1. Hard-Coding a Network. [2pts] In this problem, you need to find a set of weights and
biases for a multilayer perceptron which determines if a list of length 4 is in sorted order.
More specifically, you receive four inputs x1, . . . , x4, where xi ∈ R, and the network must
output 1 if x1 < x2 < x3 < x4, and 0 otherwise. You will use the following architecture: All of the hidden units and the output unit use a hard threshold activation function: φ(z) = 1 if z ≥ 0 0 if z < 0 Please give a set of weights and biases for the network which correctly implements this function (including cases where some of the inputs are equal). Your answer should include: • A 3 × 4 weight matrix W(1) for the hidden layer • A 3-dimensional vector of biases b (1) for the hidden layer • A 3-dimensional weight vector w(2) for the output layer • A scalar bias b (2) for the output layer You do not need to show your work. 1 https://markus.teach.cs.toronto.edu/csc321-2018-01 2 http://www.cs.toronto.edu/~rgrosse/courses/csc321_2018/syllabus.pdf 1 CSC321Homework 3 2. Backprop. Consider a neural network with N input units, N output units, and K hidden units. The activations are computed as follows: z = W(1)x + b (1) h = σ(z) y = x + W(2)h + b (2) , where σ denotes the logistic function, applied elementwise. The cost will involve both h and y: E = R + S R = r >h
ky − sk
for given vectors r and s.
• [1pt] Draw the computation graph relating x, z, h, y, R, S, and E.
• [3pts] Derive the backprop equations for computing x = ∂E/∂x. You may use σ
denote the derivative of the logistic function (so you don’t need to write it out explicitly).
3. Sparsifying Activation Function. [4pts] One of the interesting features of the ReLU
activation function is that it sparsifies the activations and the derivatives, i.e. sets a large
fraction of the values to zero for any given input vector. Consider the following network:
Note that each wi refers to the weight on a single connection, not the whole layer. Suppose
we are trying to minimize a loss function L which depends only on the activation of the
output unit y. (For instance, L could be the squared error loss 1
(y − t)
.) Suppose the unit
h1 receives an input of -1 on a particular training case, so the ReLU evaluates to 0. Based
only on this information, which of the weight derivatives
are guaranteed to be 0 for this training case? Write YES or NO for each. Justify your