Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 218 / 3

Conditional on the covariates, the binary responses are independent with Bernoulli distributions:Thus $p_i=(1+e^{-{\eta }_i}{)}^{-1}$, the logistic regression model. The maximum-likelihood estimates areUsing the reported age standard error, an approximate 95% Wald confidence interval isThis interval concerns the age coefficient in log odds per year, with income and loan amount held fixed.

The sigmoid function is strictly increasing. Therefore the risky region is $\hat{\eta }(x)\ge \log \{p/(1-p)\}$. Its decision boundary is the hyperplanePoints on or above this score threshold are assigned to the risky class. For $p=1/2$, the right side is zero.

The feedforward neural network has three input coordinates, two hidden sigmoid function units and two output softmax function units. Every input is connected to each hidden unit, and each hidden unit to each output unit; each hidden and output unit also has a bias. A layer diagram isAlgebraically, with $x_i=({\mathrm{income}}_i,{\mathrm{age}}_i,{\mathrm{amount}}_i{)}^T$,with responses conditionally independent. The fourteen displayed weight and bias parameters comprise six input weights, two hidden biases, four output weights and two output biases. The one-hot encoding supplies targets $t_{i0}=1-Y_i$, $t_{i1}=Y_i$, and the categorical cross-entropy loss is the negative of $\ell ={\sum }_i{\sum }_c{t}_{ic}\log p_{ic}$. There is softmax non-identifiability under a common shift of both output logits, but the predicted probabilities are well defined.

For one observation, let ${\ell }_i={\sum }_{c=0}^1{t}_{ic}\log p_{ic}$ and define ${\delta }_{ic}=t_{ic}-p_{ic}$. Differentiating the softmax function and its log-likelihood gives $\partial {\ell }_i/\partial o_{ic}={\delta }_{ic}$. The sigmoid-softmax network gradients areThese equations are backpropagation: use the chain rule through the output affine map, then ${\sigma }^{\prime }(u)=\sigma (u)(1-\sigma (u))$ through the hidden units, then the input affine map.

For a learning rate $\eta >0$, select one observation and update every parameter by $\theta \leftarrow \theta +\eta \nabla {\ell }_i(\theta )$, computing all entries of the gradient from the old parameters before applying the update. A uniformly selected observation gives an unbiased gradient of the average log-likelihood; multiplying by 3000 would instead estimate the gradient of the sum. Equivalently perform stochastic gradient descent on the negative log-likelihood. Batch size one makes one update per observation, and five epochs make five passes through the data, typically with a fresh random ordering on each pass.