlinear functions of $x+y$ and $x-y$ can represent effects that no single additive linear fit captures.

Nonlinearity in hidden layers is essential: with only identity activations, repeated affine maps collapse to one affine map regardless of depth. Width and depth determine available representations; regularization, validation and suitable optimization are needed to learn useful rather than overfitted interactions. A nonlinear output activation alone applied to one linear score does not give an unrestricted model of feature interactions.