1. (10 points) Derive the update equations when the hidden units
perbolic function tanh(x) in a neural network with one hidden layer,
of the sigmoid function. Use the fact that tanh'(x) = 1 - tanh²(x).
2. (30 points) Consider the Multilayer Perceptron (MLP) for binary classification
described in section 11.7.2 in the textbook. Suppose that there is a probability
€ that the class label on i.i.d. training data points has been incorrectly set.
This gives a new error function
E(W, v/X) = - r'log ft + (1 - rt) log(1 - ft),
where ft = p(rt = 1|x2 = p(rt = 1,kt = 1|xt + p(pt = 1,kt = and the
Bernoulli random variable kt is the true label. Write down the error function
corresponding to the negative log likelihood in terms of yt and € and derive
the update equations for this new error function. Note that this error function
makes the model robust to incorrectly labeled data, in contrast to the usual
error function. (Hint: You need Bayes rule to derive the marginal probabilities).
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