disc12-examprep

CS 188 Spring 2023 Introduction to Artificial Intelligence Exam Prep 12 Q1. Linear Separability (a) For each of the datasets represented by the graphs below, please select the feature maps for which the perceptron algorithm can perfectly classify the data. Each data point is in the form ( ? 1 , ? 2 ) , and has some label ? , which is either a 1 (dot) or −1 (cross). (i) □ [ ? 1 ? 2 1 ] □ [ ? 1 ? 2 ? 2 1 ] □ [ ? 1 ? 2 | ? 1 | ] □ [ ? 1 ? 2 ? ] □ [ ? 1 ? 2 ] (ii) □ [ ? 1 ? 2 1 ] □ [ ? 1 ? 2 ? 2 1 ] □ [ ? 1 ? 2 | ? 1 | ] □ [ ? 1 ? 2 ? ] □ [ ? 1 ? 2 ] (iii) □ [ ? 1 ? 2 1 ] □ [ ? 1 ? 2 ? 2 1 ] □ [ ? 1 ? 2 | ? 1 | ] □ [ ? 1 ? 2 ? ] □ [ ? 1 ? 2 ] 1

Q2. Deep Learning (a) Perform forward propagation on the neural network below for ? = 1 by filling in the values in the table. Note that (i), … , (vii) are outputs after performing the appropriate operation as indicated in the node. (i) (ii) (iii) (iv) (v) (vi) (vii) ? ∗ 2 ∗ 3 ∗ 4 ∑ max min max (i) (ii) (iii) (iv) (v) (vi) (vii) (b) [Optional] Below is a neural network with weights ?, ?, ?, ?, ?, ? . The inputs are ? 1 and ? 2 . The first hidden layer computes ? 1 = max( ? ⋅ ? 1 + ? ⋅ ? 2 , 0) and ? 2 = max( ? ⋅ ? 1 + ? ⋅ ? 2 , 0) . The second hidden layer computes ? 1 = 1 1+exp(− ? ⋅ ? 1 ) and ? 2 = 1 1+exp(− ? ⋅ ? 2 ) . The output layer computes ? = ? 1 + ? 2 . Note that the weights ?, ?, ?, ?, ?, ? are indicated along the edges of the neural network here. Suppose the network has inputs ? 1 = 1 , ? 2 = −1 . The weight values are ? = 1 , ? = 1 , ? = 4 , ? = 1 , ? = 2 , ? = 2 . Forward propagation then computes ? 1 = 2 , ? 2 = 0 , ? 1 = 0 . 9 , ? 2 = 0 . 5 , ? = 1 . 4 . Note: some values are rounded. ? 1 ? 2 ? 1 ? 2 ? 1 ? 2 ? ? ? ? ? ? ? Using the values computed from forward propagation , use backpropagation to numerically calculate the following partial derivatives. Write your answers as a single number (not an expression). You do not need a calculator. Use scratch paper if needed. Hint: For ? ( ? ) = 1 1+exp(− ? ) , the derivative is 𝜕? 𝜕? = ? ( ? )(1 − ? ( ? )) . 𝜕? 𝜕? 𝜕? 𝜕? 𝜕? 𝜕? 𝜕? 𝜕? 𝜕? 𝜕? 𝜕? 𝜕? 2