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1. (10 points) Derive the update equations when the hidden units use the hy- perbolic function tanh(x) in a neural network with one hidden layer, instead 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), t 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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